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+[    {        "title": "2402.15169v1.Platforms_for_Efficient_and_Incentive_Aware_Collaboration.pdf",        "content": "arXiv:2402.15169v1  [cs.GT]  23 Feb 2024Platforms for Efficient and Incentive-Aware Collaboration\nNika Haghtalab, Mingda Qiao, and Kunhe Yang\nUniversity of California, Berkeley\n{nika,mingda.qiao,kunheyang}@berkeley.edu\nAbstract\nCollaboration is crucial for reaching collective goals. Ho wever, its potential for effectiveness\nis often undermined by the strategic behavior of individual agents — a fact that is captured by a\nhigh Price of Stability (PoS) in recent literature [ Blum et al. ,2021a ]. Implicit in the traditional\nPoS analysis is the assumption that agents have full knowled ge of how their tasks relate to one\nanother. We offer a new perspective on bringing about efficient collaboration across strategic\nagents using information design. Inspired by the increasin gly important role collaboration plays\nin machine learning (such as platforms for collaborative fe derated learning and data coopera-\ntives), we propose a framework in which the platform possess es more information about how\nthe agents’ tasks relate to each other than the agents themse lves. Our results characterize how\nand to what degree such platforms can leverage their informa tion advantage and steer strategic\nagents towards efficient collaboration.\nConcretely, we consider collaboration networks in which ea ch node represents a task type held\nby one agent, and each task benefits from contributions made i n their inclusive neighborhood\nof tasks. This network structure is known to the agents and th e platform. On the other\nhand, the real location of each agent in the network is known t o the platform only — from\nthe perspective of the agents, their location is determined by a uniformly random permutation.\nWe employ the framework of private Bayesian persuasion and d esign two families of persuasive\nsignaling schemes that the platform can use to guarantee a sm all total workload when agents\nfollow the signal. The first family aims to achieve the minmax optimal approximation ratio\ncompared to the total workload in the optimal collaboration , which is shown to be Θ(√n)for\nunit-weight graphs, Θ(n2\n3)for graphs with edge weights lower bounded by Ω(1), andO(n3\n4)\nfor general weighted graphs. The second family ensures per- instance strict improvement in the\ntotal workload compared to scenarios with full information disclosure.\n1Contents\n1 Introduction 4\n1.1 Our Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\n1.2 Related Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7\n1.3 Structure of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8\n2 Model 8\n2.1 Collaboration system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8\n2.2 Benchmarks and the price of stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 10\n2.3 Information design and signaling schemes . . . . . . . . . . . . . . . . . . . . . . . . 10\n3 Technical Overview and Structural Results 11\n3.1 Structural results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12\n3.2 Overview of upper bounds in unit-weight graphs . . . . . . . . . . . . . . . . . . . . 13\n3.3 Overview of upper bounds in weighted graphs . . . . . . . . . . . . . . . . . . . . . . 16\n3.4 Overview of lower bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18\n4 Discussion and Future Directions 19\n5 Upper Bounds for Unit-Weight Graphs 19\n5.1 Weights of cuts and induced sub-graphs . . . . . . . . . . . . . . . . . . . . . . . . . 20\n5.2 General unit-weight graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20\n5.3 Matching the cost of full revelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22\n5.4 Strict improvement upon full revelation . . . . . . . . . . . . . . . . . . . . . . . . . 24\n6 Lower Bounds for Unit-Weight Graphs 27\n6.1 Lower bound for the double-star graph . . . . . . . . . . . . . . . . . . . . . . . . . . 27\n6.2 A more general lower bound via k-stars . . . . . . . . . . . . . . . . . . . . . . . . . . 29\n7 Upper Bounds for Weighted Graphs 30\n7.1 Upper bound for approximating OPTIR. . . . . . . . . . . . . . . . . . . . . . . . . . 30\n7.2 Strict improvement upon full revelation . . . . . . . . . . . . . . . . . . . . . . . . . 31\n8 Lower Bounds for Weighted Graphs 32\nA Omitted Proofs from Section 2 38\nA.1 Alternative Benchmarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38\nA.2 Proof of Lemma 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38\nB Omitted proofs from Section 3.1 39\nB.1 Proof of Lemma 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39\nB.2 Proof of Lemma 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40\nC Details for Section 4 40\nC.1 Conjectured impossibility of matching OPTstableusingO(1)signals. . . . . . . . . . . 40\nC.2 Conjectured Ω(n3/4)lower bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42\nD Omitted Proofs from Section 5 42\nD.1 Technical Lemmas in Section 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42\n2D.2 Proof of Lemma 5.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43\nE Omitted Proofs from Section 6 44\nF Omitted Proofs from Section 7 47\nF.1 Proof of Lemma 7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47\nF.2 Proof of Theorem 7.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50\nF.3 Proof of Theorem 7.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51\nF.4 Proof of Lemma 7.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52\nF.5 Proof of Theorem 7.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52\nG Omitted Proofs from Section 8 54\nG.1 Proof of Theorem 8.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54\nG.2 Projection to a low-dimensional space . . . . . . . . . . . . . . . . . . . . . . . . . . 56\nG.3 Choice of the test function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58\nG.4 Verify the dual feasibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60\n31 Introduction\nCollaboration is the cornerstone of modern achievements ac ross various disciplines. Effective im-\nplementation of collaborative systems has substantially i ncreased what can be accomplished by the\nlimited capabilities of individual agents. For instance, t he global collaboration between agencies\nand institutions under the Genome-Wide Association Studie s (GWAS) has enabled the decoding\nof genetic foundations of diseases [ Bergen and Petryshen ,2012]; collaborations across hundreds of\nmathematicians have led to resolving longstanding open pro blems [ Steingart ,2012]; collaboratively\nmaintained virus signature databases [ AlienVault ,2012] have made technology safer; and collabo-\nrative federated learning [ McMahan et al. ,2017] has led to the training of models with superior\nperformance on a large range of applications.\nHowever, for every highly successful collaboration, there are many others that never came to fruition.\nThis is in part due to the inherently strategic nature of part icipants in a collaboration that can\nbe a significant barrier to the realization of optimal collab oration. On the one hand, achieving\noptimal collaboration has been shown to typically require l evels of effort from some participants\nthat surpass what is individually rational (the amount of wo rk required when working alone) or\nwhat is considered stable (the amount of work deemed reasona ble given others’ contributions). On\nthe other hand, stable collaboration systems where everyon e is satisfied with their assigned effort,\nif exist, often suffer from significant inefficiency.\nThe above issues are evident even in a simple double-star net work shown in Figure 1 where each\nnode represents an agent and their task, and each agent’s tas k benefits from both their own and\nneighboring agents’ contributions1. For collaboration to be feasible, each agent’s task requir es at\nleast one unit of contribution. Ideally, in the optimal coll aboration that minimizes total workload,\nthe two central nodes take on the entire workload by each cont ributing one unit of effort to support\nall their leaf neighbors. However, when incentives are fact ored in, both center nodes have the\nincentive to unilaterally reduce their contribution, as th ey have already received enough support\nfrom each other. In fact, in any stable collaboration in this network, the total workload inevitably\nscales linearly with the number of agents, which implies tha t the benefits gained from collaboration\nare merely marginal.\nx0x1\nx2\nx3\nxky0y1\ny2\ny3\nyk\nFigure 1: Double Star Graph. Formally, let θvdenote the contribution of node v. The social optimum is achieved\nwhenθx0=θy0= 1with all other θv= 0.Feasibility requires that each agent vreceives a total contribution\nofθv+/summationtext\nu:{u,v}∈Eθu≥1.Stability requires no node can unilaterally reduce their contributio n without hurting\ntheir own feasibility. Following are typical stable soluti ons: (1) One-sided leaf contribution, where θv= 1for\nv∈ {x1,x2,...,x k,y0}andθv= 0for the rest, and (2) Full leaf contribution, with θv= 1for all leaf nodes\nv∈ {x1,x2,...,x k,y1,y2,...,y k}, whileθx0=θy0= 0. Both solutions suffer from a total workload of Ω(k).\nThis example highlights the inefficiency of collaboration in the presence of strategic agents, which\nis quantitatively reflected by a high Price of Stability (PoS ) [Anshelevich et al. ,2008,Blum et al. ,\n1A variant of this structure was used by Blum et al. [2021a ] for establishing a lower bound on the PoS in several\nabstractions of collaborative federated learning.\n42021a ], a well-established concept in game theory measuring the r atio between the total workload\nin the best stable solution and the socially optimal solutio n2.\nHowever, the traditional notion of PoS may be overly pessimi stic for analyzing collaboration net-\nworks for two primary reasons. First, it assumes that agents have full knowledge of their positions\nand roles within the collaborative system. This overlooks r eal-life and practical uncertainties that\noften exist in collaborations centered on data gathering an d usage; for instance, in data coopera-\ntives and federated learning, participants are often uncer tain of how their data contribute to the\nbigger picture, including their specific uses, their broade r implications and value, and how they can\nhelp solve tasks for other participants. Collaborations ar e often hosted on platforms, yet, platforms\nlack the leverage to steer the agents toward more efficient col laborative outcomes in the traditional\nnotion of PoS. As a result, the system is severely affected by t he misalignment between the strategic\nagents’ incentives and the platform’s goal of improving the overall efficiency.\nInspired by the increasingly important role collaboration plays in machine learning, such as platforms\nfor collaborative federated learning and data cooperative s, we revisit the problem of efficient and\nstable collaboration under a new framework. To address the a forementioned issues, it is crucial to\nrecognize that while agents typically have some understand ing of the types of tasks in the system\nand the general structure relating different task types, the y often lack precise knowledge about how\ntheir specific tasks relate to others. This precise knowledg e is usually exclusive to the platform,\nwhich highlights the platform’s crucial role in communicat ing this knowledge to the agents. This\nlevel of uncertainty challenges the traditional PoS analys is that assumes complete knowledge of one’s\nposition and role. Yet, it also opens up new possibilities: b y strategically distributing information\nabout agents’ task types, not only platforms can address the se challenges but also they can create\nopportunities for efficient collaboration that was previous ly thought to be impossible.\nReflecting this into the aforementioned double-star struct ure in Figure 1 , we now consider a sce-\nnario where agents are assigned to nodes of this network by a u niformly random permutation. That\nis, while the network structure is known to the agents, they a re unaware of their exact positions\nwithin the graph. Not knowing whether one’s task is represen ted by a central node or a leaf node\nsignificantly influences agents’ decisions and collaborati on outcomes. This is where the platform’s\nrole of strategically sharing information becomes clear. I f the platform withholds all information,\nevery agent will contribute constant effort because they fac e maximum and independent uncertainty\nabout whether their task is sufficiently covered by contribut ions of others. This high level of un-\ncertainty will lead agents to contribute an unnecessary amo unt of effort. On the other extreme,\nfull information disclosure removes all uncertainty, lead ing central nodes to reduce their contribu-\ntions (as discussed above) and pushing leaf nodes into a less efficient pattern of contribution, thus\nsignificantly increasing the PoS.\nThe crux of steering strategic agents into efficient collabor ation lies in creating the right level of\ncorrelated uncertainty . By controlling the flow of information, the platform can nega tively correlate\nthe beliefs of central and leaf nodes, while maintaining a de sirable level of uncertainty. The negative\ncorrelation between agents’ beliefs ensures that agents wo uld not perform unnecessary amount of\nwork simultaneously. Moreover, the remaining uncertainty (as well as negative correlation) leverages\nthe central agents’ fear that their tasks might have not rece ived sufficient contributions from others,\nthereby encouraging them to increase their effort.\nThis leads us to the main question of our study: Is it possible for the platform to always ensure\n2In the original definition, PoS is the ratio between the cost o f the best Nash equilibrium and that of the social\noptimum. In our collaborative context, the Nash equilibriu m analog is a stable solution conditioned on being feasible.\n5efficient, stable collaboration among strategic agents by co ntrolling the information flow about their\ntypes? Specifically, we seek to determine whether a general i nformation structure can be designed\nto:\n(1) Guarantee a sublinear3upper bound on the PoS for all collaboration instances;\n(2) Consistently improve the PoS compared to a scenario of fu ll information revelation.\nTo address these questions, our approach builds on the frame work of the private Bayesian persuasion\nmodel in information design [ Kamenica and Gentzkow ,2011,Arieli and Babichenko ,2019]. We\nallow the platform to credibly commit to a signaling scheme, which is a probabilistic mapping from\nthe realization of joint types to private signals that are se nt to each agent in the system. These\nprivate signals distill the information about joint types i nto direct recommendations on how much\neach agent should contribute. Key to this framework is the si gnaling scheme’s persuasiveness – it\nguarantees that agents are incentivized to follow the recom mendations. The platform facilitates\nefficient collaboration by committing to persuasive signali ng schemes with low total workload when\nagents follow the recommendations.\n1.1 Our Contribution\nWe initiate the study of improving the efficiency of stable col laboration through the lens of informa-\ntion design. We formalize the private Bayesian persuasion pr oblem in a linear collaborative network\nand characterize the structural properties of persuasive s ignaling schemes. For a special class of\nbinary signaling schemes, we identify how its persuasiveness rela tes to graph characteristics such as\ncuts and induced subgraphs.\nBased on these structural insights, we design two families of signaling schemes that achieve the\naforementioned goals (1) and (2), respectively. We will use nto denote the number of agents\n(nodes) in the network, and use benchmarks OPT,OPTIRandOPTstableto respectively quantify the\ntotal workload under the optimal feasible collaboration as well as the optimal feasible collaboration\nwhich also satisfies individual rationality (IR) and stabil ity constraints.\nToward our first goal of achieving a nontrivial approximatio n ratio to OPTin the worst case, we\ndesign a binary signaling scheme that achieves the approxim ation ratio of O(√n)in any unit-weight\ngraphs. In general weighted graphs, we argue that OPTIRserves as a more appropriate benchmark\nthanOPT, because OPT might require some agents to contribute more than their indi vidually\nrational amount, which rational agents cannot be persuaded to do. However, we show that binary\nsignaling schemes fail to achieve this goal. To address this , we introduce a third signal to achieve\nthe optimal approximation ratios of O(n2\n3)for graphs with lower bounded edge weights, and O(n3\n4)\nfor general weighted graphs. Our results are summarized in T heorems 1.1 and 1.2.\nTheorem 1.1. In any unit-weight graph, there exists a binary signaling sc heme that is persuasive\nand has cost O(√n·OPT). Moreover, the O(√n)approximation ratio is tight for certain graphs.\nTheorem 1.2. In any weighted graph, there exists a ternary signaling sche me that is persuasive\nand has cost O(n3/4·(OPTIR)1/2). In addition, for graphs with all edge weights bounded below byδ,\nthe cost is improved to O((n·OPTIR)2/3·δ−1/3), with the O(n2/3)approximation ratio being tight\non a family of graphs with δ= 1/2.\nWe introduce a different family of signaling schemes that ach ieve our second goal of strictly improving\nOPTstable. This approach uses only two signals in unit-weight graphs a nd at most n+ 1signals\n3Recall that worst-case PoS can be linear in the number of agen ts, as demonstrated by Figure 1\n6in general weighted graphs. We summarize the results in Theorems 1.3 and1.4. Despite the\nsubstantial increase in the signal space required for weigh ted graphs, we demonstrate the infeasibility\nof achieving this goal through binary or ternary signaling s chemes, and leave the identification of\nthe minimum number of signals needed as an open problem discu ssed in Section 4 .\nTheorem 1.3. In any unit-weight graph, whenever OPT<OPTstable, there exists a binary signaling\nscheme that is persuasive and has a cost strictly lower than OPTstable.\nTheorem 1.4. In any general weighted graph, whenever OPTIR<OPTstable, there exists a signaling\nscheme with at most n+1signals that is persuasive and has a cost strictly lower than OPTstable.\n1.2 Related Works\nCollaborative federated learning. Our work connects to the broader topic of collaborative\nand federated learning among strategic agents, where incen tives and fairness have been extensively\nexplored [ Donahue and Kleinberg ,2021a ,b,c,Blum et al. ,2021a ]. Most related to us is the work of\nBlum et al. [2021a ] that introduces a formalism for collaborative learning am ong network of agents\nand highlights the inefficiency in collaboration among strat egic agents as captured by a high PoS.\nOur work introduces a novel approach to reducing the PoS unde r a more realistic scenario where\nagents face uncertainty regarding the types of their tasks.\nRecent studies have adopted a mechanism design approach in federated learning, targeting objectives\nsuch as incentivizing data contribution and preventing fre e-riding [ Huang et al. ,2023,Karimireddy et al. ,\n2022,Chen et al. ,2023b ,Xu et al. ,2021], encouraging participation [ Hu et al. ,2023,Xu et al. ,2023,\nCohen and Shao ,2023,Han et al. ,2023], and promoting fairness [ Lin et al. ,2023,Lyu et al. ,2020].\nSee recent surveys [ Zhan et al. ,2021,Tu et al. ,2022] for a more comprehensive overview. Addi-\ntionally, research on federated bandits [ Wei et al. ,2023] explores strategies to motivate agents to\nshare their data. However, these papers primarily rely on ei ther monetary incentives or penalization\nthrough reduced model accuracy and usually operate under th e condition that agents know how\ntheir data or tasks relate to those of others. In contrast, we assume that the platform possesses\nmore information about the agents’ types but without a direc t control over the distribution of data,\ngradients, or models.\nIn addition, there is research focusing on incentives and me chanism design in data markets where\nconsumers purchase data from the market [ Agarwal et al. ,2019,2020,Jia et al. ,2019], but they do\nnot focus on incentivizing collaboration among participan ts.\nAnother line of literature focuses on the communication cos t [Blum et al. ,2021b ] or sample ef-\nficiency of collaborative learning without incentives [ Blum et al. ,2017,Nguyen and Zakynthinou ,\n2018,Chen et al. ,2018,Haghtalab et al. ,2022,Awasthi et al. ,2023,Zhang et al. ,2023,Peng,2023].\nIn contrast, we achieve both efficiency and incentive-awareness through the lens of information de-\nsign.\nInformation design. Bayesian persuasion, a seminal work of Kamenica and Gentzkow [2011]\nrooted in the context of incomplete information games [ Aumann et al. ,1995], is a canonical model\nfor information design that studies how an informed sender c an strategically share information\nto influence the actions of a single uninformed receiver. Thi s framework has been expanded\nto multi-receiver settings [ Bergemann and Morris ,2016] and explored through algorithmic per-\nspectives [ Dughmi and Xu ,2016,Dughmi ,2017], supporting its applications in diverse problems.\nThese applications include but are not limited to routing ga mes [Das et al. ,2017,Wu and Amin ,\n72019], coordination games [ Wu and Amin ,2019], voting [ Schnakenberg ,2015,Alonso and Câmara ,\n2016,Bardhi and Guo ,2018,Wang ,2013,Arieli and Babichenko ,2019], and incentivizing explo-\nration [ Mansour et al. ,2020,2022,Kremer et al. ,2014]. We refer the readers to survey papers [ Candogan ,\n2020,Kamenica ,2019] for a comprehensive overview.\nThe problem we study involves designing signaling schemes i n a specific multi-receiver setting, where\nthe literature offers two general approaches: characterizi ng and selecting Bayes Correlated Equilibria\n(BCE) based on sender utility [ Bergemann and Morris ,2013,2016,2019,Taneva ,2019], and belief-\nbased optimization [ Mathevet et al. ,2020]. However, applying these approaches directly in our\ncontext is computationally intractable due to the exponent ially large state space. Indeed, finding\noptimal signal schemes is proved hard even in some simple mul ti-receiver settings [ Bhaskar et al. ,\n2016,Rubinstein ,2017,Dughmi ,2017], emphasizing the importance of leveraging our collaborat ive\nnetwork’s structural properties.\nOur work also contributes to the recent line of work on information design in networks and plat-\nforms. In the context of social networks, Candogan and Drakopoulos [2017],Candogan [2019],\nEgorov and Sonin [2019] design optimal signaling schemes to maximize overall enga gement, while\nArieli and Babichenko [2019] studies the setting when the sender’s utility is more gener al and non-\nadditive. Mathevet and Taneva [2022] studies vertical and horizontal information transmissio n,\nidentifying scenarios where single-meeting schemes — signals shared with and observed by a spe-\ncific subset of agents — are optimal. Our analysis of signalin g schemes in the collaboration network\nalso focuses on optimality but diverges in its definition: we assess optimality not by the ability to\nimplement all BCE, as previous studies do, but by achieving an optimal asymptotic approxima-\ntion ratio compared to the socially optimal solution when ag ents are non-strategic. To the best\nof our knowledge, we are the first work that focuses on the pers pective of multi-receiver Bayesian\npersuasion to enhance and encourage collaboration between strategic agents.\nIn the context of online platforms with strategic agents, Papanastasiou et al. [2018] studies infor-\nmation provision policy for sequentially arriving consume rs who choose among alternative products\nor services, Bergemann et al. [2022],Chen et al. [2023a ] focuses on maximizing revenues in click-\nthrough auctions when the platform has exclusive knowledge about the true click-through rates,\namong many other studies in this field. Our work complements t he literature by studying the\nefficiency of collaboration platforms.\n1.3 Structure of the paper\nWe formally introduce our model in Section 2 . In Section 3 , we discuss structural results that\ncharacterize the persuasiveness of signaling schemes, and provide a technical overview of our main\nresults. In Section 4 , we discuss a few open problems and future research directio ns. In Section 5 ,\nwe focus on unit-weight graphs and prove the upper bounds in Theorems 1.1 and1.3. We prove\nthe lower bound side of Theorem 1.1 inSection 6 . For weighted graphs, we provide the upper\nbounds ( Theorems 1.2 and1.4) inSection 7 , and the lower bound (second half of Theorem 1.2 ) in\nSection 8 .\n2 Model\n2.1 Collaboration system\nTask-based contribution. LetVbe the set of task types in the collaboration system, with\n|V|=n. Letθv∈R≥0be the effort contributed directly towards solving task v, and let θ= (θv)v∈V\n8be its vector form. We assume that the total contribution tha t each type receives is a linear\ncombination of the direct contributions that are put into al l tasks in the system, with coefficients\nWu,vthat quantifies the extent to which efforts dedicated to a type -utask help complete a type- v\ntask. We assume that Wu,v=Wv,u∈[0,1]andWu,u= 1for all pairs (u,v)∈V2, i.e.,Wis an\nn×nsymmetric matrix with coefficients Wu,vand has a diagonal of all ones. For v∈V, we useuv\nto denote the total contribution entering a type- vtask, and let u= (uv)v∈Vbe its vector form. We\nthen have\nuv(θ) =/summationdisplay\nv′∈VWv,v′θv′,andu(θ) =Wθ.\nGraph representation. The task structure can be equivalently represented as a weig hted undi-\nrected graph G= (V,E,w), where each vertex is a task type in V. Each edge{v1,v2}∈Erepre-\nsents that efforts towards task v2positively impact task v1. In other words, E={{v1,v2}:v1,v2∈\nV,v1/\\e}atio\\slash=v2,Wv1,v2>0}. The weight function w:E→[0,1]maps each edge{v1,v2}∈EtoWv1,v2\nand reflects the coefficients between tasks. We also use N(v)to denote the open neighborhood of\nvertexv∈V.\nStrategic agents. Let there be nagents in the collaboration system. Each agent iis associated\nwith a unique task type ti∈Vin the network. Task types for all agents form a type profile\nt= (t1,...,tn), which is a permutation of the types in V, i.e.,t∈Sym(V)withSym(V)being the\nsymmetric group on V. Additionally, each agent ican take an action ai∈R≥0that is the effort they\nput into solving their own task. The action profile aand task type profile t∈Sym(V)naturally\ninduce a type-based direct contribution profile θ(a;t) = (θv)v∈V, whereai=θti, or more succinctly\nθ(a;t) = Π−1a. In the above equation, Π = (πi,j)is then×npermutation matrix corresponding\nto permutation t, whereπi,j= 1ifj=tiand0otherwise. We abbreviate θ(a;t)toθwhen it is\nclear from context.\nFor each agent i, the quality of a collaborative solution, denoted with qualityi, is a function of\nbothaandtand equals the total contribution entering the task of type ti. We also use quality=\n(quality1,···,qualityn)to denote the agent-based utility vector. Formally, we have\nqualityi(a;t) =uti(θ(a;t)),andquality(a;t) = Π·u(θ(a;t)).\nStrategic agents have three main concerns about the collabo ration outcome: feasibility, individual\nrationality (IR), and stability . Given a type profile t, an action profile a, and the type-based action\nprofileθthey induce, we say that a solution is feasible if all agents s ecure a quality goal of 1, i.e.,\nquality(a;t)≥1or equivalently u(θ)≥1coordinate-wise — For example, when specializing our\nmodel to the context of collaborative learning, qualityirepresents the accuracy of the final model\non agent i’s learning task, and feasibility requires the model to achi eve sufficient accuracy across\nall agents’ tasks. IR is satisfied if participating in the col laboration system is more beneficial for\neach agent than completing the task independently. Note tha t a solution satisfies IR iff a≤1(or\nequivalently θ≤1) coordinate-wise. This is because the matrix Whas an all-one diagonal, implying\nthat completing a task independently would require a unit of effort. In addition, a feasible solution\nis stable if, given the contribution of others, no agent can u nilaterally reduce their contribution\nwithout compromising their own feasibility. Formally, thi s requires\n∀i, ai= min{x≥0|qualityi(x,a−i;t)≥1} ⇐⇒ ∀ v, θv= min{x≥0|uv(x,θ−v)≥1}.\n92.2 Benchmarks and the price of stability\nOur benchmarks are the minima of the total workload (i.e., /⌊ard⌊la/⌊ard⌊l1=/⌊ard⌊lθ/⌊ard⌊l1) among solutions that\nsatisfy different combinations of the feasibility, IR, and s tability requirements. We introduce three\nbenchmarks: OPTas the optimal workload in feasible collaborations, and OPTIR,OPTstableas the\noptimal workload subject to additional IR and stability con straints. Formally, they are solutions to\nthe following programs:\nOPT= min\nθ/⌊ard⌊lθ/⌊ard⌊l1s.t.u(θ)≥1,θ≥0;\nOPTIR= min\nθ/⌊ard⌊lθ/⌊ard⌊l1s.t.u(θ)≥1,θ≥0,θ≤1;\nOPTstable= min\nθ/⌊ard⌊lθ/⌊ard⌊l1s.t.u(θ)≥1,θ≥0, θv= min{x≥0|uv(x,θ−v)≥1}(∀v∈V).\nTheprice of stability (PoS) is formally defined as the gap between the socially optimal so lution and\nthe optimal stable solution, i.e., PoS/definesOPTstable/OPT.\n2.3 Information design and signaling schemes\nPrior distribution and information asymmetry. We now introduce the information design\nperspective of our model. Let τbe the prior distribution from which the type profile tis drawn. We\nmodelτas the uniform distribution over the symmetric group, i.e., τ=Unif(Sym(V)). As a result,\nfor each fixed agent i, the marginal distribution on its type tiis uniform over V. We assume that\nthe graph structure Gand prior distribution τare common knowledge, but the realization of the\ntrue types tis exclusive knowledge held by the platform. The platform us es the private Bayesian\npersuasion protocol to strategically communicate this exc lusive knowledge to the agents.\nPrivate Bayesian persuasion. Asignaling scheme , denoted with ϕ:Sym(V)→∆(SV), is a\nmapping from the task type profile t∈Sym(V)to a correlated distribution over private signals\ns= (sv)v∈V∈SV, whereS⊂[0,1]is a finite space of signals, and each svis the signal value sent to\nthe agent with type v.4The interaction protocol between the platform and the agent s is as follows:\ninitially, both the platform and the agents start with a comm on prior τ, and the platform credibly\ncommits to a signaling scheme ϕ, which also becomes common knowledge upon commitment. Then ,\nthe true types t∼τare realized from the prior distribution and observed by the platform. With the\nknowledge of t, the platform generates a set of signals s= (sv)v∈V∼ϕ(t), and sends stiprivately\nto agent ifor alli∈[n], which represents the recommended action to be taken. When a n agenti\nreceives the private signal sti, they form a posterior belief about the type profile tand the signals\nsent to other types s−ti, which we denote with µi:\nµi(t,s−ti|sti) =τ(t)·Prϕ(t)[sti,s−ti]/summationtext\nt′,s′\n−tiτ(t′)·Prϕ(t′)/bracketleftbig\nsti,s′\n−ti/bracketrightbig. (1)\nPersuasiveness. We say that a signaling scheme is persuasive if no agent has the incentive to\nunilaterally deviate from the signal, assuming that all the other agents follow their signals. Formally,\nthis requires that for every agent i, the following two conditions hold:\n4In the original definition, the signals should be sent to each agent rather than each type, but the two definitions\nare equivalent as the platform can reassign the signals to ag ents according to Π·s.\n10• (Feasibility) For every θ∈S, conditioning on receiving a private signal sti=θ, taking action\nθis feasible in expectation:\nE\n(t,s−ti)∼µi(·|sti=θ)[uti(θ,s−ti)]≥1.\n• (Stability) For every θ∈S, conditioning on receiving a private signal of value θ, the agent\ncannot contribute strictly less than θeffort while still meeting the aforementioned feasibility\ncondition:\nθ= min/braceleftBigg\nx≥0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleE\n(t,s−ti)∼µi(·|sti=θ)[uti(x,s−ti)]≥1/bracerightBigg\n.\nEquivalently, the feasibility condition must be tight when everθ >0.\nThecostof a persuasive signaling scheme ϕis the expected total contribution assuming that all\nagents follow the signals. Formally,\nCost(ϕ)/definesE\nt∼τ,s∼ϕ(t)[/⌊ard⌊ls/⌊ard⌊l1].\nIdentity-independent signaling schemes. Note that in the above formulation, since ϕis a\nfunction of permutations of V, the signal sent to each task type could depend on the identit y of the\nagent performing the task. Nevertheless, we claim that it is sufficient to consider signaling schemes\nthat are identity-independent : one such that the signal distribution is independent of the realized\npermutation, i.e., there is a distribution Dϕ∈∆(SV)such that ϕ(t) =Dϕholds for all t∈Sym(V).\nWe prove the following lemma in Appendix A.2 .\nLemma 2.1 (Identity-independent signaling schemes) .For any persuasive signaling scheme ϕ, there\nexists an identity-independent signaling scheme /tildewideϕthat is persuasive and has Cost(/tildewideϕ) =Cost(ϕ).\nSince identity-independent signaling schemes are equival ent to joint distributions on SV, this lemma\nsets the groundwork for interpreting a signaling scheme as a “random labeling” or “random assign-\nment” that directly assigns values in Sto types in V. In the rest of the paper, we will adopt this\nterminology and refer to the signaling process in task struc ture graphs as “labeling” or “assigning”\nrandomized values to its vertices. Furthermore, Lemma 2.1 shows that, to lower bound the cost of\nall persuasive signaling schemes on a particular instance, it suffices to prove such a lower bound\nagainst identity-independent schemes.\n3 Technical Overview and Structural Results\nThis section outlines our approach for proving all of our mai n results. In Section 3.1 , we give a\nsufficient and necessary condition for a signaling scheme to b e persuasive and show that for binary\nsignaling schemes this condition reduces to a simple inequa lity involving cuts and induced sub-\ngraphs in the graph representation of the collaboration sys tem. This characterization is useful both\nfor verifying the persuasiveness of the signaling schemes t hat we design, and for proving lower bounds\nagainst persuasive schemes. With these tools in hand, we giv e an overview of our upper bounds for\nunit-weight graphs ( Section 3.2 ) and general weighted graphs ( Section 3.3 ) starting from concrete\nillustrative examples. We end the section by explaining how we use the LP duality framework to\nprove our lower bounds.\n113.1 Structural results\nConsider an identity-independent signaling scheme Dϕ∈∆(SV)on a finite signal space S⊂[0,1].\nFor each θ∈S, we introduce the concept of slack for receiving signal θas follows:\nDefinition 3.1 (Slack) .For a signaling scheme Dϕ∈∆(SV)and any θ∈S, we define Contrib θ\nas the expected total contribution entering vertices recei vingθ, and Numθas the expected number of\nvertices receiving θ:\nContrib θ/definesE\ns∼Dϕ\n/summationdisplay\nv∈V\n/BD[sv=θ]/summationdisplay\nv′∈N(v)Wv,v′sv′\n;Numθ/definesE\ns∼Dϕ/bracketleftBigg/summationdisplay\nv∈V\n/BD[sv=θ]/bracketrightBigg\n,\nwhereN(v)stands for the open neighborhood of v. The slackof receiving signal θ, denoted with ∆θ,\nis the difference between Contrib θandNumθscaled by the factor (1−θ):\n∆θ/definesContrib θ−(1−θ)·Numθ.\nLemma 3.2 (Persuasiveness of general signaling schemes) .An identity-independent signaling scheme\nDϕ∈∆(SV)with finiteSis persuasive if and only if the following conditions are met :\n•For every θ∈Swhereθ >0, the slack ∆θmust equal zero ( ∆θ= 0).\n•If the signal space includes zero ( 0∈S), the slack ∆0must be non-negative ( ∆0≥0).\nProof sketch. Recall from the definition of persuasiveness, Dϕis persuasive if and only if ∀θ∈S,\nE(t,s−ti)∼µi(·|sti=θ)[uti(θ,s−ti)]≥1, with this inequality being tight for θ >0. Therefore, to prove\nLemma 3.2 , it suffices to show that\nE\n(t,s−ti)∼µi(·|sti=θ)[uti(θ,s−ti)] =θ+Contrib θ\nNumθ. (2)\nThis is because θ+Contrib θ\nNumθ≥1⇔Contrib θ−(1−θ)·Numθ≥0when Numθ>0, which is true because\nit is without loss of generality to assume that every θ∈Sis realized with a non-zero probability.\nWe establish Equation (2) by straightforward application of Bayes’ rule in Appendix B.1 .\nA special class of binary signals Consider a binary scheme ϕwith signal space S={0,α}\nforα∈(0,1]. Recall from Lemma 2.1 thatDϕcan be interpreted as a random labeling of Vusing\neither0orα. It is useful to view the random labeling sas first selecting a random subset S⊆V\nfrom a distribution Dover all subsets of V(namely 2V), and then assigning sv=α· /BD[v∈S]for\neachv∈V. Using this view, we can rewrite Contrib 0andContrib αusing the random subset S∼D:\nContrib α=E\ns∼Dϕ\n/summationdisplay\nv∈V\n/BD[sv=α]/summationdisplay\nv′∈N(v)Wv,v′sv′\n=α·E\nS∼D\n/summationdisplay\nv∈S/summationdisplay\nv′∈S\\{v}Wv,v′\n;\nContrib 0=E\ns∼Dϕ\n/summationdisplay\nv∈V\n/BD[sv= 0]/summationdisplay\nv′∈N(v)Wv,v′sv′\n=α·E\nS∼D\n/summationdisplay\nv/\\e}atio\\slash∈S/summationdisplay\nv′∈SWv,v′\n.\nThe above two quantities have graph-theoretical interpret ation using the cut (S,V\\S)and the\nsubgraph induced by S. We define their total weight respectively as follows.\n12Definition 3.3 (Weights of Cut and Induced Subgraph) .The weight of the cut between Sand its\ncomplement V\\S, denoted Cut(S,V\\S), and the weight of the subgraph induced by S, denoted\nInduced(S), are given by\nCut(S,V\\S)/defines/summationdisplay\nu∈S/summationdisplay\nv∈(V\\S)Wu,v;\nInduced(S)/defines/summationdisplay\nu∈S/summationdisplay\nv∈SWu,v=|S|+2/summationdisplay\n{u,v}⊆SWu,v.\nHere,Cut(S,V\\S)measures the total weight of edges crossing the cut from StoV\\S, andInduced(S)\nmeasures the total weight of all edges within S, including self-loops with Wv,v= 1, and counts each\nundirected internal edge twice.\nLeveraging the characteristics of Sdefined in Definition 3.3 , we provide a graph-theoretical charac-\nterization of the persuasiveness of a binary signaling sche me in the next lemma. The detailed proof\nmirrors the arguments used in Lemma 3.2 and is deferred to Appendix B.2 .\nLemma 3.4 (Persuasiveness of binary signaling schemes) .LetD∈∆(2V)be a distribution over\nsubsets of V. IfDsatisfies the following inequality,5\nES∼D[Cut(S,V\\S)]\nES∼D[|V\\S|]≥ES∼D[Induced(S)]\nES∼D[|S|],\nforα=ES∼D[|S|]\nES∼D[Induced(S)], there exists an identity-independent persuasive signali ng schemeDϕ∈\n∆/parenleftbig\n{0,α}V/parenrightbig\nthat assigns sv=α· /BD[v∈S]according to a random subset S∼D. The cost of this\nsignaling scheme is Cost(ϕ) =(ES∼D[|S|])2\nES∼D[Induced(S)]≤ES∼D[|S|].\nCorollary 3.5 (Cost of the no-information scheme) .WhenDinLemma 3.4 is the degenerate\ndistribution at V, the resulting signaling scheme is persuasive for α=n\nInduced(V)and incurs a cost\nofn2\nInduced(V). Using m=/summationtext\n{v1,v2}∈EWv1,v2to denote the total edge weights in the graph, the cost\nsimplifies ton2\nn+2m=O(n2\nm). This scheme, by not differentiating between vertices, esse ntially conveys\nno information. Therefore, its cost is equal to the total con tribution of agents when no signal is sent.\n3.2 Overview of upper bounds in unit-weight graphs\nWe start by sketching the proof of Theorem 1.1 , which gives a binary signaling scheme with an\nO(√n·OPT)cost. Recall that by Lemma 2.1 , a signaling scheme is equivalent to a randomized\nlabeling of vertices with values between 0and1.\nA signaling scheme for the double-star graph. Our proof is best illustrated by the double-\nstar example from Figure 1 , on which the signaling scheme simply randomizes between th e following\ntwo plans for some small ǫ >0:\n• Plan A: Label the two centers with 1−ǫ. Label the 2kleaves with 0.\n• Plan B: Label the two centers with 0. Label the 2kleaves with 1−ǫ.\nPlan A is almost the same as the socially optimal solution (wi thout the stability constraint), in which\neach center contributes a unit amount. While Plan A has a low c ost, we should not always follow it,\n5WhenDis the degenerate distribution at V, the left-hand side is treated as +∞.\n13x0x1\nx2\nx3\nxky0y1\ny2\ny3\nykx0x1\nx2\nx3\nxky0y1\ny2\ny3\nyk\nFigure 2: Plan A (left) and Plan B (right) for the double-star graph. The shaded vertices are labeled\nwith1−ǫ. The empty vertices are labeled with 0.\nfor two different reasons. First, it violates the stability c ondition: conditioning on receiving signal\n1−ǫ, the agent knows for sure that one of their neighbors will pla y1−ǫ, and thus has an incentive\nto deviate and play a much lower value. Second, it violates fe asibility: conditioning on receiving 0,\nthe agent expects a total contribution of 1−ǫfrom their neighbors, so following the signal would\nnot satisfy their demand. On the other hand, while Plan B appe ars extremely inefficient at first\nglance, it does remedy both issues of Plan A by reducing the sl ack when signal 1−ǫis received,\nand increasing the total contributions from neighbors when the signal is 0.\nWith the notation from Definition 3.1 (i.e.,∆θdenotes the amount of slack corresponding to signal\nθ), always following Plan A gives\n∆1−ǫ= 2·(1−ǫ)−ǫ·2 = +Θ(1) ,∆0= 2k(1−ǫ)−1·2k=−Θ(kǫ),\nwhereas following Plan B gives\n∆1−ǫ= 0−ǫ·2k=−Θ(kǫ),∆0= 2k(1−ǫ)−1·2 = +Θ( k).\nA simple calculation shows that, for some ǫ,p= Θ(1/√\nk), we can ensure ∆1−ǫ= 0and∆0≥0by\nfollowing Plan A with probability 1−pand following Plan B with probability p. The resulting cost\nwould then be (1−p)·2·(1−ǫ)+p·2k·(1−ǫ) =O(√\nk) =O(√n)as desired.\nA binary signal perspective. To generalize this result to all unit-weight graphs, it is he lpful to\nview the signaling scheme above through the lens of Lemma 3.4 , our characterization of persuasive\nbinary signaling schemes. Recall that the lemma states that a distributionD ∈∆(2V)gives a\npersuasive binary signaling scheme if the following inequa lity holds:\nES∼D[Cut(S,V\\S)]\nES∼D[|V\\S|]≥ES∼D[Induced(S)]\nES∼D[|S|]. (3)\nFurthermore, the cost of the scheme is at most ES∼D[|S|].\nIn the double-star example, let DSbe the set of the two centers, and ISbe the set of the 2kleaves.\n(The names are justified since DSis a dominating set of the graph, and ISis an independent set.)\nWhenDis the degenerate distribution at DS,Equation (3) reduces to2k\n2k≥4\n2,which does not hold.\nIn contrast, when Dis the degenerate distribution at IS,Equation (3) gives2k\n2≥2k\n2k,which not\nonly holds, but holds with a large margin! Again, it follows f rom an elementary calculation that,\nEquation (3) can be satisfied by setting D(DS) = 1−pandD(IS) =pfor some p= Θ(1/√\nk), and\nthe resulting cost is, as expected, bounded by ES∼D[|S|] = (1−p)·|DS|+p·|IS|=O(√n).\n14Signaling scheme for general unit-weight graphs. In general, we choose DSas a minimum\ndominating set of G, and choose ISas a maximal independent set of the sub-graph induced by\nV\\DS. As in the double-star graph, DSgives a good approximation of the socially optimal\ncostOPT, but does not guarantee persuasiveness. Concretely, |DS|/OPT is upper bounded by\nthe integrality gap of the LP relaxation of the dominating se t problem, which is O(logn)(e.g.,\n[Williamson and Shmoys ,2011, Section 1.7]). To see that DSmight not be persuasive, note that\nthe left-hand side of Equation (3) ,Cut(DS,V\\DS)\n|V\\DS|, can be as small as 1, while the right-hand side\nInduced(DS)\n|DS|can be as large as |DS|, when the induced sub-graph of DSis a clique.\nOnce again, the other set IScomes to the rescue of persuasiveness. By the independence of IS, the\nright-hand side of Equation (3) givesInduced(IS)\n|IS|= 1. For the left-hand side, we note that\nCut(IS,V\\IS) =Cut(IS,DS)+Cut(IS,V\\(IS∪DS))≥|IS|+|V\\(IS∪DS)|=n−|DS|,\nV\\(DS∪IS)\nISDS\nFigure 3: Cut(IS,V\\IS). Every vertex in IShas a neighbor\ninDS; every vertex in V\\(DS∪IS)has a neighbor in IS.The first step holds since DSandV\\(IS∪DS)\npartitions V\\IS(seeFigure 3 for a pictorial\nillustration). In the second step, Cut(IS,DS)≥\n|IS|sinceDSis a dominating set of G, which\nguarantees that every vertex in IS⊆V\\DShas\na neighbor in DS. Meanwhile, the maximality of\nISimplies that every vertex in V\\(DS∪IS)has a\nneighbor in IS. This gives Cut(IS,V\\(IS∪DS))≥\n|V\\(IS∪DS)|. Therefore, the left-hand side\nofEquation (3) ,Cut(IS,V\\IS)\n|V\\IS|, is at leastn−|DS|\nn−|IS|,\nwhich is typically strictly larger than 1. This\nshows that Equation (3) holds with a margin\nforIS.\nBy carefully randomizing between DSandIS, we obtain a persuasive binary signaling scheme with\ncostO(√n)·|DS|, which is an O(√n·logn)-approximation of OPT. To shave this extra lognfactor,\nour actual proof replaces DSwith a distribution over sets obtained from an independent r ounding\nof the optimal solution. This essentially preserves all the desirable properties of DS, while reducing\nthe cost from|DS|toOPT.\nStrict improvement upon OPTstable.We sketch the proof of Theorem 1.3 , which states that\nthere is a binary signaling scheme with a strictly lower cost thanOPTstable, whenever OPT<\nOPTstable. A key ingredient of the proof is the construction of a binary scheme with cost exactly\nOPTstable, which is based on the following structural result regardin g stable solutions in unit-weight\ngraphs.\nLemma 3.6. Letθ∈[0,1]Vbe an arbitrary stable solution on a unit-weight graph G= (V,E).\nThere exists a distribution D∈∆(2V)supported over the independent sets of G, such that for every\nv∈V,PrS∼D[v∈S] =θv.\nWe prove Lemma 3.6 inSection 5 by rounding the fractional solution θ(which satisfies stability)\ninto an integral solution (i.e., a subset of V), via a rounding scheme known as competing exponential\nclocks. Given the lemma, it then follows quite easily that such a dis tributionDsatisfies Equation (3) ,\nand gives a binary signaling scheme of cost exactly OPTstable. To see this, note that since every\nset in the support of Dis an independent set, we have ES∼D[Induced(S)] =ES∼D[|S|], i.e., the\nright-hand side of Equation (3) is1. Then, using the assumption that θis a stable solution, along\n15···\nFigure 4: A weighted graph on which all binary schemes fail. There are ktriangles in total. Each thick line indicates\nthe edges between a center and all the three vertices in a tria ngle. Every edge is of weight 1/2.\nwith the fact that the marginal of Dexactly matches θ, we can prove that ES∼D[Cut(S,V\\S)]≥\nn−/⌊ard⌊lθ/⌊ard⌊l1=ES∼D[|V\\S|]. This lower bounds the left-hand side of Equation (3) by1.\nThe other ingredient of our proof is an argument based on LP du ality, which shows that, whenever\nOPT<OPTstable, there exists an optimal stable solution θ∈[0,1]Vsuch that the rounding of\nθactually satisfies Equation (3) with a positive margin. This allows us to randomize between\nthis scheme (with cost OPTstable) and an independent rounding of the optimal solution (with c ost\nOPT<OPTstable), and achieve a cost strictly below OPTstable.\n3.3 Overview of upper bounds in weighted graphs\nRecall that for unit-weight graphs, our binary signaling sc heme is based on sending a positive signal\nto either DS(a minimum dominating set) or IS(a maximal independent set of the sub-graph induced\nbyV\\DS), each with a carefully chosen probability. Sending signal 1to the vertices in DSgives a\nlow cost, guarantees feasibility, but might not be stable. T he purpose of the set ISis to reduce the\nslack introduced by DS.\nWhat goes wrong when we repeat this strategy on weighted grap hs? The argument about DS\ngeneralizes easily. On unit-weight graphs, a dominating se t is equivalent to an integral solution, in\nwhich the contribution from each vertex is either 0or1. Naturally, for weighted graphs, we choose\nDSas the set corresponding to the optimal integral solution. T owards satisfying Equation (3) , the\nfeasibility of the integral solution guarantees Cut(DS,V\\DS)≥|V\\DS|, i.e., the left-hand side of\nEquation (3) is lower bounded by 1. Under mild assumptions on the edge weights, we can still sho w\nthat|DS|is not much larger than OPTIR, the optimal fractional solution subject to IR.\nUnfortunately, as we demonstrate in the example below, it mi ght be impossible to find an analogue\nofISfor our purpose.\nA concrete example where binary schemes fail. The graph is shown in Figure 4 . Every edge\nin the graph has a weight of 1/2. The graph contains two “centers” and ktriangles (consisting of\n3k“leaves”). Each center is connected to all other vertices in the graph.\nOn this graph, the optimum subject to IR, OPTIR= 2, is achieved when each center plays 1. As\ndiscussed earlier, we pick DSas the set of the two centers. If we, as in unit-weight graphs, pickIS\nas a maximal independent set of the induced sub-graph of V\\DS,ISwould contain exactly one\nvertex in each triangle. However, it can be verified that no di stributionDsupported over{DS,IS}\nsatisfies Equation (3) . Therefore, we cannot derive a persuasive binary signaling scheme following\nthe previous approach.\nIn fact, this is not even a problem with this specific choice of DSandIS— inTheorem 8.1 , we show\nthatevery persuasive binary signaling scheme has a cost of Ω(n)on the instance above!\n16Solution: a third signal. While binary schemes are insufficient for weighted graphs, a m inimal\nfix — adding a third signal — would solve the problem. Concrete ly, for the graph in Figure 4 , there\nis a non-trivial signaling scheme via randomizing among the three plans below:\n• Plan A: Label the centers with 1−ǫ. Label the leaves with 0.\n• Plan B: Label one vertex from each triangle with 1−ǫ. Label all other vertices with 0.\n• Plan C: Label the leaves with α. Label the centers with 0.\n··· ··· ···\nFigure 5: The three plans (left: Plan A; middle: Plan B; right: Plan C) o f signaling for the graph from Figure 4 .\nShaded circle vertices are labeled with 1−ǫ. Empty circle vertices are labeled with 0. Shaded square vertices are\nlabeled with α.\nTo ensure persuasiveness, we go back to the characterizatio n inLemma 3.2 . The three plans, when\nwe follow each of them alone, give\n• Plan A: ∆1−ǫ= (1−ǫ)−ǫ·2 = +Θ(1) and∆0= 3k(1−ǫ)−1·3k=−Θ(kǫ).\n• Plan B: ∆1−ǫ= 0−ǫ·k=−Θ(kǫ)and∆0= 2k(1−ǫ)−1·(2k+2) =−Θ(1+kǫ).\n• Plan C: ∆0= 3kα−1·2 = 3kα−2and∆α= 3kα−(1−α)·3k= 6kα−3k.\nTo satisfy the constraint ∆α= 0, we setα= 1/2. The effect of Plan C is then simplified to\n∆0= +Θ(k)and∆1/2= 0.\nIn hindsight, the introduction of Plan C and the third signal αis natural. Our previous strategy,\nwhich corresponds to randomizing between only Plans A and B, c annot guarantee both ∆1−ǫ= 0\nand∆0≥0— enforcing ∆1−ǫ= 0inevitably results in ∆0<0. The purpose of Plan C is then to\nbring∆0back to0, without introducing additional slacks.\nFormally, it can be verified that, for some ǫ,p,q= Θ(1/√\nk), following the three plans with prob-\nability1−p−q,p, andqrespectively gives a persuasive scheme. The resulting cost is given\nby\n(1−p−q)·2(1−ǫ)+p·k(1−ǫ)+q·3kα=O(√\nk) =O(√n).\nUpper bound for general graphs. For general weighted graphs, generalizing the ternary sig-\nnaling scheme above gives a cost of n|DS|//radicalbig\n|IS|, whereISis an independent set of the graph. In\nparticular, the first two plans still correspond to assignin g signal1−ǫtoDSandIS, respectively,\nwhile Plan C assigns a different non-zero value to V\\DS, in the hope of making the slack at 0\nnon-negative. When we can find a large IS, this would give a non-trivial approximation of |DS|,\nwhich in turn approximates OPTIR. What if there are no large independent sets, e.g., when the\ngraph is dense?\nAssuming a lower bound on the non-zero edge weights, this iss ue can be resolved via a win-win\nargument. Suppose that the edge weights are at least δ >0. Letmbe the total edge weight in the\ngraph. If mis large, by Corollary 3.5 , we have a persuasive signaling scheme with cost O(n2/m).\nIfmis small, by our assumption on the edge weights, there are at m ostm/δedges in the graph.\n17Intuitively, this sparse graph must have a large independen t set. Indeed, we can lower bound |IS|\nbyΩ(δn2/m). Then, regardless of the value of m, the better between the two schemes gives an\nO((n·OPTIR)2/3·δ−1/3)cost.\nWithout such a lower bound δ, we still have a non-trivial approximation of OPTIR. The key obser-\nvation is that ISdoes not need to be an independent set — the same proof strateg y goes through\nas long as, in the sub-graph induced by IS, every vertex has a (weighted) degree of O(ǫ). Going one\nstep further, we do not need ISto be a deterministic set at all! In our actual proof, we repla ceIS\nwith a random set /tildewideIS⊆V, obtained from including each vertex in the graph with a fixed , carefully\nchosen probability. This gives the O(n3/4·/parenleftbig\nOPTIR/parenrightbig1/2)bound for the general case.\n3.4 Overview of lower bounds\nTo prove the tightness of our approximation guarantees, we s tart by reverse-engineering the proofs\nof the upper bounds, and identifying graphs on which the anal yses are tight. For unit-weight graphs,\nthe hard instance is exactly the double-star graph in Figure 1 . For graphs with edge weights bounded\nbyΩ(1), we modify the construction in Figure 4 by replacing the Θ(n)triangles with Θ(n2/3)copies\nof a clique of size Θ(n1/3).\nThe more difficult part is, of course, to lower bound the cost of allpersuasive signaling schemes on\nthese graphs. To this end, we revisit the characterization o f persuasiveness in Lemma 3.2 :Dϕ∈\n∆(SV)is persuasive if and only if the induced slacks satisfy ∆0≥0and∆θ= 0for allθ∈S\\{0}. By\nDefinition 3.1 , each∆θcan be expressed as an expectation over the distribution Dϕ. Therefore, the\nfamily of persuasive signaling schemes with signal space Sis exactly a subset of ∆(SV)defined by\nfinitely many linear constraints. Consequently, the minimu m-cost persuasive scheme is characterized\nby a linear program.\nThis simple observation leads us to lower bound the cost of pe rsuasive signaling schemes by con-\nstructing a feasible solution of the dual LP. Concretely, su ppose that we could find a function\nf: [0,1]→Rsuch that f(0)≥0and, for anySand degenerate distribution at s∈SV(i.e., a\ndeterministic labeling of Vusing values inS), the resulting slacks satisfy\n/⌊ard⌊ls/⌊ard⌊l1≥/summationdisplay\nθ∈Sf(θ)��∆θ+C. (4)\nThen, for any persuasive signaling scheme Dϕ∈∆(SV), the linearity of expectation gives\nE\ns∼Dϕ[/⌊ard⌊ls/⌊ard⌊l1]≥/summationdisplay\nθ∈Sf(θ)·∆θ+C≥C.\nThe last step above holds since when θ= 0,f(0)·∆0≥0holds as both f(0)and∆0are non-negative.\nWhenθ/\\e}atio\\slash= 0,∆θ= 0impliesf(θ)·∆θ= 0.\nOur proof of the Ω(n2/3)lower bound (for graphs with edge weights lower bounded by Ω(1)) proceeds\nby carefully choosing the function f(θ), and proving Equation (4) for a sufficiently large Cvia an\ninvolved case analysis. For unit-weight graphs, while the Ω(√n)lower bound admits a simpler proof,\nin the proof we implicitly consider a dual solution, in which fis a piece-wise constant function f\nthat takes a negative value with a large magnitude when θis close to 1.\n184 Discussion and Future Directions\nIn this section, we highlight a few concrete open problems an d discuss a few natural extensions of\nour model.\nThe power of simple signaling schemes. A recurring theme in our positive results is the\nsurprising effectiveness of simple signaling schemes that only use a few different signal values . The\nonly exception is Theorem 1.4 : we use Ω(n)different signals to achieve a cost below OPTstable. Can\nwe reduce the number of signals to O(1)? InAppendix C.1 , we explain the technical difficulties in\nproving such a result, and give instances which suggest that the binary and ternary schemes that\nwe consider are insufficient in general.\nTight approximation ratio for weighted graphs. In terms of worst-case approximation guar-\nantees, the only result that is not shown to be optimal is the O(n3/4)approximation on general\nweighted graphs. In Appendix C.2 , we give a concrete instance on which the n3/4ratio is conjec-\ntured to be tight, and discuss why the conjecture does not fol lowing easily from our current proof\nstrategy.\nAlternative collaboration systems. Our work focuses on collaboration systems in which the\nquality of each agent’s task is a linear combination of the ag ent’s own contribution and the amounts\ncontributed by the neighbors. In the context of collaborati ve federated learning, this corresponds\nto the random discovery model proposed by Blum et al. [2021a ]. A non-linear version of the col-\nlaboration system is another model of Blum et al. [2021a ], termed random coverage , in which each\nagent is associated with a (discrete) data distribution, an d the accuracy of each agent’s task is linear\nin the total probability mass of the elements sampled by the a gent themself and their neighbors.\nThis formulation brings more structure to the relation betw een each pair of agents’ tasks and might\nallow us to circumvent some hard instances in the linear mode l. For instance, agent i’s data are\nmaximally effective for another agent jonly if they share the same distribution, and this property\nwould be transitive. In contrast, in our current model, ther e might exist task types i,j, andksuch\nthatWi,j=Wj,k= 1yetWi,k= 0.\nAlternative models of incentives. Another modeling assumption that we made is regarding\nthe agents’ incentives. Implicit in the definition of persua siveness ( Section 2.3 ) is that each agent\nprioritizes satisfying feasibility, and exactly minimize s their own effort subject to the feasibility con-\nstraint. A natural relaxation is to allow approximate stability , i.e., a signaling scheme is considered\nstable as long as no agent has the incentive to decrease their action by some ǫ >0. We may also\nconsider alternative models in which each agent maximizes t he expected quality of their task minus\na penalization term that depends on their own workload.\n5 Upper Bounds for Unit-Weight Graphs\nIn this section, we prove the upper bounds in Theorems 1.1 and1.3. To this end, we first state\na few bounds on the cuts and induced sub-graphs in the unit-we ight graph representation of the\ncollaboration system. We then present a binary signaling sc heme that is shown to be persuasive and\nachieve an O(√n)approximation of OPT. Towards proving Theorem 1.3 , we give a binary scheme\nwith a cost of exactly OPTstable, which, under the additional assumption that OPT<OPTstable,\neasily implies a strict improvement upon OPTstable.\n195.1 Weights of cuts and induced sub-graphs\nWe start by stating several elementary bounds on the cuts and induced edge weights (from Definition 3.3 )\nfor dominating sets and maximal independent sets in a graph. These bounds will be used towards\nconstructing binary signaling schemes via Lemma 3.4 . We defer the proofs to Appendix D .\nThe first bound is regarding maximal independent sets in an in duced sub-graph obtained from\nremoving a dominating set.\nLemma 5.1. LetDSdenote a dominating set of a unit-weight graph G= (V,E), andISbe an\narbitrary maximal independent set of the sub-graph induced byV\\DS. Then, Cut(IS,V\\IS)≥\n|V|−|DS|.\nThe second lemma considers a random set /tildewiderDSobtained from the independent rounding of a feasible\nsolution that satisfies the IR requirement as de���ned in Section 2.2 . This lemma holds for weighted\ngraphs as well.\nLemma 5.2. LetG= (V,E,w)be a weighted graph, and Wbe the corresponding matrix. Let\nθ∈[0,1]Vdenote a feasible solution subject to IR, i.e., Wθ≥1andθ≤1hold coordinate-\nwise. LetD∈∆(2V)denote the distribution of a random subset of Vthat includes each v∈V\nindependently with probability θv. Then, the following bounds hold:\n•ES∼D[|S|] =/⌊ard⌊lθ/⌊ard⌊l1.\n•ES∼D[Induced(S)]≤/⌊ard⌊lθ/⌊ard⌊l2\n1+/⌊ard⌊lθ/⌊ard⌊l1.\n•ES∼D[Cut(S,V\\S)]≥|V|−2/⌊ard⌊lθ/⌊ard⌊l1.\n5.2 General unit-weight graphs\nIn this section, we prove the O(√n·OPT)upper bound in Theorem 1.1 by designing a distribution\nD ∈∆(2V)that satisfies Lemma 3.4 . Our proof closely relies on the properties developed in\nSection 5.1 . Before the proof, we first present a lemma on the integrality g ap ofOPT. Note\nthat the program for computing OPTis essentially the relaxed program determining the minimum\ndominating set of a graph. Therefore, our lemma follows from the integrality gap upper bound for\nthe dominating set problem, e.g., see [ Williamson and Shmoys ,2011, Section 1.7].\nLemma 5.3 (Integrality gap) .There exists a dominating set DS⊆Vsuch that|DS|=O(logn·\nOPT). Moreover, the set DScan be computed efficiently through randomized rounding of OPT.\nWe are now ready to prove the upper bound theorem.\nTheorem 5.4 (Upper bound part of Theorem 1.1 ).In any unit-weight graph, there is a persuasive\nsignaling scheme with a cost of O(√n·OPT).\nProof of Theorem 5.4 .We start with constructing a family of distributions Dp∈∆(2V)parametrized\nbyp∈[0,1], and then show that there exists a choice of psuch that the binary signaling scheme\ninduced byDpis persuasive and has a small cost.\nLetθ⋆∈[0,1]Vbe a socially optimal solution that achieves /⌊ard⌊lθ⋆/⌊ard⌊l1=OPT. We defineDA∈∆(2V)\nto be the distribution of a random subset /tildewiderDS⊆Vthat includes each v∈Vindependently with\nprobability θ⋆\nv— we call/tildewiderDS∼ DAanindependent rounding ofθ⋆. In addition, let DSbe a\ndominating set satisfying Lemma 5.3 . Then,DB∈∆(2V)is defined as the degenerate distribution\nonISwhereISis a maximal independent set of V\\DS.\n20We set the probability mass function of the distribution Dpto be\n∀S⊆V,Dp(S) = (1−p)·DA(S)+p·DB(S).\nRecall from Lemma 3.4 thatDpinduces a persuasive binary signaling scheme if it satisfies\nES∼Dp[Cut(S,V\\S)]\nES∼Dp[|V\\S|]≥ES∼Dp[Induced(S)]\nES∼Dp[|S|]. (5)\nWe analyze each term separately to obtain a sufficient conditi on for eq. (5) to hold. For brevity, we\ndenoteβ/defines|IS|.\n• Applying the first property in Lemma 5.2 toDAgives\nE\nS∼Dp[|S|] = (1−p)·E\nS∼DA[|S|]+p·E\nS∼DB[|S|] = (1−p)·OPT+p·β,\nand thus, ES∼Dp[|V\\S|] =n−ES∼Dp[|S|] = (1−p)·(n−OPT)+p·(n−β).\n• For the expected size of the cut, we can lower bound it as belo w.\nE\nS∼Dp[Cut(S,V\\S)] =(1−p)·E\nS∼DA[Cut(S,V\\S)]+p·E\nS∼DB[Cut(S,V\\S)]\n≥(1−p)·(n−2OPT)+p·(n−|DS|)\n≥(1−p)·(n−2OPT)+p·(n−ι·OPT).\nIn the above equations, the second step uses the third proper ty ofLemma 5.2 andLemma 5.1 .\nIn the last step, we denote |DS|byι·OPTwithι=O(logn)according to Lemma 5.3 .\n• For the expected size of induced subgraphs, we upper bound i t using the second property of\nLemma 5.2 and the independence of IS,\nE\nS∼Dp[Induced(S)] =(1−p)·E\nS∼DA[Induced(S)]+p·E\nS∼DB[Induced(S)]\n≤(1−p)/parenleftbig\nOPT2+OPT/parenrightbig\n+pβ.\nPlugging the above bounds into eq. (5) , we obtain a sufficient condition for persuasiveness:\n(1−p)(n−2OPT)+p(n−ι·OPT)\n(1−p)(n−OPT)+p·(n−β)≥(1−p)(OPT2+OPT)+pβ\n(1−p)OPT+pβ. (6)\nFor the remaining proof, we assume OPT≤√n/Candβ≥C√n·OPT for a constant C=\n10. The first assumption is WLOG because if OPT>√n/C,Corollary 3.5 guarantees a cost of\nO(n2\nn+2m) =O(n)by not sending any signals, which is already an O(√n)approximation to OPT.\nThe second assumption is WLOG because if β=|IS|< C√n·OPT, we can expand ISinto a\nmaximal independent set of Vby including some additional vertices from DS. The size of the\nresulting maximal independent set is at most |IS|+|DS|=β+ι·OPT=O(√n+ logn)·OPT.\nTherefore, sending signal 1to the resulting maximal independent set gives a persuasive binary\nsignaling scheme with cost O(√n·OPT).\nUnder these two additional assumptions, there always exist sp∈[0,1]that satisfies eq. (6) — for\nexample, taking p= 1satisfies eq. (6) by making the left-hand side >1and the right-hand side\n= 1. However, since\nCost(Dp)≤E\nS∼Dp[|S|] = (1−p)·OPT+p·β,\n21the larger pis, the more costly the signaling scheme would be. Therefore , we seek the smallest p\nthat satisfies eq. (6) . To do so, we continue simplifying eq. (6) by subtracting 1from both sides:\n(6)⇐⇒(1−p)(−OPT)+p(β−ι·OPT)\n(1−p)(n−OPT)+p·(n−β)≥(1−p)OPT2\n(1−p)OPT+pβ.\nDividing (1−p)on both denominators and enumerators and setting κ=p/(1−p)gives us:\n(6)⇐⇒−OPT+κ(β−ι·OPT)\n(n−OPT)+κ(n−β)≥OPT2\nOPT+κβ\n⇐⇒(−OPT+κ(β−ι·OPT))(OPT+κβ)≥((n−OPT)+κ(n−β))OPT2\n⇐⇒κ2β(β−ι·OPT)≥OPT2(n−OPT+κ(n−β))+OPT2−κ((β−ι·OPT)OPT−βOPT)\n⇐⇒κ2β(β−ι·OPT)≥OPT2(n−OPT+κ(n−β+ι)+1).\nIn the last equation above, we have β−ι·OPT≥β\n2for all sufficiently large n, by the assumption\nthatβ≥C√n·OPTand the fact that ι=O(logn). In addition, we upper bound the right-hand\nside by dropping the negative terms. This gives the followin g sufficient condition:\n(6)⇐=κ2β2\n2≥(2+2κ)nOPT2,\nwhich can be satisfied by choosing κ= Θ(√nOPT/β). In particular, under this choice, we have\nκ=O(1)due to the assumption that β≥C√nOPT, so both the left- and right-hand sides are of\nthe order Θ(nOPT2). This gives us the choice of pviaκ=p\n1−p.\nFinally, by Lemma 3.4 , the cost of the resulting signaling scheme satisfies\nCost(Dp)≤E\nS∼Dp[|S|] = (1−p)·OPT+p·β≤OPT+κβ≤O(√n·OPT).\n5.3 Matching the cost of full revelation\nTowards proving Theorem 1.3 , in this section, we give a binary signaling scheme with a cos t of\nexactlyOPTstable.\nTheorem 5.5. For any unit-weight graph G, there is a binary signaling scheme with cost OPTstable.\nNote that Theorem 5.5 would be trivial if we allow signaling schemes with a large si gnal space. This\nis because, for any optimal stable solution θ⋆∈[0,1]V, labeling each vertex v∈Vwithθ⋆\nvgives\na persuasive signaling scheme with cost OPTstable. Indeed, the interesting aspect of Theorem 5.5\nis that, while θ⋆might contain many different entries, it is possible to achie ve the same cost, via\nsignaling, using only two different values.\nRecall that Lemma 3.6 states that any stable solution in a unit-weight graph can be written as the\nmarginal of a distribution over independent sets. We first sh ow that the lemma immediately implies\nTheorem 5.5 .\nProof of Theorem 5.5 .Letθ⋆∈[0,1]Vbe an optimal stable solution for graph G= (V,E)with\n/⌊ard⌊lθ⋆/⌊ard⌊l1=OPTstable. We consider the binary signaling scheme defined by distribu tionD∈∆(2V)\nobtained from θ⋆viaLemma 3.6 .\n22ByLemma 3.4 , to verify the persuasiveness of the binary signaling schem e, it suffices to show that\nES∼D[Cut(S,V\\S)]\nES∼D[|V\\S|]≥ES∼D[Induced(S)]\nES∼D[|S|].\nSince every Sin the support of Dis an independent set, we have ES∼D[Induced(S)] =ES∼D[|S|].\nIt remains to show that ES∼D[Cut(S,V\\S)]≥ES∼D[|V\\S|].\nAgain, since Sis always an independent set, the cut size |Cut(S,V\\S)|is equal to/summationtext\nv∈Sdeg(v).\nThen, we have\nE\nS∼D[|Cut(S,V\\S)|] =/summationdisplay\nv∈VPr\nS∼D[v∈S]·deg(v) (Sis an independent set)\n=/summationdisplay\nv∈Vθ⋆\nv·deg(v) (property ofDfromLemma 3.6 )\n=/summationdisplay\nv∈V/summationdisplay\nu∈N(v)θ⋆\nv=/summationdisplay\nv∈V/summationdisplay\nu∈N(v)θ⋆\nu\n≥/summationdisplay\nv∈V(1−θ⋆\nv) (θ⋆is a feasible solution)\n=n−/⌊ard⌊lθ⋆/⌊ard⌊l1=E\nS∼D[|V\\S|].\nThe fourth step holds since both the third line and the fourth line are equal to/summationtext\n{u,v}∈E(θ⋆\nu+θ⋆\nv).\nThis proves persuasiveness.\nByLemma 3.4 , the cost of this scheme is exactly\n(ES∼D[|S|])2\nES∼D[Induced(S)]=E\nS∼D[|S|] =/summationdisplay\nv∈VPr\nS∼D[v∈S] =/summationdisplay\nv∈Vθ⋆\nv=/⌊ard⌊lθ⋆/⌊ard⌊l1=OPTstable.\nThe third step above applies the property of Dguaranteed by Lemma 3.6 .\nNow we prove Lemma 3.6 via rounding the stable solution through a correlated round ing method.\nThe method is referred to as the competing exponential clocks in the context of approximation\nalgorithms (e.g., [ Buchbinder et al. ,2013]).\nProof of Lemma 3.6 .LetV′:={v∈V:θv>0}be the support of the stable solution θ∈[0,1]V.\nWe defineD∈∆(2V)as the distribution of the set S⊆Vgenerated by the following process:\n• For each v∈V′, independently draw Xv∼Exponential (θv).\n• Choose the set Sas/braceleftbig\nv∈V′:Xv<minu∈N(v)∩V′Xv/bracerightbig\n.\nIt is clear that Smust be an independent set: if Scontains two neighbouring vertices uandv, we\nmust have both Xu< XvandXv< Xu, a contradiction.\nFor every v∈V\\V′, we clearly have PrS∼D[v∈S] = 0 =θv. To see that PrS[v∈S] =θvholds\nfor every v∈V′, we use the condition θv+/summationtext\nu∈N(v)∩V′θu= 1, which follows from stability. Then,\nthe property of the exponential distribution shows that minu∈N(v)∩V′Xufollows the exponential\ndistribution Exponential (1−θv)and is independent of Xu. Therefore, we have\nPr[v∈S] = Pr/bracketleftbigg\nXv<min\nu∈N(v)∩V′Xu/bracketrightbigg\n23= E\nXv∼Exponential (θv)[exp(−(1−θv)Xv)]\n=/integraldisplay+∞\n0θvexp(−θvx)·exp(−(1−θv)x) dx=θv,\nas desired.\n5.4 Strict improvement upon full revelation\nIn this section, we build on Theorem 5.5 and characterize conditions under which signaling results\nin astrict improvement compared to OPTstableand the amount of improvement achieved.\nTheorem 5.6 (Formal version of Theorem 1.3 ).On any unit-weight graph that satisfies OPTstable>\nOPT, there exists a binary signaling scheme DϕwithCost(ϕ) =OPTstable−ǫ(OPTstable−OPT),\nwhere\nǫ= max/braceleftBigg\nmin/braceleftBigg\n1\n2,PoS−1\nPoS+2n\nPoS+1/bracerightBigg\n,PoS−1\nPoS+(n−OPT)/bracerightBigg\n>0, (7)\nandPoS=OPTstable/OPTis the price of stability.\nTo prove Theorem 5.6 , we first present two technical lemmas that connect the “wast efulness” of the\noptimal stable solution to the gap OPTstable−OPT. Formally, we call a feasible solution θ∈[0,1]V\nwasteful if it satisfies Wθ/\\e}atio\\slash=1. Recall that feasibility requires Wθ≥1coordinate-wise. Therefore,\na wasteful solution is one in which at least one of the tasks re ceives strictly more contribution than\nwhat is required. Our first lemma states that, even on general weighted graphs, every non-wasteful\nfeasible solution must be optimal. In particular, assuming OPTstable>OPT, the lemma below\nimplies that the optimal stable solution must be wasteful, a nd this wastefulness allows us to design\na signaling scheme with a cost strictly below OPTstable.\nLemma 5.7. In a general weighted graph, if there exists a non-wasteful f easible solution θ∈[0,1]V,\nit holds that OPT=/⌊ard⌊lθ/⌊ard⌊l1.\nProof. Recall from the definition that OPTcan be computed by the following linear program:\nmin\nθ∈Rn1⊤θ (primal LP)\ns.t.Wθ≥1;\nθ≥0.\nSinceW=W⊤, the dual linear program of ( primal LP ) is given by:\nmax\nφ∈Rn1⊤φ (dual LP)\ns.t.Wφ≤1;\nφ≥0.\nSinceθis feasible and non-wasteful, we have Wθ=1, which implies that θis feasible solution for\nboth ( primal LP ) and ( dual LP ), and achieves the same objective value of 1⊤θ=/⌊ard⌊lθ/⌊ard⌊l1. Therefore,\nθmust be an optimal solution for ( primal LP ), and we have OPT=/⌊ard⌊lθ/⌊ard⌊l1.\n24The second lemma quantitatively captures the gap OPTstable−OPTin terms of the “wastefulness”\nof the optimal stable solution θ⋆, i.e., the gap/⌊ard⌊lWθ⋆/⌊ard⌊l1−n, in the case of unit-weight graphs. Again,\nwe state a more general result, and will apply it specifically to the optimal stable solution.\nLemma 5.8. Letθ∈[0,1]Vbe a feasible solution on a unit-weight graph. Then,\n/⌊ard⌊lWθ/⌊ard⌊l1≥n+/⌊ard⌊lθ/⌊ard⌊l1−OPT.\nWe prove Lemma 5.8 inAppendix D.2 . The high-level idea of the proof is to convert θ, which is\na feasible solution of ( primal LP ), into a feasible solution φ⋆of (dual LP ) by lowering the contri-\nbutions that enter wasteful coordinates. Through our caref ul construction, we ensure the resulting\ndecrease in the objective value (i.e., 1⊤θ−1⊤φ⋆) is upper bounded by the wastefulness of θ(i.e.,\n/⌊ard⌊lWθ/⌊ard⌊l1−n). Finally, we establish the lemma by invoking weak duality t o show that 1⊤φ⋆≤OPT.\nNow we prove Theorem 5.6 by constructing a signaling scheme that randomizes between an inde-\npendent rounding of OPTand the signaling scheme in Theorem 5.5 .\nProof of Theorem 5.6 .Letθ⋆∈[0,1]Vbe an optimal solution with cost OPT, andθ∈[0,1]Vbe an\noptimal stable solution with cost OPTstable. Then, letDAbe the distribution defined by rounding\nθ⋆independently, and DBbe the distribution obtained from rounding θaccording to Lemma 3.6 .\nWe define a parametrized family of distributions Dǫ∈∆(2V)forǫ∈[0,1]as\n∀S⊆V,Dǫ(S) =ǫ·DA(S)+(1−ǫ)·DB(S).\nRecall from Lemma 3.4 that the distribution Dǫinduces a persuasive binary signaling scheme if the\nfollowing inequality holds:\nES∼Dǫ[Cut(S,V\\S)]\nES∼Dǫ[|V\\S|]≥ES∼Dǫ[Induced(S)]\nES∼Dǫ[|S|]. (8)\nWe analyze each term separately to obtain a sufficient conditi on for eq. (8) to hold.\n• For the expected size of |S|and|V\\S|, we combine Lemma 3.6 with the first property of\nLemma 5.2 to get\nE\nS∼Dǫ[|S|] =ǫE\nS∼DA[|S|]+(1−ǫ)E\nS∼DB[|S|] =ǫOPT+(1−ǫ)OPTstable;\nE\nS∼Dǫ[|V\\S|] =n−E\nS∼Dǫ[|S|] =n−OPTstable+ǫ(OPTstable−OPT).\n• For the expected size of the cut, we have\nE\nS∼DB[Cut(S,V\\S)] =E\nS∼DB/bracketleftBigg/summationdisplay\nv∈Sdeg(v)/bracketrightBigg\n(S∼DBis an independent set)\n=/summationdisplay\nv∈VPr\nS∼DB[v∈S]·deg(v) (linearity of expectation)\n=/summationdisplay\nv∈Vθv·deg(v) (property ofDBfromLemma 3.6 )\n=θ⊤(W1−1) (Wis the adjacency matrix of G)\n=/⌊ard⌊lWθ/⌊ard⌊l1−/⌊ard⌊lθ/⌊ard⌊l1 (θ⊤W1= (Wθ)⊤1=/⌊ard⌊lWθ/⌊ard⌊l1)\n25≥(n+OPTstable−OPT)−OPTstable(Lemmas 3.6 and5.8)\n=n−OPT.\nCombining the above inequalities with the third property of Lemma 5.2 gives us\nE\nS∼Dǫ[Cut(S,V\\S)] =ǫE\nS∼DA[Cut(S,V\\S)]+(1−ǫ)E\nS∼DB[Cut(S,V\\S)]\n≥ǫ(n−2OPT)+(1−ǫ)(n−OPT)\n=n−(1+ǫ)OPT.\n• For the expected size of induced subgraphs, we upper bound i t using the second property of\nLemma 5.2 and the independence of subsets drawn from DB(Lemma 3.6 ).\nE\nS∼Dǫ[Induced(S)] =ǫE\nS∼DA[Induced(S)]+(1−ǫ)E\nS∼DB[Induced(S)]\n≤ǫ/parenleftbig\nOPT2+OPT/parenrightbig\n+(1−ǫ)OPTstable.\nPlugging the above bounds into Equation (8) , we obtain its sufficient condition:\nn−(1+ǫ)OPT\nn−OPTstable+ǫ(OPTstable−OPT)≥ǫ/parenleftbig\nOPT2+OPT/parenrightbig\n+(1−ǫ)OPTstable\nǫOPT+(1−ǫ)OPTstable(9)\nWe first show the existence of ǫ∈(0,1]to make eq. (9) hold. Note that when ǫ= 0, it reduces to\nn−OPT\nn−OPTstable≥OPTstable\nOPTstable= 1,\nwhich holds with a strict inequality because OPTstable>OPT. Since both sides are continuous in ǫ,\nthere should exist a small neighborhood of ǫaround0that all satisfies eq. (9) .\nMoreover, according to Lemma 3.4 , the cost of the signaling scheme induced by Dǫis\nCost(Dǫ)≤E\nS∼Dǫ[|S|] =ǫOPT+(1−ǫ)OPTstable=OPTstable−ǫ(OPTstable−OPT).\nTherefore, to prove Theorem 5.6 , it remains to show that the choice of ǫineq. (7) satisfies eq. (9) .\nWe start with simplifying the condition by subtracting 1on both sides:\n(9)⇐⇒OPTstable−OPT−ǫOPTstable\nn−OPTstable+ǫ(OPTstable−OPT)≥ǫOPT2\nǫOPT+(1−ǫ)OPTstable.\nOn the one hand, since lower bounding the left-hand side and u pper bounding the right-hand side\nresults in a sufficient condition, we use ǫOPT+(1−ǫ)OPTstable≥OPTand obtain\n(9)⇐=OPTstable−OPT−ǫOPTstable\nn−OPT≥ǫOPT2\nOPT\n⇐⇒OPTstable−OPT\nn−OPT≥ǫ/parenleftBigg\nOPT+OPTstable\nn−OPT/parenrightBigg\n⇐⇒ǫ≤OPTstable−OPT\nOPT(n−OPT)+OPTstable=PoS−1\nPoS+(n−OPT),\n26wherePoS=OPTstable/OPT is the price of stability. This justifies the second choice of ǫin\nEquation (7) .\nOn the other hand, under the condition of ǫ≤1\n2, we have ǫOPT+(1−ǫ)OPTstable≥OPT+OPTstable\n2.\nWe use this to provide an alternative sufficient condition to eq. (9) .\n(9)⇐=OPTstable−OPT−ǫOPTstable\nn≥ǫOPT2\nOPT+OPTstable\n2\n⇐⇒OPTstable−OPT≥ǫ/parenleftbigg2nOPT2\nOPT+OPTstable+OPTstable/parenrightbigg\n⇐⇒ǫ≤OPTstable−OPT\n2nOPT2\nOPT+OPTstable+OPTstable=PoS−1\nPoS+2n\nPoS+1,\nwhich justifies the first option of ǫwhenǫ≤1\n2.\n6 Lower Bounds for Unit-Weight Graphs\nIn this section, we prove the lower bound part of Theorem 1.1 by showing that on the double-star\ngraph in Figure 1 , every persuasive signaling scheme must have an Ω(√n)cost, while the optimal\ntotal workload is O(1). Therefore, the O(√n)approximation guarantee in Theorem 1.1 cannot be\nsignificantly improved, even when non-binary schemes are al lowed.\nThen, we prove a more general lower bound, which states that f or anynandk∈[2,n], there is\nan instance on which OPT= Θ(k),OPTstable= Θ(n), and no persuasive signaling scheme can\nachieve a cost that is much better than min{k√n,n}. Recall that by Theorems 1.1and1.3, on this\ninstance there is a persuasive binary signaling scheme with costO(min{OPT·√n,OPTstable}) =\nO(min{k√n,n}). Therefore, for a wide range of OPT, between the two signaling schemes that we\ndevelop for unit-weight graphs, the better one is essential ly optimal.\n6.1 Lower bound for the double-star graph\nTheorem 6.1 (Lower bound part of Theorem 1.1 ).On the double-star graph with n= 2k+ 2\nvertices, every persuasive signaling scheme has a cost of Ω(√n).\nProof. For clarity, we rename the vertices in the graph with V={(c,1),(c,2)}∪{(i,j) :i∈[2],j∈\n[k]}, where(c,i)is the center of the i-th star, and (i,j)is thej-th leaf in the i-th star.\nFix a persuasive signaling scheme for the graph. By Lemma 2.1 , it is without loss of generality to\nassume that the scheme is identity-independent and is speci fied byDϕ∈∆(SV)on signal space\nS. In other words, the signaling scheme samples (sc,1,sc,2,s1,1,...,s1,k,s2,1,...,s2,k)∈SVfrom\nDϕ, labels the two centers with sc,1andsc,2, and labels the j-th leaf in the i-th star with si,j.\nFor brevity, we use xandyas shorthands for sc,1andsc,2, and let random variable sumdenote\nx+y+/summationtext2\ni=1/summationtextk\nj=1si,j. Then, the cost of the scheme is given by EDϕ[sum].\nLetC:= 3. We will show that, under the assumption that EDϕ[sum]≤√\nk/C, the persuasiveness\nofDϕimpliesEDϕ[sum] = Ω(√\nk). Therefore, EDϕ[sum] = Ω(√\nk) = Ω(√n)always holds.\n27ByLemma 3.2 , the persuasiveness of Dϕimplies that for every θ∈S,\n0≤∆θ=E\nDϕ\n/summationdisplay\nv∈V\n/BD[sv=θ]\n/summationdisplay\nu∈N(v)su−(1−sv)\n\n.\nSumming over all θ∈Sgives\n0≤/summationdisplay\nθ∈S∆θ=E\nDϕ\n/summationdisplay\nv∈V\nsv+/summationdisplay\nu∈N(v)su−1\n\n=E\nDϕ/bracketleftBigg/summationdisplay\nu∈Vsu(1+|N(u)|)/bracketrightBigg\n−n.\nSince|N(u)|= 1whenuis a leaf and|N(u)|=k+1whenuis a center, we have\nn≤E\nDϕ/bracketleftBigg/summationdisplay\nu∈Vsu(1+|N(u)|)/bracketrightBigg\n= 2E\nDϕ/bracketleftBigg/summationdisplay\nv∈Vsv/bracketrightBigg\n+k·E\nDϕ[sc,1+sc,2] = 2E\nDϕ[sum]+k·E\nDϕ[x+y],\nwhich further implies EDϕ[x+y]≥n−2EDϕ[sum]\nk. By the assumption that EDϕ[sum]≤√\nk/C, we\nhave\nE\nDϕ[x+y]≥2k+2−2√\nk/C\nk≥2−2\nC√\nk.\nLetǫ:= 1/√\nk. Since2−(x+y)is always non-negative, by Markov’s inequality, we have\nPr\nDϕ[x+y <2−ǫ] = Pr\nDϕ[2��(x+y)> ǫ]≤EDϕ[2−(x+y)]\nǫ\n≤2−/parenleftBig\n2−2\nC√\nk/parenrightBig\n1/√\nk=2\nC. (EDϕ[x+y]≥2−2/(C√\nk))\nEquivalently, we have PrDϕ[x+y≥2−ǫ]≥1−2/C.\nAs long as k≥2, we have 1−ǫ= 1−1/√\nk >0, soLemma 3.2 implies/summationtext\nθ∈S∆θ· /BD[θ≥1−ǫ] = 0,\nwhich is equivalent to\nE\nDϕ\n/summationdisplay\nv∈V\n/BD[sv≥1−ǫ]\nsv+/summationdisplay\nu∈N(v)su−1\n\n= 0.\nThe left-hand side above can be written as the sum of the follo wing four terms:\n•T1:=EDϕ/bracketleftBig/BD[x+y≥2−ǫ]·/parenleftBig\n2(x+y−1)+/summationtext2\ni=1/summationtextk\nj=1si,j/parenrightBig/bracketrightBig\n, the contribution from the\ntwo centers when x+y≥2−ǫ. Note that this condition implies both x≥1−ǫandy≥1−ǫ.\n•T2:=EDϕ/bracketleftBig/BD[x+y <2−ǫ∧x≥1−ǫ]/parenleftBig\nx+y−1+/summationtextk\ni=1s1,i/parenrightBig/bracketrightBig\n, the contribution from the\ncenter of the first star, when x+y <2−ǫ.\n•T3:=EDϕ/bracketleftBig/BD[x+y <2−ǫ∧y≥1−ǫ]/parenleftBig\nx+y−1+/summationtextk\ni=1s2,i/parenrightBig/bracketrightBig\n, the contribution from the\ncenter of the second star, when x+y <2−ǫ.\n•T4:=EDϕ/bracketleftBig/summationtext\nv∈V\\{(c,1),(c,2)}\n/BD[sv≥1−ǫ]/parenleftBig\nsv+/summationtext\nu∈N(v)su−1/parenrightBig/bracketrightBig\n, the contribution from the\nleaves.\n28Recall that we showed PrDϕ[x+y≥2−ǫ]≥1−2/Cearlier, so the first term T1satisfies\nT1≥E\nDϕ[ /BD[x+y≥2−ǫ]·2(x+y−1)]≥E\nDϕ[ /BD[x+y≥2−ǫ]·2(2−ǫ−1)]\n= Pr\nDϕ[x+y≥2−ǫ]·(2−2ǫ)≥(1−2/C)·2·(1−ǫ)≥2·(1−2/C−ǫ).\nFor the second term, note that when x≥1−ǫholds,x+y−1+/summationtextk\ni=1s1,i≥x−1≥−ǫ. Thus,\nT2≥−ǫ·Pr\nDϕ[x+y <2−ǫ∧x≥1−ǫ]≥−ǫ·Pr\nDϕ[x+y <2−ǫ]≥−2ǫ/C.\nBy symmetry, we also have T3≥−2ǫ/C. For the same reason, the last term can be lower bounded\nby−ǫ·EDϕ/bracketleftBig/summationtext\nv∈V\\{(c,1),(c,2)}\n/BD[sv≥1−ǫ]/bracketrightBig\n. Note that\nE\nDϕ[sum]≥/summationdisplay\nv∈V\\{(c,1),(c,2)}E\nDϕ[sv]≥/summationdisplay\nv∈V\\{(c,1),(c,2)}(1−ǫ)·Pr\nDϕ[sv≥1−ǫ]\n= (1−ǫ)·E\nDϕ\n/summationdisplay\nv∈V\\{(c,1),(c,2)}\n/BD[sv≥1−ǫ]\n,\nso we have T4≥−ǫ\n1−ǫEDϕ[sum].\nRecall that T1+T2+T3+T4= 0. Therefore, combining the four lower bounds gives\n0≥2·(1−2/C−ǫ)−2ǫ/C−2ǫ/C−ǫ\n1−ǫE\nDϕ[sum],\nwhich is equivalent to\nE\nDϕ[sum]≥1−ǫ\nǫ·[2·(1−2/C−ǫ)−4ǫ/C].\nBy our choice of C= 3andǫ= 1/√\nk, the right-hand side above is Ω(√\nk) = Ω(√n)for all\nsufficiently large k. This establishes the Ω(√n)lower bound on the cost.\n6.2 A more general lower bound via k-stars\nThe previous lower bound gives an instance where OPT=O(1)and the optimal persuasive signaling\nscheme has an Ω(√n)cost. Now, we present a more general lower bound in Theorem 6.2 that holds\nfor a wider range of OPT. The proof of Theorem 6.2 is deferred to Appendix E .\nTheorem 6.2. For any n≥k≥2, there is a unit-weight graph with ≤nvertices, on which the\nfollowing hold simultaneously:\n•OPT= Θ(k).\n•OPTstable= Ω(n).\n•Every persuasive signaling scheme has a cost of Ω(min{k√n,n}).\nWhennis sufficiently large, the Θ(·)andΩ(·)notations above hide universal constants that do not\ndepend on nandk.\n29Whenk=O(√n), we consider a generalized version of the double-star const ruction, in which there\narekdisjoint stars, each with Θ(n/k)leaves. The centers of the kstars form a clique. This gives an\ninstance on which OPT=kandOPTstable= Ω(n). By a similar analysis to that of the double-star\ngraph, we show that every persuasive signaling scheme has an Ω(k√n)cost on this k-star graph.\nWhenk≫√n, the above construction cannot be directly used for establi shing an Ω(n)lower\nbound. This is because the k-clique formed by the centers already contains Ω(k2)≫nedges; by\nCorollary 3.5 , anO(n2/m) =o(n)cost can be achieved. Instead, we apply this construction on\nn′= (n/k)2andk′=n/k(so that k′=O(√\nn′)still holds), and construct n/n′=k2/ndisjoint\ncopies of such a graph. Intuitively, the optimal solutions i n the large graph must be n/n′times\nthose for a single copy. This gives OPT= (n/n′)·k′=k,OPTstable= (n/n′)·n′=n, and the cost\nof every persuasive scheme is lower bounded by (n/n′)·Ω(n′) = Ω(n).\n7 Upper Bounds for Weighted Graphs\nIn this section, we prove upper bounds in weighted graphs usi ng the ternary signaling approach\noutlined in Section 3.3 . Together, they give the upper bound part of Theorem 1.2 . We also prove\nTheorem 1.4 , which states that we can achieve a cost strictly lower than OPTstablewhenever OPTIR<\nOPTstable.\n7.1 Upper bound for approximating OPTIR\nBefore presenting the formal theorem statements, we first est ablish the following technical lemma,\nwhich shows that if the graph contains a large set that is “alm ost independent”, we have a good\napproximation of OPTIR.\nLemma 7.1. Suppose that distribution D���∆(2V)satisfies\nE\nS∼D[Induced(S)]≤(1+γ)·E\nS∼D[|S|].\nThen, there is a persuasive signaling scheme with a cost of\nO/parenleftBigg\nOPTIR+γn+n·OPTIR\n/radicalbig\nES∼D[|S|]/parenrightBigg\n.\nNote that if γ= 0, the condition ES∼D[Induced(S)]≤(1+γ)·ES∼D[|S|]is equivalent to that Dis\nsupported over independent sets of V. Whenγ >0, we allow some edges (with a small total weight)\nin the sub-graph induced by a random S∼D, at the cost of an additional γnterm in the cost of\nthe resulting signaling scheme.\nWe include a proof of Lemma 7.1 inAppendix F.1 . The proof is based on optimizing the probabilities\nof sending signals to /tildewiderDS,/tildewideIS∼D, andV\\/tildewiderDSas well as the values of the three signals, as described\ninSection 3.3 . In particular, let θ⋆∈[0,1]Vbe an optimal solution subject to IR with cost /⌊ard⌊lθ⋆/⌊ard⌊l1=\nOPTIR. We consider the following family of ternary signaling sche mes with parameters p,q≥0and\nǫ,α∈[0,1]:\n• With probability1\n1+p+q, draw a random set /tildewiderDS⊆Vthat includes each v∈Vindependently\nwith probability θ⋆\nv. Label (the vertices in) /tildewiderDSwith1−ǫ, and label V\\/tildewiderDSwith0.\n30• With probabilityp\n1+p+q, draw a random set /tildewideIS∼D. Label/tildewideISwith1−ǫ, and label V\\/tildewideISwith\n0.\n• With probabilityq\n1+p+q, draw a random set /tildewiderDS⊆Vthat includes each v∈Vindependently\nwith probability θ⋆\nv. LabelV\\/tildewiderDSwithα, and label/tildewiderDSwith0.\nIn the next section, we prove the upper bound side of Theorem 1.2 by instantiating Lemma 7.1 with\nspecific choices of D.\nInTheorem 7.2 , we provide the O(n3/4)approximation guarantee for general weighted graphs. The\ndistributionDis chosen as the distribution of a random subset of the vertic es sampled with a\ncarefully chosen probability. The proof of Theorem 7.2 is inAppendix F.2 .\nTheorem 7.2 (The first upper bound in Theorem 1.2 ).In any weighted graph, there exists a ternary\nsignaling scheme that is persuasive and has cost O/parenleftBig\nn3/4·/parenleftbig\nOPTIR/parenrightbig1/2/parenrightBig\n.\nWhen the edges in the graph have weights lower bounded by δ >0, we apply Lemma 7.1 with\nγ= 0and chooseDas a degenerate distribution at an independent set of the gra ph. The proof of\nTheorem 7.3 is inAppendix F.3 .\nTheorem 7.3 (The second upper bound in Theorem 1.2 ).In any weighted graph, if every edge has a\nweight of at least δ, there is a persuasive ternary signaling scheme with a cost o fO/parenleftBig/parenleftbig\nn·OPTIR/parenrightbig2/3·δ−1/3/parenrightBig\n.\n7.2 Strict improvement upon full revelation\nIn this section, we state and prove a more formal version of Theorem 1.4 regarding strict improve-\nment upon OPTstable.\nTheorem 7.4 (Formal version of Theorem 1.4 ).In any weighted graphs, if OPTIR<OPTstable,\nthere exists a persuasive signaling scheme that uses at most (n+1)signals and achieves a cost of\nCost=OPTstable−ǫ(OPTstable−OPTIR),whereǫ=PoS−1\nPoS−1+OPTIR(n−OPTIR+1)\nOPTIR+1>0,(10)\nandPoS=OPTstable/OPTis the price of stability.\nSimilar to Theorem 5.6 , our proof relies on quantifying the wastefulness of any feasible solution.\nWe first present a quantitative version of Lemma 5.7 in the case of weighted graphs.\nLemma 7.5. Letθbe a feasible solution in a general weighted graph. Then, we h ave\n/⌊ard⌊lWθ/⌊ard⌊l1≥n+/⌊ard⌊lθ/⌊ard⌊l1−OPT\nOPT.\nThe proof of Lemma 7.5 (deferred to Appendix F.4 ) mirrors that of Lemma 5.8 and follows from a\ndifferent construction of the feasible dual variables. Leve raging this lemma, Theorem 7.4 is achieved\nby randomizing between the independent rounding of the opti mal IR solution and the degener-\nate scheme of deterministically sending the optimal stable solution. We prove this theorem in\nAppendix F.5 .\n318 Lower Bounds for Weighted Graphs\nIn this section, we prove the lower bound part of Theorem 1.2 . Even when all the edges are of\nweight1/2, there are graphs on which OPTIRis a constant, yet no signaling scheme achieves a cost\nlower than n2/3. In addition, when restricted to binary signaling schemes, the gap becomes Ω(n).\nWe start with the lower bound against binary signaling schem es. In contrast to the unit-weight\ncase, we cannot achieve any non-trivial approximation via a binary signaling scheme as before. This\nshows that the introduction of a third signal in Theorem 1.2 is necessary. The proof of the following\ntheorem is in Appendix G.1 .\nTheorem 8.1. There exists a family of weighted graphs on which (1) all edge weights are 1/2; (2)\nOPTIR=O(1); (3) every persuasive binary signaling scheme has a cost of Ω(n).\nThe rest of this section is devoted to proving the lower bound part of Theorem 1.2 .\nTheorem 8.2 (Lower bound part of Theorem 1.2 ).There exists a family of weighted graphs on\nwhich (1) all edge weights are 1/2; (2)OPTIR=O(1); (3) every persuasive signaling scheme has a\ncost ofΩ(n2/3).\nRecall that when the edge weights are 1/2, our proof of Theorem 7.3 implies a signaling scheme\nwith a cost of\nO/parenleftbigg\nmin/braceleftbigg\nOPTIR√\nn+m,n2\nm/bracerightbigg/parenrightbigg\n,\nwheremis the total edge weight in the graph. In order to establish an Ω(n2/3)gap, we need\nOPTIR= Θ(1) andm= Θ(n4/3). This naturally leads to the following construction, which is a\nmodified version of the graph in Figure 4 :\n• The graph contains n=k3+2vertices: two centers and k3leaves.\n• Thek3leaves form k2disjoint cliques, each of size k.\n• Each of the two centers is adjacent to every other vertex in t he graph.\nWe indeed have OPTIR= 2 =O(1)(achieved by letting both centers play 1) andm= Θ(k4) =\nΘ(n4/3). In the rest of this section, we show that any persuasive sign aling scheme for this graph\nmust have an Ω(k2) = Ω(n2/3)cost.\nOur proof of Theorem 8.2 can be divided into the following three steps. We provide a sk etch in the\nremainder of this section and offer more details in Appendices G.2 toG.4.\nStep 1. Dimensionality reduction. ByLemma 2.1 , it suffices to consider an identity-independent\npersuasive signaling scheme specified by Dϕ∈∆(SV)with signal space S. According to Lemma 3.2 ,\nthe persuasiveness of such a signaling scheme is characteri zed by|S|constraints on the slack for\neach signal, which are linear in the n-dimensional probability simplex. As a result, the optimal\nsignaling scheme with minimum cost can be characterized by a linear program. In Appendix G.2 ,\nwe show that this is equivalent to considering the projectio n ofDϕonto a lower-dimensional prob-\nability simplex supported on k+ 2vertices — two centers and a random clique of size k. As a\nresult, we obtain a k+2-dimensional distribution D′and significantly simplify the constraints for\npersuasiveness.\n32Step 2. Choosing test functions. In the second step, we aim to use the duality of linear\nprograms to lower bound the cost of D′. We consider two choices of test functions f:S→R≥0: a\nconstant function f1(θ) = 1, and a function that puts more weight on larger signal values f2(θ) =θ\n1−θ.\nSince the second test function is ill-defined at θ= 1, we additionally show that it is without loss of\ngenerality to consider signaling schemes without the signa l1, as the restriction of distribution D′\ntoS\\{1}incurs at most a constant blow-up in the cost. With this, the o utputs of the two test\nfunctions essentially serve as two sets of dual variables — t hey give rise to/summationtext\nθ∈Sf1(θ)∆θ≥0and/summationtext\nθ∈Sf2(θ)∆θ= 0.See more details in Appendix G.3 .\nStep 3. Verify the dual feasibility In the final step, we consider the dual variables β1f1(θ)−\nβ2f2(θ)with carefully chosen constants β1,β2, and verify the feasibility by establishing the inequality\nE\nD′[sum]≥/summationdisplay\nθ∈S(β1f1(θ)−β2f2(θ))∆θ+Ω(1)≥Ω(1),\nwheresumis the sum of the signals sent to the randomly selected clique . 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In this section, we discuss alternative benchmarks for the Bayesian setting where\nthe agents only know the prior distribution τoftbut not its actual realization. In this case,\nagents evaluate the quality of a collaborative solution in e xpectation under the type distribution\nτ. We start by defining the alternative benchmarks OPT,OPTIR,OPTstable, then show how they\ncan be easily achieved by a no-information signaling scheme and discuss the comparison with the\nbenchmarks defined in Section 2.2 .\nWe define benchmarks OPT as the optimal total workload of a feasible action profile a, and\nOPTIR,OPTstableas the optimal workload under additional IR and stability co nstraints.\nOPT= min\na/⌊ard⌊la/⌊ard⌊l1s.t.E\nt∼τ[quality(a;t)]≥1,a≥0;\nOPTIR= min\na/⌊ard⌊la/⌊ard⌊l1s.t.E\nt∼τ[quality(a;t)]≥1,a≥0,a≤1;\nOPTstable= min\na/⌊ard⌊la/⌊ard⌊l1s.t.E\nt∼τ[quality(a;t)]≥1,a≥0;\n∀i∈[n], ai= min/braceleftBig\nx≥0/vextendsingle/vextendsingle/vextendsingleE\nt∼τ[qualityi(x,a−i;t)]≥1/bracerightBig\n.\nIn is not hard to see that because qualityi(t,a) = ΠWΠ−1afor the permutation matrix that cor-\nresponds to t, the benchmarks OPT,OPTIR,OPTstableforWequals the corresponding benchmarks\nunder the original definition for a different problem instanc eWwith entries\nWij=E\nt∼τ/bracketleftbig\nWti,tj/bracketrightbig\n=/braceleftBigg\n1, i =j,\nInduced(V)−n\nn(n−1), i/\\e}atio\\slash=j,\nin which the unique optimal solution for all three benchmark s isn2\nInduced(V). According to Corollary 3.5 ,\none can achieve this cost by either incorporating a degenera te signaling scheme or not sending any\nsignals at all. Therefore, we do not use these alternative be nchmarks to evaluate our signaling\nschemes.\nNext, we use the comparison of OPTandOPTin unit-weight graphs as an example to show that\nthe two sets of benchmarks are incomparable in general.\n• Consider a unit-weight graph that consists of a complete su bgraphKn\n2andn\n2isolated vertices.\nIn this graph, we have OPT= Θ(n)as all isolated vertices have to contribute unit effort.\nOn the other hand, OPT= Θ(1) becauseInduced(V) = Θ(n2). This shows the possibility of\nhavingOPT≪OPT.\n• Consider a star graph with n−1leaves. We have OPT= 1, achieved by having the center\nnode contribute unit effort. However, OPT= Θ(n)asInduced(V) = Θ(n). Therefore, it is\nalso possible to have OPT≫OPT.\nA.2 Proof of Lemma 2.1\nLetϕ:Sym(V)→∆(SV)be a persuasive signaling scheme that is not necessarily ide ntity-\nindependent. Consider the following identity-independen t signaling scheme /tildewideϕthat always chooses\n38/tildewideϕ(t) =D/tildewideϕto be the mixture of {ϕ(t)|t∈Sym(V)}, i.e.,\nD/tildewideϕ(s) =1\n|Sym(V)|/summationdisplay\nt∈Sym(V)ϕ(t)(s).\nWe show that /tildewideϕis persuasive. Let µi,/tildewideµibe the posterior distribution of agent iunder signaling\nschemes ϕand/tildewideϕ, respectively. Since /tildewideϕ(t)is defined to be independent of t, we have\n/tildewideµi(t,s−ti|sti) =τ(t)·/summationtext\nt′∈Sym(V)Prϕ(t′)/bracketleftBig\nst′\ni,s−t′\ni/bracketrightBig\n/summationtext\ns′\n−t′\ni/summationtext\nt′∈Sym(V)Prϕ(t′)/bracketleftBig\nst′\ni,s′\n−t′\ni/bracketrightBig=1\n|Sym(V)|/summationdisplay\nt′∈Sym(V)µi(t′,s−t′\ni|st′\ni).\nwhere the last step follows from τ(t) =1\n|Sym(V)|and the definition of µiinEquation (1) . Therefore,\nfor the expected quality under the posterior distribution /tildewideµi, we have\nE\n(t,s−ti)∼/tildewideµi(·|sti=θ)[uti(θ,s−ti)] =1\n|Sym(V)|E\nt∼τ\n/summationdisplay\nt′∈Sym(V)E\n(t′,s−t′\ni)∼µi(·|st′\ni=θ)/bracketleftBig\nut′\ni(θ,s−t′\ni)/bracketrightBig\n≥1,\nwhere the inequality is because the persuasiveness of ϕimplies that\nE\n(t,s−ti)∼µi(·|sti=θ)[uti(θ,s−ti)]≥1\nfor any agent i∈[n]and anyθ∈S, which holds with equality when θ >0. We have thus established\nthe persuasiveness of /tildewideϕ.\nFinally, as for the cost, we have\nCost(/tildewideϕ) =E\ns∼D/tildewideϕ[/⌊ard⌊ls/⌊ard⌊l1] =1\n|Sym(V)|/summationdisplay\nt∈Sym(V)E\ns∼ϕ(t)[/⌊ard⌊ls/⌊ard⌊l1] =E\nt∼τ,s∼ϕ(t)[/⌊ard⌊ls/⌊ard⌊l1] =Cost(ϕ).\nB Omitted proofs from Section 3.1\nB.1 Proof of Lemma 3.2\nIn this subsection, we finish the proof of Lemma 3.2 by establishing eq. (2) , which is restated as\nfollows:\nE\n(t,s−ti)∼µi(·|sti=θ)[uti(θ,s−ti)] =θ+Contrib θ\nNumθ.\nWe begin by substituting the expression for the posterior µi(·|sti=θ)fromeq. (1) :\n/summationdisplay\nt,s−tiµi(t,s−ti|sti=θ)·uti(θ,s−ti)(a)=/summationdisplay\nt,s−tiτ(t)Dϕ(θ,s−ti)/summationtext\nt′,s′\n−tiτ(t′)Dϕ(θ,s′\n−ti)·uti(θ,s−ti)\n(b)=/summationtext\nt,sτ(t)Dϕ(s)· /BD[sti=θ]·uti(s)\n/summationtext\nt′,s′τ(t′)Dϕ(s′)· /BD/bracketleftbig\ns′\nti=θ/bracketrightbig=Es∼Dϕ[/summationtext\ntτ(t)· /BD[sti=θ]·uti(s)]\nEs′∼Dϕ/bracketleftbig/summationtext\nt′τ(t′)· /BD/bracketleftbig\ns′\nti=θ/bracketrightbig/bracketrightbig\n(c)=Es∼Dϕ/bracketleftbig/summationtext\nv∈V1\nn· /BD[sv=θ]·uv(s)/bracketrightbig\nEs′∼Dϕ/bracketleftbig/summationtext\nv∈V1\nn· /BD[s′v=θ]/bracketrightbig(d)=Es∼Dϕ/bracketleftBig/summationtext\nv∈V\n/BD[sv=θ]·/parenleftBig\nθ+/summationtext\nv′∈N(v)Wv,v′sv′/parenrightBig/bracketrightBig\nEs′∼Dϕ/bracketleftbig/summationtext\nv∈V\n/BD[s′v=θ]/bracketrightbig\n39=θ+Contrib θ\nNumθ.\nIn the above equations, step (a) uses the expression of poste riorµiineq. (1) together with the\nidentity-independent property of scheme Dϕ. Step (b) follows from Dϕ(θ,s−ti) =/summationtext\nstiDϕ(s)·/BD[sti=θ], and that uti(θ,s−ti) =uti(s)whensti=θ. Step (c) is because/summationtext\nt:ti=vτ(t) =1\nn\nsince the marginal distribution of τontiis uniform over V. Step (d) follows from uv(s) =θ+/summationtext\nv′∈N(v)Wv,v′sv′whensv=θ. The final step uses the linearity of expectations.\nB.2 Proof of Lemma 3.4\nWe prove Lemma 3.4 using the characterizations in Lemma 3.2 . If the signaling scheme Dϕinduced\nby sending αto a random subset of vertices S∼Dis persuasive, its slack (defined in Definition 3.1 )\nmust satisfy ∆0≥0and∆α= 0for allα∈(0,1]. We have\n∆α=Contrib α−(1−α)Numα\n=E\ns∼Dϕ\n/summationdisplay\nv∈V\n/BD[sv=α]\n/summationdisplay\nv′∈N(v)Wv,v′sv′−(1−α)\n\n\n=E\nS∼D\n/summationdisplay\nv∈S\n/summationdisplay\nv′∈S\\{v}Wv,v′·α−1+α\n\n (sv′=α· /BD[v′∈S])\n=E\nS∼D/bracketleftBigg\nα·/summationdisplay\nv∈S/summationdisplay\nv′∈SWv,v′−|S|/bracketrightBigg\n(Wv,v= 1)\n=α·E\nS∼D[Induced(S)]−E\nS∼D[|S|].\nTherefore, to make Dϕpersuasive, we should set α=ES∼D[|S|]\nES∼D[Induced(S)].\nSimilarly, for ∆0, we have\n∆0=Contrib 0−Num0=E\nS∼D\n/summationdisplay\nv/\\e}atio\\slash∈S/parenleftBigg/summationdisplay\nv′∈Sα·Wv,v′−1/parenrightBigg\n=α·E\nS∼D[Cut(S,V\\S)]−E\nS∼D[|V\\S|].\nPlugging in the choice of α, to make ∆0≥0, the distribution Dshould satisfy\nES∼D[Cut(S,V\\S)]\nES∼D[|V\\S|]≥ES∼D[Induced(S)]\nES∼D[|S|],\nwhich is exactly the inequality in Lemma 3.4 . Finally, for the cost of Dϕ, we have\nCost(Dϕ) =α·E\nS∼D[|S|] =(ES∼D[|S|])2\nES∼D[Induced(S)].\nC Details for Section 4\nC.1 Conjectured impossibility of matching OPTstableusingO(1)signals.\nWe conjecture that, in general, ω(1)different signals are needed to achieve a cost below OPTstable.\nOn a technical level, even if it is possible to achieve a cost o fOPTstableusingO(1)different signals,\n40such a result is unlikely to follow from a “natural” proof str ategy, because signaling schemes with\nsmall signal spaces do not compose well.\nOn a graph Gthat consists of several disjoint components G1,G2,..., a natural proof strategy\nwould first derive a signaling scheme ϕiwith a constant-size signal space for each Gi(such that\nCost(ϕi) =OPTstable(Gi)), and then appropriately combine them into a scheme for G. While the\ndirect product of ϕ1,ϕ2,...does give a persuasive scheme for the larger graph G, the number of\ndifferent signals also gets added up, and may scale with the si ze ofG.\nAnother natural idea is to mimic our approach to designing bi nary schemes — we view the signaling\nscheme as a randomized partitioning of GiintoO(1)parts6, and take the direct product of each\nDiinstead. Unfortunately, this breaks even for the case of bin ary schemes: Recall from Lemma 3.4\nthatDi∈∆(2Vi)gives a persuasive binary scheme for Giif and only if\nES∼Di[Cut(S,Vi\\S)]\nES∼Di[|Vi\\S|]≥ES∼Di[Induced(S)]\nES∼Di[|S|]. (11)\nIf we chooseDas the direct product of D1,D2,..., the corresponding condition reduces to\n/summationtext\niES∼Di[Cut(S,Vi\\S)]/summationtext\niES∼Di[|Vi\\S|]≥/summationtext\niES∼Di[Induced(S)]/summationtext\niES∼Di[|S|]. (12)\nPerhaps counter-intuitively, even if Equation (11) holds for every i, the combined inequality in\nEquation (12) might nothold!7\nBased on this technical insight, we found a simple instance on which a numerical optimization\nover ternary signaling schemes suggests that three differen t signals are not enough to achieve a\ncost ofOPTstable. We consider a graph that consists of multiple connected com ponentsG1,G2,...\nwith vertex sets V1,V2,.... Each graph Giadmits a persuasive binary signaling scheme with cost\nOPTstable(Gi)induced by a distribution Di∈∆(2Vi)that satisfies Equation (11) . Towards ensuring\nthat the direct product of D1,D2,...does not give a persuasive signaling scheme, we choose these\ncomponents such that: (1) Equation (11) is tight for each i; (2) the two sides of Equation (11) take\ndifferent values for different values of i.\nConcretely, we consider the graph that consists of:\n•Component 1: Three vertices that form a path with two unit-weight edges.\n•Component 2: Two additional vertices connected by an edge of weight 1/2.\nIn the optimal stable solution, the vertex in the middle of th e3-path plays 1, while both vertices in\nthe other component play 2/3. Recall that a binary signaling scheme is specified by a distr ibution\nover2V, which is a 31-dimensional simplex on this specific graph. Therefore, the optimal binary\nsignaling scheme is characterized by a constrained continu ous optimization problem on this simplex,\nin which the constraint is given by Lemma 3.4 . We relax this optimization problem to multiple linear\nprograms, and numerically verified that no binary signaling scheme achieves a cost of OPTstableor\nlower.\nThe same idea gives another graph that consists of:\n6For instance, the distribution D ∈∆(2V)that specifies a binary scheme is essentially a randomized 2-partition\nofV.\n7This is similar to Simpson’s paradox in statistics. For unit-weight graphs, we avoided this issu e by showing that\nthe left-hand side of Equation (11) is always lower bounded by 1, which in turn upper bounds the right-hand side.\nThis stronger condition doesimply Equation (12) when multiple components are combined.\n41•Component 1: Three vertices that form a path with two unit-weight edges.\n•Component 2: Two vertices connected by an edge of weight 2/3.\n•Component 3: Three vertices that form a triangle with three edges of weigh ts1/2,3/4, and\n3/4.\nA similar but more complicated numerical optimization on te rnary signaling schemes indicates that\nno ternary scheme achieves a cost of OPTstableor lower.\nDespite this negative evidence, it might still be possible to achieve an O(OPTstable)cost using\nO(1)signals. However, such a constant factor approximation, in conjunction with our proof of\nTheorem 1.4 , would not guarantee a strict improvement upon OPTstable.\nC.2 Conjectured Ω(n3/4)lower bound\nWe conjecture that the n3/4ratio is tight on the following instance: The graph consists of two\ncenters connected by a unit-weight edge, along with n−2leaves that form a clique with edge weight\nn−3/4. In addition, each center is connected to each leaf by an edge of weight 1/2.\nWhen only the centers play 1each, we obtain a feasible IR solution with cost 2 =O(1), whereas\nour approach at best gives an O(n3/4)cost on this graph. For a sanity check, note that the total\nedge weight in the graph is m= Θ(n2·n−3/4+n) = Θ(n5/4). Our proof of Theorem 1.2 (more\nspecifically, Theorem 7.2 ) gives a persuasive signaling scheme with cost upper bounde d by a constant\nfactor times\nmin/braceleftbigg\n(OPTIR)2/3m1/3n1/3+OPTIR·√n,n2\nm/bracerightbigg\n,\nyet both terms reduce to Θ(n3/4)when we plug in OPTIR= Θ(1) andm= Θ(n5/4).\nThe graph above resembles the hard instances for the other ca ses on which we manage to prove\ntight lower bounds, so why is it harder to prove an Ω(n3/4)bound? Recall from Section 3.4 that we\nprove lower bounds by carefully choosing a dual solution f: [0,1]→Rto the LP that characterizes\nthe optimal scheme. On the lower bounds that we managed to pro ve, we could choose fas a linear\ncombination of the constant function and the function f(θ) =θ/(1−θ). This choice gives a simple\nclosed-form expression, and satisfies concavity and Lipsch itz continuity (except when θis close to\n1). These make the choice of frelatively amenable to formal proofs. However, in the hard i nstance\ndefined above, a numerical computation suggests that the opt imal dual solution is much more ill-\nbehaved — it is non-convex, non-concave, and has a large deri vative around 1/2. Therefore, we\nexpect that, even if the instance above does witness an Ω(n3/4)bound, the proof would involve a\nmuch more complicated choice of f, and a more involved analysis.\nD Omitted Proofs from Section 5\nD.1 Technical Lemmas in Section 5.1\nWe prove Lemmas 5.1 and5.2stated in Section 5.1 .\nProof of Lemma 5.1 .SinceDSandV\\(IS∪DS)partitions V\\IS, we have\nCut(IS,V\\IS) =Cut(IS,DS)+Cut(IS,V\\(IS∪DS)).\n42SinceDSis a dominating set of G, every vertex in IS⊆V\\DShas a neighbour in DS, so we have\nCut(IS,DS)≥|IS|. SinceISis maximal with respect to the induced sub-graph of V\\DS, every\nvertexv∈V\\(DS∪IS)has a neighbour in IS; otherwise, vcan be added to ISto give a larger\nindependent set. This gives Cut(IS,V\\(IS∪DS))≥|V\\(IS∪DS)|. Therefore, we conclude that\nCut(IS,V\\IS)≥|IS|+|V\\(IS∪DS)|=|V|−|DS|.\nProof of Lemma 5.2 .The first bound follows easily from the fact that each vertex i s included inde-\npendently with probability θ⋆\nv. For the second bound, note that\nE\nS∼D[Induced(S)]≤/summationdisplay\nu,v∈VPr\nS∼D[u,v∈S].\nBy definition ofD,PrS∼D[u,v∈S]is given by θuθvifu/\\e}atio\\slash=v, andθuifu=v. It follows that\nE\nS∼D[Induced(S)]≤/summationdisplay\nu,v∈V:u/\\e}atio\\slash=vθuθv+/summationdisplay\nu∈Vθu≤/parenleftBigg/summationdisplay\nu∈Vθu/parenrightBigg2\n+/parenleftBigg/summationdisplay\nu∈Vθu/parenrightBigg\n=/⌊ard⌊lθ/⌊ard⌊l2\n1+/⌊ard⌊lθ/⌊ard⌊l1.\nFor the third bound, note that\nE\nS∼D[Cut(S,V\\S)] =/summationdisplay\nu,v∈VWu,v·Pr\nS∼D[u /∈S∧v∈S] =/summationdisplay\nu∈V/summationdisplay\nv∈N(u)Wu,v·(1−θu)·θv.\nFor each u∈V, the condition Wθ≥1implies/summationtext\nv∈N(u)Wu,vθv≥1−θu, and it follows that\nE\nS∼D[Cut(S,V\\S)] =/summationdisplay\nu∈V(1−θu)/summationdisplay\nv∈N(u)Wu,vθv≥/summationdisplay\nu∈V(1−θu)2≥/summationdisplay\nu∈V(1−2θu) =|V|−2/⌊ard⌊lθ/⌊ard⌊l1.\nD.2 Proof of Lemma 5.8\nProof of Lemma 5.8 .Recall from the proof of Lemma 5.7 thatOPTis characterized by the linear\nprogram in ( primal LP ), and its dual is given by ( dual LP ). In the following, we construct a feasible\nsolution φ⋆to the dual LP, such that 1⊤(θ−φ⋆)≤/⌊ard⌊lWθ/⌊ard⌊l1−n. By the weak duality of LP, we\nhave1⊤φ⋆≤OPT. It follows that\n/⌊ard⌊lθ/⌊ard⌊l1−OPT≤1⊤θ−1⊤φ⋆≤/⌊ard⌊lWθ/⌊ard⌊l1−n,\nwhich is equivalent to the desired bound.\nNow we show how to construct φ⋆. Without loss of generality, the vertices of the graph are la beled\nwithV= [n]. Letθ(0)=θ. Fori= 1,2,...,n , we examine the i-th coordinate of Wθ(i−1). If\n(Wθ(i−1))i≤1, we set θ(i)=θ(i−1)and move on to the next coordinate. If (Wθ(i−1))i>1, we\nchoose vector θ(i)such that:\n•θ(i)\ni+/summationtext\nj∈N(i)θ(i)\nj= 1, andθ(i)\nj∈/bracketleftBig\n0,θ(i−1)\nj/bracketrightBig\nholds for every j∈N(i)∪{i}. This is possible\nsinceθ(i−1)\ni+/summationtext\nj∈N(i)θ(i−1)\nj= (Wθ(i−1))i>1.\n•θ(i)\nj=θ(i−1)\njfor each j /∈N(i)∪{i}.\n43If(Wθ(i−1))i≤1, we immediately obtain (Wθ(i))i= (Wθ(i−1))i≤1; otherwise, our choice of θ(i)\nguarantees\n(Wθ(i))i=θ(i)+/summationdisplay\nj∈N(i)θ(i)\nj= 1,\nThus, in both cases, the dual constraint Wφ≤1is satisfied by φ=θ(i)on thei-th coordinate.\nFurthermore, it is clear from our procedure that 0≤θ(n)≤θ(n−1)≤ ··· ≤ θ(0)=θholds\ncoordinate-wise. Since all entries in Ware non-negative, Wθ(n)≤Wθ(n−1)≤···≤Wθ(0)=Wθ\nalso holds coordinate-wise.\nTherefore, if we let φ⋆=θ(n), it holds for every i∈[n]that\n(Wφ⋆)i= (Wθ(n))i≤(Wθ(i))i= 1.\nIn other words, φ⋆is a feasible solution to ( dual LP ). Moreover, for each i∈[n], ifθ(i)is different\nfromθ(i−1), we have\n1⊤/parenleftBig\nθ(i−1)−θ(i)/parenrightBig\n=\nθ(i−1)\ni+/summationdisplay\nj∈N(i)θ(i−1)\nj\n−\nθ(i)\ni+/summationdisplay\nj∈N(i)θ(i)\nj\n= (Wθ(i−1))i−1≤(Wθ)i−1.\nIfθ(i)remains the same as θ(i−1), we trivially have\n1⊤/parenleftBig\nθ(i−1)−θ(i)/parenrightBig\n= 0≤(Wθ)i−1,\nwhere the last step follows from the feasibility of θ. Therefore, we have 1⊤/parenleftBig\nθ(i−1)−θ(i)/parenrightBig\n≤\n(Wθ)i−1in both cases, and the desired bound follows from\n1⊤(θ−φ⋆) =n/summationdisplay\ni=11⊤/parenleftBig\nθ(i−1)−θ(i)/parenrightBig\n≤n/summationdisplay\ni=1[(Wθ)i−1] =/⌊ard⌊lWθ/⌊ard⌊l1−n.\nE Omitted Proofs from Section 6\nProof of Theorem 6.2 .We start with the case that k=O(√n). Thek≫√ncase would easily\nfollow from the lower bound for the first case.\nThek=O(√n)case. Concretely, we assume that k≤√n/2. Consider the following graph:\nThere are kstar graphs, each with a center connected to l:=⌊n\nk⌋−1leaves. The kcenters of\nthe stars form a clique. For this graph, we have OPT=k(achieved when each of the kcenters\ncontributes 1) andOPTstable= Ω(n)(e.g., achieved when the center in one of the stars and the\nleaves in all the other stars contribute 1each). We will show that every persuasive signaling scheme\nmust have a cost of Ω(k√n) = Ω(min{k√n,n}).\nAgain, we assume that there is a signaling scheme with cost ≤k√n/CwhereC:= 20, and we will\nstill derive a lower bound of Ω(k√n)on the cost under this assumption. By Lemma 2.1 , without\nloss of generality, the signaling scheme is identity-indep endent and is specified by a distribution\nDϕ∈∆(SV). In other words, the scheme samples ((sc,1,...,sc,k),(si,j)i∈[k],j∈[l])∈SVfromDϕ,\n44labels the center of the i-th star with sc,iand labels the j-th leaf in the i-th star with si,j. For\nbrevity, we shorthand xiforsc,i, and let sum:=/summationtextk\ni=1xi+/summationtextk\ni=1/summationtextl\nj=1si,jdenote the sum of the\nsignals.\nWe first use the persuasiveness of Dϕto derive a lower bound on EDϕ/bracketleftBig/summationtextk\ni=1xi/bracketrightBig\n, the expected total\nsignal received by the kcenters. By Lemma 3.2 , it holds for every θ∈Sthat\n0≤∆θ=E\nDϕ\n/summationdisplay\nv∈V\n/BD[sv=θ]\n/summationdisplay\nu∈N(v)su−(1−sv)\n\n.\nSumming over θ∈Sgives\n0≤/summationdisplay\nθ∈S∆θ=E\nDϕ\n/summationdisplay\nv∈V\nsv+/summationdisplay\nu∈N(v)su−1\n\n=E\nDϕ/bracketleftBigg/summationdisplay\nu∈Vsu(1+|N(u)|)/bracketrightBigg\n−k(l+1).\nSince|N(u)|= 1whenuis a leaf and|N(u)|=l+k−1whenuis a center, we have\nk(l+1)≤E\nDϕ/bracketleftBigg/summationdisplay\nu∈Vsu(1+|N(u)|)/bracketrightBigg\n= 2E\nDϕ[sum]+(l+k−2)·E\nDϕ/bracketleftBiggk/summationdisplay\ni=1xi/bracketrightBigg\n.\nApplying the assumption that EDϕ[sum]≤k√n/Cgives the desired lower bound on EDϕ/bracketleftBig/summationtextk\ni=1xi/bracketrightBig\n:\nE\nDϕ/bracketleftBiggk/summationdisplay\ni=1xi/bracketrightBigg\n≥k(l+1)−2EDϕ[sum]\nl+k−2≥k(l+1)−2k√n/C\nl+k.\nLety:=√n√n+2kand define the random variable X:=/summationtextk\ni=1\n/BD[xi≥y]. Next, we use the lower\nbound on EDϕ/bracketleftBig/summationtextk\ni=1xi/bracketrightBig\nto lower bound the expectation of XbyΩ(k). Applying the inequality/BD[xi≥y]≥xi−y\n1−ygives\nE\nDϕ[X]≥ED/bracketleftBig/summationtextk\ni=1xi/bracketrightBig\n−ky\n1−y≥k(l+1)−2k√n/C\nl+k−ky\n1−y\n=kl(1−y)+k−2k√n/C−yk2\n(1−y)(l+k)\n≥kl\nl+k−2k√n\nC(1−y)(l+k)−yk2\n(1−y)(l+k).\nPlugging y=√n√n+2kinto the last term above gives\nkl\nl+k−n+2k√n\nC(l+k)−k√n\n2(l+k).\nRecall that k≤√n/2,l=⌊n/k⌋−1 = (n/k)·(1+on(1))andC= 20. For all sufficiently large n,\nthe three terms above can be bounded as follows:\nkl\nl+k=k\n1+k/l=k\n1+k2/n·(1+on(1))≥k\n1+1/4+on(1)≥k\n2,\n45n+2k√n\nC(l+k)≤2n\nCl=k\n10(1+on(1))≤k\n8,\nand\nk√n\n2(l+k)=k\n2(√n/k+k/√n)·(1+on(1))≤k\n2·(2+1/2)·(1+on(1))≤k\n4.\nTherefore, we conclude that EDϕ[X]≥k/2−k/8−k/4 =k/8.\nSincek−Xis always non-negative, by Markov’s inequality, we have\nPr\nDϕ/bracketleftbigg\nX≤k\n10/bracketrightbigg\n= Pr\nDϕ/bracketleftbigg\nk−X≥9k\n10/bracketrightbigg\n≤EDϕ[k−X]\n9k/10≤7k/8\n9k/10=35\n36,\nwhich implies PrDϕ/bracketleftbig\nX≥k\n10/bracketrightbig\n≥1\n36.\nApplying Lemma 3.2 again gives\n0 =/summationdisplay\nθ∈S∆θ· /BD[θ≥y] =E\nDϕ\n/summationdisplay\nv\n/BD[sv≥y]\n/summationdisplay\nu∈N(v)su−(1−sv)\n\n\n≥E\nDϕ\nk/summationdisplay\ni=1\n/BD[xi≥y]·/parenleftBiggk/summationdisplay\ni=1xi−1/parenrightBigg\n+k/summationdisplay\ni=1l/summationdisplay\nj=1\n/BD[si,j≥y](y−1)\n\n=E\nDϕ/bracketleftBigg\nX/parenleftBiggk/summationdisplay\ni=1xi−1)/parenrightBigg/bracketrightBigg\n−(1−y)E\nDϕ\nk/summationdisplay\ni=1l/summationdisplay\nj=1\n/BD[si,j≥y]\n.(13)\nIn the second line above, we drop the contributions from the l eaves when xi≥y, and drop the\ncontribution from the incident center when a leaf satisfies si,j≥y.\nIn the rest of the proof, we will first lower bound the EDϕ/bracketleftBig\nX/parenleftBig/summationtextk\ni=1xi−1)/parenrightBig/bracketrightBig\nterm. By Equation (13) ,\nthis gives a lower bound on (1−y)EDϕ/bracketleftBig/summationtextk\ni=1/summationtextl\nj=1\n/BD[si,j≥y]/bracketrightBig\n, which, in turn, lower bounds the\ncost.\nNote that\nX·/parenleftBiggk/summationdisplay\ni=1xi−1/parenrightBigg\n≥X·/parenleftBiggk/summationdisplay\ni=1y· /BD[xi≥y]−1/parenrightBigg\n=X·(yX−1),\nand the minimum of the quadratic function x/ma√sto→yx2−xis−1/(4y). By the assumption that\nk≤√n/2, we have y=√n√n+2k∈[1/2,1], which implies X·/parenleftBig/summationtextk\ni=1xi−1/parenrightBig\n≥−1/(4y)≥−1/2.\nWhenX≥k/10, we have a stronger lower bound of X·/parenleftBig/summationtextk\ni=1xi−1/parenrightBig\n≥(k/10)·[y·(k/10)−1].\nIt follows that\nE\nDϕ/bracketleftBigg\nX/parenleftBiggk/summationdisplay\ni=1si−1)/parenrightBigg/bracketrightBigg\n≥E\nDϕ/bracketleftbigg/BD/bracketleftbigg\nX≥k\n10/bracketrightbigg\n·k\n10·/parenleftbiggky\n10−1/parenrightbigg\n+ /BD/bracketleftbigg\nX <k\n10/bracketrightbigg\n·(−1/2)/bracketrightbigg\n≥Ω(k2)·Pr\nDϕ/bracketleftbigg\nX≥k\n10/bracketrightbigg\n−1\n2≥Ω(k2).\nThe last step follows from the inequality PrDϕ[X≥k/10]≥1/36that we proved earlier. Plugging\nthe above into Equation (13) gives(1−y)EDϕ/bracketleftBig/summationtextk\ni=1/summationtextl\nj=1\n/BD[si,j≥y]/bracketrightBig\n≥Ω(k2).\n46Finally, we conclude that\nE\nDϕ[sum]≥y·E\nDϕ\nk/summationdisplay\ni=1l/summationdisplay\nj=1\n/BD[si,j≥y]\n≥Ω(k2)·y\n1−y≥Ω(k√n),\nwhere the last step follows from y=√n√n+2k. This completes the proof for the k≤√n/2case.\nThek≫√ncase. We handle the k >√n/2case by a reduction to the first case. Let k′:=\n⌊n/(4k)⌋andn′= 4(k′)2. Sincek′≤√\nn′/2, our proof for the first case gives a graph G′with\nn′vertices, on which OPT=k′,OPTstable= Ω(n′), and every persuasive signaling scheme has an\nΩ(k′√\nn′) = Ω(n′)cost.\nLetm:=⌊n/n′⌋= Θ(k2/n). Consider the graph Gthat consists of mdisjoint copies of G′, denoted\nbyG′\n1through G′\nm. It is clear that the benchmarks OPTandOPTstableare additive on a graph\nconsisting of multiple connected components. Thus, on the g raphG, we have OPT=m·k′= Θ(k)\nandOPTstable=m·Ω(n′) = Ω(n).\nIt remains to show that the lower bound on the cost of persuasi ve signaling schemes also composes.\nSuppose thatDϕis a persuasive signaling scheme for graph G, with a cost of C. Consider the\nfollowing induced signaling scheme D′\nϕfor graph G′: Picki∈[m]uniformly at random, draw\ns∼Dϕ, and choose the signal as the restriction of stoG′\ni. Let∆and∆′denote the slacks induced\nby the signaling schemes DϕandD′\nϕ, respectively. Note that D′\nϕis still persuasive, since for every\nθ∈[0,1], we have\n∆′\nθ=1\nmm/summationdisplay\ni=1E\nDϕ\n/summationdisplay\nv∈G′\ni\n/BD[sv=θ]\n/summationdisplay\nu∈N(v)su−(1−sv)\n\n=∆θ\nm≥0,\nand the inequality is tight for all θ >0. Moreover, the cost of the signaling scheme D′\nϕis given by\nC/m.\nNow, applying the lower bound for graph G′shows that C/m≥Ω(k′√\nn′) = Ω(n′). Therefore,\nwe have C≥Ω(mn′) = Ω(n). In other words, every persuasive signaling scheme must hav e an\nΩ(n) = Ω(min{k√n,n})cost on graph G. This completes the proof.\nF Omitted Proofs from Section 7\nF.1 Proof of Lemma 7.1\nProof of Lemma 7.1 .Letθ⋆∈[0,1]Vbe an optimal solution subject to IR with cost /⌊ard⌊lθ⋆/⌊ard⌊l1=OPTIR.\nConsider the following signaling scheme with parameters p,q≥0andǫ,α∈[0,1].\n• With probability1\n1+p+q, draw a random set /tildewiderDS⊆Vthat includes each v∈Vindependently\nwith probability θ⋆\nv. Label (the vertices in) /tildewiderDSwith1−ǫ, and label V\\/tildewiderDSwith0.\n• With probabilityp\n1+p+q, draw a random set /tildewideIS∼D. Label/tildewideISwith1−ǫ, and label V\\/tildewideISwith\n0.\n• With probabilityq\n1+p+q, draw a random set /tildewiderDS⊆Vthat includes each v∈Vindependently\nwith probability θ⋆\nv. LabelV\\/tildewiderDSwithα, and label/tildewiderDSwith0.\n47Properties of /tildewiderDSand/tildewideIS.We will use the following properties of /tildewiderDSand/tildewideIS:\n•E/bracketleftBig\n|/tildewiderDS|/bracketrightBig\n=/⌊ard⌊lθ⋆/⌊ard⌊l1=OPTIR.\n•E/bracketleftBig\nInduced(/tildewiderDS)/bracketrightBig\n≤(OPTIR)2+OPTIR.\n•E/bracketleftBig\nCut(/tildewiderDS,V\\/tildewiderDS)/bracketrightBig\n≥n−2OPTIR.\n•E/bracketleftBig\nInduced(/tildewideIS)/bracketrightBig\n≤(1+γ)E/bracketleftBig\n|/tildewideIS|/bracketrightBig\n.\nThe first three bounds follow from Lemma 5.2 . The last follows from the assumption on D.\nConditions for persuasiveness. ByDefinition 3.1 andLemma 3.2 , in order for the signaling\nscheme above to be persuasive, we need to satisfy:\n• Forθ∈{α,1−ǫ},\n(1−θ)·Numθ=Contrib θ (14)\n• Forθ= 0,Equation (14) holds with “ =” replaced with “≤”.\nIn the rest of the proof, we carefully pick the parameters of t he signaling scheme to satisfy the\nconditions above, while ensuring that the resulting cost sa tisfies the desired upper bound.\nPickαto handle θ=α.We first examine Equation (14) whenθ=α. Since we send αtoV\\/tildewiderDS\nwith probabilityq\n1+p+q, the left-hand side of Equation (14) is given by\n(1−α)·q\n1+p+qE/bracketleftBig\n|V\\/tildewiderDS|/bracketrightBig\n= (1−α)·q\n1+p+q(n−OPTIR).\nThe right-hand side, on the other hand, is given by\nq\n1+p+q·α·E/bracketleftBig\nInduced(V\\/tildewiderDS)−(n−|/tildewiderDS|)/bracketrightBig\n=q\n1+p+q·α·/bracketleftBig\nE/bracketleftBig\nInduced(V\\/tildewiderDS)/bracketrightBig\n−(n−OPTIR)/bracketrightBig\n.\nTherefore, for θ=α,Equation (14) reduces to\n(1−α)(n−OPTIR) =α·[E[Induced(V\\DS)]−(n−OPTIR)],\nwhich holds if we pick α=n−OPTIR\nE/bracketleftBig\nInduced(V\\/tildewiderDS)/bracketrightBig. Note that this choice of αis valid, since we have\nE/bracketleftBig\nInduced(V\\/tildewiderDS)/bracketrightBig\n≥E/bracketleftBig\n|V\\/tildewiderDS|/bracketrightBig\n=n−OPTIR≥0,\nwhich implies α∈[0,1].\nPickǫto handle θ= 1−ǫ.Whenθ= 1−ǫ, the left-hand side of Equation (14) is\nǫ·/bracketleftbigg1\n1+p+q·E/bracketleftBig/vextendsingle/vextendsingle/vextendsingle/tildewiderDS/vextendsingle/vextendsingle/vextendsingle/bracketrightBig\n+p\n1+p+q·E/bracketleftBig\n|/tildewideIS|/bracketrightBig/bracketrightbigg\n=1\n1+p+q·ǫ·/parenleftBig\nOPTIR+pE/bracketleftBig\n|/tildewideIS|/bracketrightBig/parenrightBig\n.\nThe right-hand side is equal to\n(1−ǫ)·/bracketleftbigg1\n1+p+q·E/bracketleftBig\nInduced(/tildewiderDS)−|/tildewiderDS|/bracketrightBig\n+p\n1+p+q·E/bracketleftBig\nInduced(/tildewideIS)−|/tildewideIS|/bracketrightBig/bracketrightbigg\n.\n48Then, Equation (14) atθ= 1−ǫis equivalent to\nǫ·/parenleftBig\nOPTIR+pE/bracketleftBig\n|/tildewideIS|/bracketrightBig/parenrightBig\n= (1−ǫ)·/bracketleftBig\nE/bracketleftBig\nInduced(/tildewiderDS)−|/tildewiderDS|/bracketrightBig\n+p·E/bracketleftBig\nInduced(/tildewideIS)−|/tildewideIS|/bracketrightBig/bracketrightBig\n,\nwhich is satisfied if we pick\nǫ=E/bracketleftBig\nInduced(/tildewiderDS)/bracketrightBig\n−OPTIR+p·E/bracketleftBig\nInduced(/tildewideIS)−|/tildewideIS|/bracketrightBig\nE/bracketleftBig\nInduced(/tildewiderDS)/bracketrightBig\n+p·E/bracketleftBig\nInduced(/tildewideIS)/bracketrightBig∈[0,1].\nPickqto handle θ= 0case. Finally, at θ= 0, the left-hand side of Equation (14) is equal to\n1\n1+p+q·/bracketleftBig\nE/bracketleftBig\n|V\\/tildewiderDS|/bracketrightBig\n+p·E/bracketleftBig\n|V\\/tildewideIS|/bracketrightBig\n+q·E/bracketleftBig\n|/tildewiderDS|/bracketrightBig/bracketrightBig\n=1\n1+p+q·/bracketleftBig\n(n−OPTIR)+p/parenleftBig\nn−E/bracketleftBig\n|/tildewideIS|/bracketrightBig/parenrightBig\n+qOPTIR/bracketrightBig\n.\nThe right-hand side is given by\n1\n1+p+q·/bracketleftBig\n(1−ǫ)·E/bracketleftBig\nCut(/tildewiderDS,V\\/tildewiderDS)/bracketrightBig\n+p·(1−ǫ)·E/bracketleftBig\nCut(/tildewideIS,V\\/tildewideIS)/bracketrightBig\n+q·α·E/bracketleftBig\nCut(/tildewiderDS,V\\/tildewiderDS)/bracketrightBig/bracketrightBig\n.\nRecall that we have E/bracketleftBig\nCut(/tildewiderDS,V\\/tildewiderDS)/bracketrightBig\n≥n−2OPTIR. We also trivially relax the E/bracketleftBig\nCut(/tildewideIS,V\\/tildewideIS)/bracketrightBig\nterm to0. Recall that at θ= 0, it suffices for the left-hand side of Equation (14) to be upper bounded\nby the right-hand side. This can be guaranteed if we have\n(n−OPTIR)+p/parenleftBig\nn−E/bracketleftBig\n|/tildewideIS|/bracketrightBig/parenrightBig\n+qOPTIR≤(1−ǫ)·(n−2OPTIR)+q·α·(n−2OPTIR),\nwhich is equivalent to\nq·[α(n−2OPTIR)−OPTIR]≥(n−OPTIR)+p/parenleftBig\nn−E/bracketleftBig\n|/tildewideIS|/bracketrightBig/parenrightBig\n−(1−ǫ)·(n−2OPTIR).\nWe claim that it is without loss of generality to assume that α(n−2OPTIR)−OPTIR≥αn/2;\notherwise, as we show at the end of the proof, a cost of O(OPTIR)can be trivially achieved. Under\nthis additional assumption, it is, in turn, sufficient to sati sfy\n1\n2qαn≥ǫn+(1−2ǫ)OPTIR+p/parenleftBig\nn−E/bracketleftBig\n|/tildewideIS|/bracketrightBig/parenrightBig\n.\nTherefore, we can always pick qto satisfy the condition at θ= 0, such that qαnis at most O(OPTIR+\n(ǫ+p)n).\nUpper bound the cost by optimizing p.The cost of our signaling scheme is clearly\n1\n1+p+q·/bracketleftBig\n(1−ǫ)E/bracketleftBig\n|/tildewiderDS|/bracketrightBig\n+p·(1−ǫ)E/bracketleftBig\n|/tildewideIS|/bracketrightBig\n+q·αE/bracketleftBig\n|V\\/tildewiderDS|/bracketrightBig/bracketrightBig\n≤OPTIR+pE/bracketleftBig\n|/tildewideIS|/bracketrightBig\n+qαn.\nBy our choice of q, theqαnterm is at most O(OPTIR+(ǫ+p)n), so the cost is also upper bounded\nbyO(OPTIR+(ǫ+p)n).\n49Now, recall our choice of\nǫ=E/bracketleftBig\nInduced(/tildewiderDS)/bracketrightBig\n−OPTIR+p·E/bracketleftBig\nInduced(/tildewideIS)−|/tildewideIS|/bracketrightBig\nE/bracketleftBig\nInduced(/tildewiderDS)/bracketrightBig\n+p·E/bracketleftBig\nInduced(/tildewideIS)/bracketrightBig .\nAlso recall that E/bracketleftBig\nInduced(/tildewiderDS)/bracketrightBig\n≤(OPTIR)2+OPTIRandE/bracketleftBig\nInduced(/tildewideIS)/bracketrightBig\n≤E/bracketleftBig\n|/tildewideIS|/bracketrightBig\n·(1+γ). Note\nthat the denominator above is trivially lower bounded by p·E/bracketleftBig\nInduced(/tildewideIS)/bracketrightBig\n≥pE/bracketleftBig\n|/tildewideIS|/bracketrightBig\n. This gives\nǫ≤(OPTIR)2+p·/parenleftBig\nE/bracketleftBig\n|/tildewideIS|/bracketrightBig\n·(1+γ)−E/bracketleftBig\n|/tildewideIS|/bracketrightBig/parenrightBig\npE/bracketleftBig\n|/tildewideIS|/bracketrightBig =γ+(OPTIR)2\npE/bracketleftBig\n|/tildewideIS|/bracketrightBig.\nIt follows that our cost is at most\nO(OPTIR+(ǫ+p)n)/√re⌋edesequalOPTIR+γn+pn+n(OPTIR)2\npE/bracketleftBig\n|/tildewideIS|/bracketrightBig/√re⌋edesequalOPTIR+γn+n·OPTIR\n/radicalbigg\nE/bracketleftBig\n|/tildewideIS|/bracketrightBig,\nwhere the second step holds if we set p=OPTIR/radicalBig\nE[|/tildewideIS|].\nWhenα(n−2OPTIR)−OPTIR≥αn/2does not hold. Finally, we show that if the assumption\nα(n−2OPTIR)−OPTIR≥αn/2is violated, we can achieve an O(OPTIR)cost easily.\nNote that we must have OPTIR> αn/6in this case; otherwise, we would have\nα(n−2OPTIR)−OPTIR≥α[n−2·(αn/6)]−αn/6≥αn−αn/3−αn/6 =αn/2,\na contradiction. The second step above holds since α∈[0,1]. Also, we may assume that OPTIR≤\nn/2; otherwise, any persuasive signaling scheme would give a co st of at most n=O(OPTIR).\nNow, recall that α=n−OPTIR\nE/bracketleftBig\nInduced(V\\/tildewiderDS)/bracketrightBig. Then,OPTIR> αn/6andOPTIR≤n/2together imply\nOPTIR>n(n−OPTIR)\n6·E/bracketleftBig\nInduced(V\\/tildewiderDS)/bracketrightBig≥n2\n12·E/bracketleftBig\nInduced(V\\/tildewiderDS)/bracketrightBig≥n2\n12(n+2m),\nwheremis the total edge weight in the graph. Then, by Corollary 3.5 , we can achieve a cost of\nn2\nn+2m≤12OPTIR=O(OPTIR).\nF.2 Proof of Theorem 7.2\nProof of Theorem 7.2 .Letmbe the total edge weight in the graph, and γ >0be a parameter to\nbe chosen later. Define D∈∆(2V)as the distribution of the random set obtained from includin g\neachv∈Vindependently with probability r:= min/braceleftbigγn\n2m,1/bracerightbig\n. Clearly, we have ES∼D[|S|] =rnand\nE\nS∼D[Induced(S)] =rn+2r2m=/parenleftbigg\n1+2rm\nn/parenrightbigg\n·E\nS∼D[|S|]≤(1+γ)E\nS∼D[|S|].\n50ByLemma 7.1 , there is a persuasive signaling scheme with a cost of, up to a constant factor, at\nmost\nOPTIR+γn+n·OPTIR\n/radicalbigg\nmin/braceleftBig\nγn2\n2m,n/bracerightBig/√re⌋edesequalOPTIR+γn+OPTIR·max/braceleftBig/radicalbig\nm/γ,√n/bracerightBig\n.\nUnder this choice of γ= (OPTIR)2/3m1/3n−2/3, that above cost reduces to\n(OPTIR)2/3m1/3n1/3+OPTIR·√n.\nIn addition, recall from Corollary 3.5 that we can easily achieve a cost of O(n2/m). Therefore, we\ncan always achieve the minimum between these two and obtain a n upper bound of\nmin/braceleftbigg\n(OPTIR)2/3m1/3n1/3+OPTIR·√n,n2\nm/bracerightbigg\n≤min/braceleftbigg\n(OPTIR)2/3m1/3n1/3,n2\nm/bracerightbigg\n+OPTIR·√n\n/√re⌋edesequaln3/4·(OPTIR)1/2+OPTIR·√n.\nNote that the second term above dominates the first only if OPTIR= Ω(√n), at which point\nboth terms are greater than the trivial cost of n. Therefore, we can further simplify the bound to\nO/parenleftbig\nn3/4·(OPTIR)1/2/parenrightbig\n.\nF.3 Proof of Theorem 7.3\nBefore proving the theorem, we first present a lemma that lower bounds the size of the maximum\nindependent set in a graph.\nLemma F.1. A graph with nvertices and medges contains an independent set of size Ω/parenleftbig\nmin{n2/m,n}/parenrightbig\n.\nProof. Consider the greedy algorithm that keeps adding the vertex w ith the smallest degree, and\nthen removing the vertex along with its neighbors. As long as the number of remaining vertices,\nn′, is at least n/2, the smallest degree is at most2m\nn′≤4m/n, so each iteration removes at most\n4m/n+ 1vertices. The maximal independent set obtained from this gr eedy approach has size at\nleastn/2\n4m/n+1= Ω(min{n2/m,n}).\nNow we are ready to prove Theorem 7.3 .\nProof of Theorem 7.3 .Letmdenote the total edge weight of the graph. Since the edge weig hts are\nlower bounded by δ, there are at most m/δedges in the graph. By Lemma F.1 , the graph contains\nan independent set ISof sizeΩ(min{n2/(m/δ),n}) = Ω(min{δn2/m,n}).\nLetD∈∆(2V)be the degenerate distribution at IS. We have ES∼D[Induced(S)] =ES∼D[|S|] =|IS|.\nThen, applying Lemma 7.1 withγ= 0gives a persuasive signaling scheme with a cost of\nO/parenleftBigg\nOPTIR+n·OPTIR\n/radicalbig\nmin{δn2/m,n}/parenrightBigg\n=O/parenleftBig\nOPTIR·/parenleftBig√n+/radicalbig\nm/δ/parenrightBig/parenrightBig\n.\nAgain, we use the fact that we can achieve a cost of O(n2/m)(Corollary 3.5 ). This give an upper\nbound of\nO/parenleftbigg\nmin/braceleftbigg\nOPTIR·/parenleftBig√n+/radicalbig\nm/δ/parenrightBig\n,n2\nm/bracerightbigg/parenrightbigg\n=O/parenleftbigg/parenleftBig\nn·OPTIR/parenrightBig2/3\nδ−1/3+OPTIR·√n/parenrightbigg\n.\n51Note that the OPTIR·√nterm dominates the first term only if OPTIR= Ω(√n), at which point the\nfirst term is already Ω(n). Therefore, the upper bound can be simplified to O/parenleftbig\n(n·OPTIR)2/3δ−1/3/parenrightbig\n.\nF.4 Proof of Lemma 7.5\nProof of Lemma 7.5 .For the sake of contradiction, assume /⌊ard⌊lWθ/⌊ard⌊l1< n+/bardblθ/bardbl−OPT\nOPT. Therefore, it\nmust be the case that for all i∈[n],\n1≤(Wθ)i<1+/⌊ard⌊lθ/⌊ard⌊l1−OPT\nOPT=/⌊ard⌊lθ/⌊ard⌊l1\nOPT.\nRecall from the proof of Lemma 5.7 thatOPTis characterized by the linear program in eq. (primal LP) ,\nand its dual is given by eq. (dual LP) . We construct dual variables φ⋆as follows, where Rmaxis the\nlargest value among all coordinates of Wθ:\nφ⋆/defines1\nRmaxθ, R max/definesmax\ni∈[n](Wθ)i</⌊ard⌊lθ/⌊ard⌊l1\nOPT.\nNote that φ⋆is a feasible solution of eq. (dual LP) because\nWφ⋆=1\nRmax(Wθ)≤1.\nConnecting to the optimal objective value OPT= minθ1⊤θofeq. (primal LP) , we have\nOPT=min\nθ1⊤θ≥max\nφ1⊤φ≥1⊤φ⋆(weak duality)\n=1\nRmax/parenleftBig\n1⊤θ/parenrightBig\n=1\nRmax·/⌊ard⌊lθ/⌊ard⌊l1 (definition of φ⋆)\n>OPT, (Rmax</bardblθ/bardbl1\nOPT)\na contradiction! Therefore, we must have /⌊ard⌊lWθ/⌊ard⌊l1≥n+/bardblθ/bardbl1−OPT\nOPT.\nF.5 Proof of Theorem 7.4\nProof of Theorem 7.4 .Letθ⋆∈[0,1]Vbe an optimal IR solution with cost OPTIR,θ∈[0,1]Vbe\nthe optimal stable solution with cost OPTstable, andα∈[0,1]be a signal value to be defined later.\nByLemma 5.7 , the fact that/⌊ard⌊lθ/⌊ard⌊l1=OPTstable>OPTIR≥OPTimplies that θmust be wasteful.\nLetDA∈∆({0,α}V)be the distribution defined by first sampling a random set S⊆Vby including\neach vertex vindependently with probability θ⋆\nv, and then labeling each v∈Vwithsv=α· /BD[v∈S].\nLetDθ⋆∈∆(2V)denote the distribution of such a random set S. In addition, we define DB∈\n∆([0,1]V)as the degenerate distribution supported on {θ}. Let\nS={θv:v∈V}∪{α}\nbe the signal space including all distinct values in θandα. Note that we must have 0∈S, because\na stable solution can only be wasteful when there exists vsuch that θv= 0; otherwise, the stability\ncondition implies Wθ=1, i.e.,θis not wasteful. Thus, we have |S|≤n+1.\nWe define a parametrized family of distributions Dǫ∈∆(SV)forǫ∈[0,1]as\n∀s∈SV,Dǫ(s) =ǫ·DA(s)+(1−ǫ)·DB(s).\n52Recall from Lemma 3.2 that the signaling scheme Dǫis persuasive if for all θ∈S, we have\n∆θ=E\ns∼Dϕ\n/summationdisplay\nv∈V\n/BD[sv=θ]\n/summationdisplay\nv′∈N(v)Wv,v′sv′−(1−θ)\n\n≥0, (15)\nand the above is tight for all θ∈S\\{0}. We now analyze ∆θfor each θ∈S.\n•Case 1. θ∈S\\{0,α}. Sinceθis stable, for any vsuch that θ=θv>0, the feasibility\ncondition must be tight, i.e.,\nuv(θ) =θ+/summationdisplay\nv′∈N(v)Wv,v′θv′= 1.\nThis, together with the fact that DBis a degenerate distribution on θ, implies\n∆DB\nθ=/summationdisplay\nv∈V\n/BD[θv=θ]\n/summationdisplay\nv′∈N(v)Wv,v′θv′−(1−θ)\n= 0.\nTherefore, we have\n∆Dǫ\nθ= (1−ǫ)∆DB\nθ= 0.\n•Case 2. θ=α. Ifα=θvfor some v, the analysis in the previous case gives us ∆DBα= 0. On\nthe other hand,\n∆DAα=E\ns∼DA\n/summationdisplay\nv∈V\n/BD[sv=α]\n/summationdisplay\nv′∈N(v)Wv,v′sv′−(1−α)\n\n\n=E\nS∼Dθ⋆/bracketleftBigg/summationdisplay\nv∈S/parenleftBigg\nα/summationdisplay\nv′∈SWv,v′−1/parenrightBigg/bracketrightBigg\n(sv=α· /BD[v∈S])\n=α·E\nS∼Dθ⋆[Induced(S)]−OPTIR. (ES∼Dθ⋆[|S|] =OPTIR)\nTherefore, to guarantee ∆DAα= 0, we set\nα=OPTIR\nES∼Dθ⋆[Induced(S)]. (16)\nNote that ��≤1becauseES∼Dθ⋆[Induced(S)]≥ES∼Dθ⋆[|S|] =OPTIR. In addition, we have\n∆Dǫα=ǫ∆DAα+(1−ǫ)∆DBα= 0.\n•Case 3. θ= 0. We first analyze the slack in DA.\n∆DA\n0=E\ns∼DA\n/summationdisplay\nv∈V\n/BD[sv= 0]\n/summationdisplay\nv′∈N(v)Wv,v′sv′−1\n\n\n=E\nS∼Dθ⋆\n/summationdisplay\nv∈V\\S/parenleftBigg\nα/summationdisplay\nv′∈SWv,v′−1/parenrightBigg\n (sv=α· /BD[v∈S])\n53=α·E\nS∼Dθ⋆[Cut(S,V\\S)]−(n−OPTIR) (ES∼Dθ⋆[|V\\S|] =n−OPTIR)\n=OPTIR·ES∼Dθ⋆[Cut(S,V\\S)]\nES∼Dθ⋆[Induced(S)]−(n−OPTIR) (choice of αineq. (16) )\n≥OPTIR·n−2OPTIR\n(OPTIR)2+OPTIR−(n−OPTIR)(last two properties in Lemma 5.2 )\n=−OPTIR(n−OPTIR+1)\nOPTIR+1.\nFor∆DB\n0, since(Wθ)v−1 = 0 for all coordinates where θv>0, we have\n∆DB\n0=/summationdisplay\nv∈V\n/BD[θv= 0]\nθv+/summationdisplay\nv′∈N(v)Wv,v′θv′−1\n\n=/summationdisplay\nv∈V\n/BD[θv= 0]((Wθ)v−1)\n=/summationdisplay\nv∈V\n/BD[θv= 0]((Wθ)v−1)+/summationdisplay\nv∈V\n/BD[θv>0]((Wθ)v−1)(θv>0 =⇒(Wθ)v= 1)\n=/⌊ard⌊lWθ/⌊ard⌊l1−n\n≥/⌊ard⌊lθ/⌊ard⌊l1−OPT\nOPT=OPTstable−OPT\nOPT,\nwhere the inequality follows from Lemma 7.5 . Therefore, for the mixed distribution Dǫ, we\nhave\n∆Dǫ\n0=ǫ∆DA\n0+(1−ǫ)∆DB\n0\n≥ −ǫ·OPTIR(n−OPTIR+1)\nOPTIR+1+(1−ǫ)OPTstable−OPT\nOPT.\nTherefore, for the signaling scheme Dǫto be persuasive, it suffices to choose ǫ∈[0,1]that\nsuch that\n−ǫ·OPTIR(n−OPTIR+1)\nOPTIR+1+(1−ǫ)·OPTstable−OPT\nOPT≥0\n⇐⇒ǫ≤OPTstable−OPT\nOPTstable−OPT+OPT·OPTIR(n−OPTIR+1)\nOPTIR+1=PoS−1\nPoS−1+OPTIR(n−OPTIR+1)\nOPTIR+1,\nwhich justifies the choice of ǫinEquation (10) . The cost of this signaling scheme is\nCost(Dǫ) =ǫαE\nS∼DA[|S|]+(1−ǫ)OPTstable≤OPTstable−ǫ(OPTstable−OPTIR),\nwhere we have used α≤1andES∼DA[|S|] =OPTIR.\nG Omitted Proofs from Section 8\nG.1 Proof of Theorem 8.1\nProof of Theorem 8.1 .We prove the theorem using the example from Figure 4 : Setn= 3k+2for\nsome integer k. LetC1,C2,...,C kbekcliques, each of size 3. Letv1,v2be two additional vertices.\n54Eachviis connected to each other and all the vertices in C1:k. All edges have weight 1/2. The\noptimal solution is 2, achieved when both v1andv2play1while the other vertices play 0.\nFix any binary signaling scheme D∈∆(2[n]). Consider the distribution D′over/vector x= (x3,x2,x1,x0,y)∈\n{0,1,...,k}4×{0,1,2}induced byDthrough the following procedure:\n• Sample S∼D.\n• Fori∈{0,1,2,3}, letxibe the number of cliques in C1,...,C kthat contain exactly ivertices\ninS.\n• Lety=|S∩{v1,v2}|.\nNote thatD′is a sufficient statistics of Din terms of the persuasiveness of the induced binary sig-\nnaling scheme: When Dis the degenerate distribution at S⊆[n], the resulting /vector x= (x3,x2,x1,x0,y)\nsatisfies\n\n\n|S|= 3x3+2x2+x1+y\nCut(S,V\\S) = 3x3+3x2+2x1−y\n2(3x3+x2−x1−3x0)+\n/BD[y=1]\n2\nInduced(S) =|S|+3x3+x2+y(3x3+2x2+x1)+ /BD[y= 2](17)\nByLemma 3.4 , we have\nED[Cut(S,V\\S)]\nn−ED[|S|]≥ED[Induced(S)]\nED[|S|]=1\nα,\nwhereαis the value of the non-zero signal.\nSubstituting Equation (17) into the above condition, subtracting 1from both sides, and using the\nfact that n= 3(x0+x1+x2+x3)+2, we obtain\nED′/bracketleftBig\n3x3+2x2+x1y\n2+y−3x0(1−y\n2)−y\n2(3x3+x2)−/parenleftBig\n2−\n/BD[y=1]\n2/parenrightBig/bracketrightBig\nn−ED[|S|]\n≥ED′[3x3+x2+y(3x3+2x2+x1)+ /BD[y= 2]]\nED[|S|]. (18)\nWe compare the numerators of both sides of Equation (18) when(x3,x2,x1,x0,y)is deterministic.\nWheny∈{0,2}, we have 2· /BD[y= 2]≥y, and it follows that\n2·[3x3+x2+y(3x3+2x2+x1)+ /BD[y= 2]]≥3x3+2x2+x1y\n2+y,\nwhich is, in turn, lower bounded by the numerator on the left- hand side of Equation (18) . When\ny= 1andx1+x2+x3≥1, we have\n2·[3x3+x2+y(3x3+2x2+x1)+ /BD[y= 2]] = 12 x3+6x2+2x1\n= (11x3+5x2)+x1+(x1+x2+x3)\n≥3x3+2x2+x1y\n2+y,\nwhich is again lower bounded by the numerator on the left-han d side of Equation (18) . Finally, in\nthe remaining case that x1=x2=x3= 0,x0=k, andy= 1, the numerator on the right-hand side\nofEquation (18) is0, while the numerator on the left is given by\ny−3x0(1−y/2)−(2− /BD[y= 1]/2) =−3\n2k−1\n2<0.\n55LetLHSandRHSdenote the left- and right-hand sides of Equation (18) , respectively. Then, the\nabove case analysis shows that\n2·ED[|S|]\nn−ED[|S|]·RHS≥LHS≥RHS,\nwhich implies ED[|S|]≥n/3 = Ω(n).\nRecall that by Lemma 3.4 , the cost of the binary scheme is given by α·ED[|S|]. Therefore, it is left\nto showα= Ω(1) . To prove α= Ω(1) , it suffices to upper boundED[Induced(S)]\nED[|S|]by a constant. We\nhave\nED[Induced(S)]\nED[|S|]−1 =ED′[3x3+x2+y(3x3+2x2+x1)+ /BD[y= 2]]\nED[|S|]\n≤ED′[9x3+5x2+2x1+ /BD[y= 2]]\nED′[3x3+2x2+x1+y]≤3.\nTherefore, α=ED[|S|]\nED[Induced(S)]≥1\n4= Ω(1) . This lower bounds the cost of the binary scheme by\nα·ED[|S|] = Ω(n).\nG.2 Projection to a low-dimensional space\nConsider an identity-independent persuasive signaling sc heme specified by Dϕ∈∆(SV)with signal\nspaceS. The scheme naturally induces a distribution D′over(x,y,α1,α2,...,α k)∈Sk+2in the\nfollowing way:\n• First, we sample a set of signals from Dϕ.\n• Letxandybe the signals sent to the two center vertices.\n• Choose one of the k2cliques uniformly at random, and set α1,...,α kto the signals sent to\nthekvertices in that clique.\nDefine random variable sum:=/summationtextk\ni=1αi. It is easy to verify that the cost of the signaling scheme is\ngiven by\nE\n(x,y,α)∼D′/bracketleftbig\nx+y+k2·sum/bracketrightbig\n,\nso our goal is to prove that ED′[sum] = Ω(1) .\nRecall from Definition 3.1 that∆θis the following quantity:\n∆θ=Contrib θ−(1−θ)·Numθ,\nwhich is the total amount of slack at signal value θ.Lemma 3.2 states that a valid scheme must\nsatisfy∆0≥0and∆θ= 0for every θ >0.\nOur first step is to re-write each ∆θas an expectation over the distribution of (x,y,α1,...,α k).\nThis reduces the dimensionality of the signaling scheme fro mn=k3+ 2tok+ 2, and slightly\nsimplifies the notations in the remainder of the proof.\nLemma G.1 (informal) .The slacks (∆θ)θ∈Scan be equivalently defined as the expected outcome of\nthe following procedure:\n•Draw(x,y,α1,...,α k)∼D′.\n56•∆x←∆x+y+k2sum\n2−(1−x).\n•∆y←∆y+x+k2sum\n2−(1−y).\n•For every i∈[k],∆αi←∆αi+k2·/bracketleftbigsum+x+y+αi\n2−1/bracketrightbig\n.\nInformally, the k2factors in the above account for the fact that every clique is only sampled with\nprobability 1/k2.\nBelow is a more formal statement of the lemma above.\nLemma G.2 (Formal version of Lemma G.1 ).LetDϕ∈∆(SV)be the distribution that specifies a\nsignaling scheme for the graph. Formally, the scheme draws\ns= (sc,1,sc,2,s1,1,...,s1,k,s2,1,...,s2,k,...,sk2,1,...,sk2,k)∼Dϕ,\nlabels the two centers with sc,1andsc,2, and labels the j-th vertex in the i-th clique with si,j.\nLetD′∈∆(Sk+2)be the projection of Dϕ, i.e.,D′is the distribution of\n(sc,1,sc,2,si,1,si,2,...,si,k)\nwhens∼Dϕandiis drawn uniformly at random from [k2].\nThen, for any θ∈S, we have\nE\ns∼Dϕ\n/summationdisplay\nv∈V\n/BD[sv=θ]·1\n2/summationdisplay\nu∈N(v)su\n−(1−θ)·E\ns∼Dϕ/bracketleftBigg/summationdisplay\nv∈V\n/BD[sv=θ]/bracketrightBigg\n=E\n(x,y,α)∼D′/bracketleftbigg/BD[x=θ]·/parenleftbiggy+k2sum\n2−(1−x)/parenrightbigg/bracketrightbigg\n+E\n(x,y,α)∼D′/bracketleftbigg/BD[y=θ]·/parenleftbiggx+k2sum\n2−(1−y)/parenrightbigg/bracketrightbigg\n+k/summationdisplay\ni=1E\n(x,y,α)∼D′/bracketleftbigg/BD[αi=θ]·k2·/parenleftbiggsum+x+y+αi\n2−1/parenrightbigg/bracketrightbigg\n.\nProof. It suffices to prove the identity for deterministic values of sc,1,sc,2, and(si,j)(i,j)∈[k2]×[k], i.e.,\nwhenDϕis degenerate. The general case then follows from taking an e xpectation. To this end, we\nshow that each sc,1,sc,2, andsi,jcontributes the same amount to both sides of the equation. Fo r\nsimplicity, we will assume that the k3+2entries of sare distinct; the general case follows from the\nsame argument.\nContribution from sc,1andsc,2.Whenθ=sc,1, the left-hand side is given by\n1\n2/summationdisplay\nu∈N((c,1))su−(1−sc,1) =sc,2\n2+1\n2k2/summationdisplay\ni=1k/summationdisplay\nj=1si,j−(1−sc,1).\nThe right-hand side is\nE\n(x,y,α)∼D′/bracketleftbiggy+k2sum\n2−(1−x)/bracketrightbigg\n=sc,2\n2−(1−sc,1)+k2\n2E\n(x,y,α)∼D′[sum]\n=sc,2\n2−(1−sc,1)+k2\n2·1\nk2k2/summationdisplay\ni=1k/summationdisplay\nj=1si,j,\n57which is equal to the left-hand side.\nBy symmetry, the contributions from sc,2to both sides are also equal.\nContribution from si,j.Whenθ=si,j, the left-hand side reduces to\n1\n2/summationdisplay\nu∈N((i,j))su−(1−si,j) =sc,1+sc,2\n2+/summationdisplay\nj′∈[k]\\{j}si,j′\n2−1+si,j=sc,1+sc,2+si,j\n2+/summationdisplay\nj′∈[k]si,j′\n2−1,\nwhile the right-hand side is given by\nE\n(x,y,α)∼D′/bracketleftbigg/BD[αj=si,j]·k2·/parenleftbiggsum+x+y+αj\n2−1/parenrightbigg/bracketrightbigg\n.\nNote that αj=si,jholds only when α1,...,α kare equal to si,1,...,si,k, which happens with\nprobability 1/k2by definition ofD′. Thus, the expression above can be simplified to\n1\nk2·k2·/parenleftBigg/summationtextk\nj′=1si,j′+sc,1+sc,2+si,j\n2−1/parenrightBigg\n=sc,1+sc,2+si,j\n2+/summationdisplay\nj′∈[k]si,j′\n2−1,\nwhich is exactly the left-hand side. This completes the proo f.\nG.3 Choice of the test function\nRecall that our goal is to show that any distribution D′over(x,y,α1,...,α k)induced by a valid\nsignaling schemeDϕmust satisfy ED′[sum] = Ω(1) . In order forD′to be induced by a valid scheme,\nit must satisfy ∆0≥0and∆θ= 0for every θ∈S\\{0}, where(∆θ)θ∈Sare obtained from D′\nviaLemma G.1 . While the number of constraints might be large, we will care fully take only a few\nlinear combinations of them, such that they are sufficient for lower bounding ED′[sum].\nIn particular, we note that for every function f:S→Rwithf(0)≥0, we must have\n/summationdisplay\nθ∈Sf(θ)·∆θ≥0.\nThe hope is that we can choose a few simple functions fsuch that, after plugging x,y, andαiinto\nthe inequality above, we obtain a good lower bound on ED′[sum].\nThe constant function. We start with the most simple choice of f(θ)≡1. ByLemma G.1 , the\ncontribution of the combination (x,y,α1,...,α k)to/summationtext\nθ∈Sf(θ)·∆θ=/summationtext\nθ∈S∆θis given by:\n/bracketleftbiggy+k2sum\n2−(1−x)/bracketrightbigg\n+/bracketleftbiggx+k2sum\n2−(1−y)/bracketrightbigg\n+k2·k/summationdisplay\ni=1/bracketleftbiggsum+x+y+αi\n2−1/bracketrightbigg\n=3\n2(x+y)+k2sum−2+k3·/bracketleftbiggsum+x+y\n2−1/bracketrightbigg\n+k2\n2sum\n=/parenleftbigg1\n2k3+3\n2k2/parenrightbigg\nsum+/parenleftbigg1\n2k3+3\n2/parenrightbigg\n(x+y)−(k3+2).\nBy linearity, the condition/summationtext\nθ∈S∆θ≥0can be written as\n/parenleftbigg1\n2k3+3\n2k2/parenrightbigg\nE\nD′[sum]+/parenleftbigg1\n2k3+3\n2/parenrightbigg\nE\nD′[x+y]≥(k3+2).\nNote that this condition alone is not enough for lower boundi ngED′[sum], since the above can be\neasily satisfied by setting sum≡0andx,y≡1.\n58Getting rid of signal 1.Later, we will consider another choice of f(θ) =θ\n1−θ, which is ill-defined\natθ= 1. To avoid this issue, we now take a detour and show that sendin g signal 1is essentially\nuseless, so it is without loss of generality that 1/∈S.\nNote that whenever signal 1is sent to a vertex, all the neighbouring vertices must recei ve signal\n0. This is because, according to Lemma G.1 , whenever ∆1is changed, the change is always non-\nnegative. Therefore, to ensure ∆1= 0, there cannot be any non-zero contribution to vertices that\nreceive signal 1. In other words, all neighbours of such vertices must receiv e signal0.\nTherefore, the support of D′must be contained in (S\\{1})k+2∪{e1,e2,...,ek+2}, whereeiis the\nvector with 1at thei-th coordinate and zeros elsewhere. Using Lemma G.1 , we can verify that the\ncontribution of each eito∆0is non-positive, while ∆θis unaffected for θ >0. In particular, when\neitherx= 1ory= 1,∆0increases by\n/parenleftbigg1\n2−1/parenrightbigg\n+k·k2·/parenleftbigg1\n2−1/parenrightbigg\n=−k3+1\n2<0.\nWhen one of the αi’s is equal to 1,∆0gets increased by\n2·/parenleftbiggk2\n2−1/parenrightbigg\n+(k−1)·k2·/parenleftbigg1\n2−1/parenrightbigg\n=−1\n2k3+3\n2k2−2≤0.\nTherefore, we may let D′′be the restriction of D′to(S\\{1})k+2. As argued above, D′′is still valid\nin the sense that (∆θ)θ∈S\\{1}induced byD′′(according to Lemma G.1 ) still satisfies ∆0≥0and\n∆θ= 0for allθ >0.\nWe still need to show that the restriction to (S\\{1})k+2does not significantly increase the cost,\ni.e.,ED′′[sum]should be O(1)·ED′[sum]. To this end, it suffices to prove that D′((S\\{1})k+2)is\nlower bounded by Ω(1), so that the normalization does not blow up the cost. We start by claiming\nthat\nD′({e3,e4,...,ek+2}) = Pr\n(x,y,α)∼D′[α1= 1∨α2= 1∨···∨αk= 1]≤1\n10.\nThis holds because, otherwise, we have E(x,y,α)∼D′[sum]≥1\n10·1 = Ω(1) , and we are done. Further-\nmore, we claim that\nD′({e1,e2}) = Pr\n(x,y,α)∼D′[x= 1∨y= 1]≤1\n10.\nEarlier, we showed that when either x= 1ory= 1,/summationtext\nθ∈S∆θis decreased byk3+1\n2. Furthermore,\nin general, each fixed value of (x,y,α1,...,α k)increases/summationtext\nθ∈S∆θby\n/parenleftbigg1\n2k3+3\n2k2/parenrightbigg\nsum+/parenleftbigg1\n2k3+3\n2/parenrightbigg\n(x+y)−(k3+2)≤/parenleftbigg1\n2k3+3\n2k2/parenrightbigg\nsum+1.\nThis gives\n0≤/summationdisplay\nθ∈S∆θ≤ −Pr\n(x,y,α)∼D′[x= 1∨y= 1]·k3+1\n2\n+E\n(x,y,α)∼D′/bracketleftbigg/BD[x/\\e}atio\\slash= 1∧y/\\e}atio\\slash= 1]·/bracketleftbigg/parenleftbigg1\n2k3+3\n2k2/parenrightbigg\nsum+1/bracketrightbigg/bracketrightbigg\n≤ −Pr\n(x,y,α)∼D′[x= 1∨y= 1]·k3+1\n2+/parenleftbigg1\n2k3+3\n2k2/parenrightbigg\nE\n(x,y,α)∼D′[sum]+1,\n59which, together with Pr[x= 1∨y= 1]≥1/10, would imply E[sum] = Ω(1) . Therefore, without\nloss of generality, we may assume that D′((S\\{1})k+2)≥1−1\n10−1\n10=4\n5. Therefore,\nE\n(x,y,α)∼D′′[sum]≤5\n4E\n(x,y,α)∼D′[sum].\nIn particular, proving ED′′[sum] = Ω(1) would imply ED′[sum] = Ω(1) .\nAnother choice of f.Now, we assume that 1/∈S, and examine/summationtext\nθ∈Sf(θ)·∆θwhenf(θ) =\nθ\n1−θ. Again, Lemma G.1 implies that the contribution of a fixed set of values (x,y,α1,...,α k)to/summationtext\nθ∈Sθ\n1−θ·∆θis given by\nx\n1−x·/bracketleftbiggy+k2sum\n2−(1−x)/bracketrightbigg\n+y\n1−y·/bracketleftbiggx+k2sum\n2−(1−y)/bracketrightbigg\n+k/summationdisplay\ni=1αi\n1−αi·k2/bracketleftbiggsum+x+y+αi\n2−1/bracketrightbigg\n.\nThe sum of the first two terms above can be simplified into\nxy\n2·/parenleftbigg1\n1−x+1\n1−y/parenrightbigg\n+k2sum\n2·/parenleftbiggx\n1−x+y\n1−y/parenrightbigg\n−(x+y),\nwhile the last summation can be re-written as\nk2k/summationdisplay\ni=1αi\n1−αi·/bracketleftbiggsum+x+y−1\n2−1−αi\n2/bracketrightbigg\n=k2\n2(sum+x+y−1)k/summationdisplay\ni=1αi\n1−αi−k2\n2sum.\nG.4 Verify the dual feasibility\nWe will show that, for any finite signal space S⊂[0,1)and every fixed choice of (x,y,α1,...,α k)∈\nSk+2, it holds that\nsum+β2·/summationdisplay\nθ∈Sθ\n1−θ·∆θ≥Ω(1)+β1·/summationdisplay\nθ∈S∆θ, (19)\nwhereβ2=1\n4k,β1=1\n2k2, and the Ω(1)notation hides a positive universal constant that does not\ndepend on (x,y,α1,...,α k), as long as kis sufficiently large.\nAssuming that eq. (19) holds, taking an expectation and rearranging shows that\nE\nD′[sum]≥Ω(1)+β1·E/bracketleftBigg/summationdisplay\nθ∈S∆θ/bracketrightBigg\n−β2·E/bracketleftBigg/summationdisplay\nθ∈Sθ\n1−θ·∆θ/bracketrightBigg\n≥Ω(1),\nwhich implies the Ω(k2) = Ω(n2/3)lower bound on the cost and proves Theorem 8.2 .\nNow we plug β1=1\n4k,β2=1\n2k2, as well as the expressions for/summationtext\nθ∈S∆θand/summationtext\nθ∈Sθ\n1−θ·∆θinto\neq. (19) . It is sufficient to prove that\nsum+xy\n8k/parenleftbigg1\n1−x+1\n1−y/parenrightbigg\n+k\n8sum·/parenleftbiggx\n1−x+y\n1−y/parenrightbigg\n−x+y\n4k+k(sum+x+y−1)\n8k/summationdisplay\ni=1αi\n1−αi−k\n8sum\n≥Ω(1)+/parenleftbiggk\n4+3\n4/parenrightbigg\nsum+/parenleftbiggk\n4+3\n4k2/parenrightbigg\n(x+y)−/parenleftbiggk\n2+1\nk2/parenrightbigg\n.\n60Sincex+y≤2 =O(1), every additive term of form (x+y)/poly(k)or1/poly(k)can be absorbed\ninto theΩ(1)gap (for all sufficiently large k). Thus, it suffices to show\nk\n2+xy\n8k/parenleftbigg1\n1−x+1\n1−y/parenrightbigg\n+k\n8sum·/parenleftbiggx\n1−x+y\n1−y/parenrightbigg\n+k(sum+x+y−1)\n8k/summationdisplay\ni=1αi\n1−αi\n≥Ω(1)+/parenleftbigg3\n8k−1\n4/parenrightbigg\nsum+k\n4(x+y).(20)\nWe will consider the following three different cases:\n•Case 1: x+y <1andsum+x+y−1≥0. Sincex,y,α1,...,α k∈[0,1), we have\nx\n1−x+y\n1−y≥x+yandk/summationdisplay\ni=1αi\n1−αi≥k/summationdisplay\ni=1αi=sum.\nFurthermore, we relax thexy\n8k/parenleftBig\n1\n1−x+1\n1−y/parenrightBig\nterm in eq. (20) to0. Then, it suffices to prove\nthe following:\nk\n2+k\n8sum·(x+y)+k(sum+x+y−1)\n8·sum≥Ω(1)+/parenleftbigg3\n8k−1\n4/parenrightbigg\nsum+k\n4(x+y).(21)\nFor fixed sum, both sides of eq. (21) are affine in x+y, so it suffices to verify it at x+y= 1\nandx+y= max{1−sum,0}. Atx+y= 1,eq. (21) gets reduced to\nk\n8sum2−/parenleftbiggk\n4−1\n4/parenrightbigg\nsum+k\n4≥Ω(1),\nwhich is true since the left-hand side is equal tok\n8(sum−1)2+sum\n4+k\n8≥k\n8≥1\n8.\nAtx+y= max{1−sum,0}, ifsum≤1(so that x+y= 1−sum),eq. (21) is equivalent to\nsum\n4+k\n4/parenleftbigg\n1−sum2\n2/parenrightbigg\n≥Ω(1),\nwhich is true since sum≤1implies1−sum2\n2≥1/2. In the other case that sum>1, we have\nx+y= 0, and this reduces eq. (21) to\nk\n2+k\n8sum(sum−1)−/parenleftbigg3\n8k−1\n4/parenrightbigg\nsum≥Ω(1),\nwhich always holds since the left-hand side is equal tok\n8(sum−2)2+sum\n4≥sum\n4>1\n4.\n•Case 2: x+y <1andsum+x+y−1<0. Again, we may apply the relaxation\nxy\n8k/parenleftbigg1\n1−x+1\n1−y/parenrightbigg\n+k\n8sum·/parenleftbiggx\n1−x+y\n1−y/parenrightbigg\n≥k\n8sum·(x+y).\nHowever, since the factor sum+x+y−1is now negative, the last term on the left-hand\nside of eq. (20) (namely,k(sum+x+y−1)\n8/summationtextk\ni=1αi\n1−αi) is minimized when αis a permutation of\n(sum,0,0,...,0). Thus, we need to prove the following inequality:\nk\n2+k\n8sum(x+y)+k(sum+x+y−1)\n8·sum\n1−sum≥Ω(1)+/parenleftbigg3\n8k−1\n4/parenrightbigg\nsum+k\n4(x+y).(22)\n61Again, it suffices to verify the above at x+y= 0andx+y= 1−sum, respectively. At\nx+y= 0,eq. (22) reduces to\nk\n2·(1−sum)+1\n4sum≥Ω(1),\nwhich is true sincek\n2·(1−sum)+1\n4sum≥1\n4·(1−sum)+1\n4sum=1\n4.\nAtx+y= 1−sum, we get exactly the same inequality as the “ x+y= max{1−sum,0}and\nsum≤1” part of Case 1, which has already been verified.\n•Case 3: x+y≥1(and thus, sum+x+y−1≥0). In this case, we claim that the minimum\nof bothxy\n8k/parenleftBig\n1\n1−x+1\n1−y/parenrightBig\nandx\n1−x+y\n1−yare achieved at x=y≥1/2. Writex=µ+δand\ny=µ−δforµ=x+y\n2∈[1/2,1). We have\nxy·/parenleftbigg1\n1−x+1\n1−y/parenrightbigg\n= (µ2−δ2)·2−(µ+δ)−(µ−δ)\n(1−µ−δ)(1−µ+δ)= (2−2µ)·/bracketleftbigg\n1+µ2−(1−µ)2\n(1−µ)2−δ2/bracketrightbigg\n,\nwhich is minimized at δ= 0, sinceµ∈[1/2,1)guarantees that 2−2µ >0andµ2−(1−µ)2≥0.\nSimilarly,x\n1−x+y\n1−ycan be written as\n1\n1−x+1\n1−y−2 =2−2µ\n(1−µ)2−δ2−2,\nwhich is also minimized at δ= 0.\nThus, it suffices to prove eq. (20) for thex=ycase, i.e., for all x∈[1/2,1),\nk\n2+1\n4k·x2\n1−x+k\n4sum·x\n1−x+k(sum+2x−1)\n8·sum≥Ω(1)+/parenleftbigg3\n8k−1\n4/parenrightbigg\nsum+k\n2x.\nConsider the function\ng(x,s):=k\n2+1\n4k·x2\n1−x+k\n4s·x\n1−x+k(s+2x−1)\n8·s−/parenleftbigg3\n8k−1\n4/parenrightbigg\ns−k\n2x\n=k\n8s2−/parenleftbigg1\n2k−1\n4−k\n4x−k\n4·x\n1−x/parenrightbigg\ns+k\n2(1−x)+1\n4k·x2\n1−x.\nWe need to prove that g(x,s)≥Ω(1)holds for all x∈[1/2,1)ands≥0. For any fixed x,\ng(x,s)is quadratic in s, and we have\ninf\ns≥0g(x,s) =/braceleftBigg\ng(x,0), s∗(x)<0,\ng(x,s∗(x)), s∗(x)≥0,\nwhere\ns∗(x):=1\nk/4/parenleftbigg1\n2k−1\n4−k\n4x−k\n4·x\n1−x/parenrightbigg\n= 2−x−x\n1−x−1\nk.\nThe case that s∗(x)<0is easy, since for any x≥1/2,\ng(x,0) =k\n2(1−x)+1\n4k·x2\n1−x≥2/radicalbigg\nk\n2(1−x)·1\n4k·x2\n1−x=2x√\n8≥1\n2√\n2= Ω(1).\n62To handle the s∗(x)≥0case, we note that\ninf\ns≥0g(x,s) =g(x,s∗(x)) =g(x,0)−k\n8[s∗(x)]2,\nsince the minimum of the quadratic function f(x) =ax2−bx+cwherea >0is given by\nc−b2\n4a=f(0)−a(x∗)2, achieved at x∗=b\n2a. We will prove that whenever s∗(x)≥0,\ng(x,0)≥k\n4[s∗(x)]2. (23)\nThis then implies\ninf\ns≥0g(x,s) =g(x,0)−k\n8[s∗(x)]2≥g(x,0)−g(x,0)/2 = Ω(1) ,\nwhere the last step follows from our previous argument for g(x,0) = Ω(1) .\nSince2−1/k−x−x/(1−x) =s∗(x)≥0, we have x+x/(1−x)≤2, which further implies\nx≤2−√\n2≤3/5. Then,g(x,0)can be lower bounded as follows:\ng(x,0) =k\n2(1−x)+1\n4k·x2\n1−x≥k\n2(1−x)≥k\n5.\nFurthermore, x∈[1/2,1)implies that\ns∗(x) = 2−1\nk−x−x\n1−x≤2−0−1\n2−1 =1\n2.\nTherefore, eq. (23) follows from\ng(x,0)≥k\n5>k\n4·(1/2)2≥k\n4[s∗(x)]2.\nThis finishes the proof for Case 3.\n63"    },    {        "title": "2402.15188v1.Parameter_Free_Algorithms_for_Performative_Regret_Minimization_under_Decision_Dependent_Distributions.pdf",        "content": "Parameter-Free Algorithms for Performative Regret\nMinimization under Decision-Dependent Distributions\nSungwoo Park1s.park@kaist.ac.kr\nJunyeop Kwon1junyeopk@kaist.ac.kr\nByeongnoh Kim2b-n.kim@samsung.com\nSuhyun Chae2suhyun.chae@samsung.com\nJeeyong Lee2jiyong.lee@samsung.com\nDabeen Lee1,†dabeenl@kaist.ac.kr\n1Department of Industrial and Systems Engineering, KAIST, Daejeon 34141, South Korea\n2Device Solutions Research, Samsung Electronics, Hwaseong, Gyeonggi 18448, South Korea\n†corresponding author\nAbstract\nThis paper studies performative risk minimization, a formulation of stochastic optimiza-\ntion under decision-dependent distributions. We consider the general case where the per-\nformative risk can be non-convex, for which we develop efficient parameter-free optimistic\noptimization-based methods. Our algorithms significantly improve upon the existing Lips-\nchitz bandit-based method in many aspects. In particular, our framework does not require\nknowledge about the sensitivity parameter of the distribution map and the Lipshitz con-\nstant of the loss function. This makes our framework practically favorable, together with\nthe efficient optimistic optimization-based tree-search mechanism. We provide experimen-\ntal results that demonstrate the numerical superiority of our algorithms over the existing\nmethod and other black-box optimistic optimization methods.\nKeywords: Decision-Dependent Distributions, Performative Risk Minimization, Opti-\nmistic Optimization, Black-Box Optimization, Stochastic Non-Convex Optimization\n1 Introduction\nIn the realm of stochastic optimization, where navigating uncertainty is paramount, dis-\ntributional shifts stand out as a significant challenge. Among the various sources of these\nshifts, one particularly intriguing phenomenon stems from feedback mechanisms intricately\nlinked to decision-making processes. This feedback loop alters the distribution that governs\nthe stochastic environment of the system, creating a dynamic landscape where decisions\nshape and are shaped by distributions. For example, the decisions made by a dynamic\nresource allocation algorithm for a renewable energy grid not only influence the immedi-\nate allocation of resources but also affect the underlying distribution of factors like energy\ndemand and supply. Classifiers, such as insurance underwriting systems, often promote a\nshift in behavior within the population to improve their labels. Predictions of stock prices\nwield significant influence over trading decisions. Moreover, election predictions have the\npotential to shape and influence voter behavior, which in turn can impact voting results.\n©2022 Author One and Author Two.\nLicense: CC-BY 4.0, see https://creativecommons.org/licenses/by/4.0/ .arXiv:2402.15188v1  [cs.LG]  23 Feb 2024Parameter-Free Performative Regret Minimization\nDecision-making processes under such phenomena can be formulated as stochastic opti-\nmization under decision-dependent distributions . Perdomo et al. (2020) proposed the notion\nof the distribution map to consider decision-dependent distributions for stochastic optimiza-\ntion models. That is, the distribution D(θ) of the parameter zcapturing the stochastic\nenvironment depends on the decision θ. Here, θmay encode the resource allocation deci-\nsion for a renewable energy grid and the election prediction, in which case zcorresponds\nto the energy demand and the voting results, respectively. For machine learning, we can\nassociate θwith predictive models and zwith data. Then the objective is to minimize the\nperformative risk under a loss function f, defined as\nPR(θ) :=Ez∼D(θ)[f(θ, z)].\nThe expression performative comes from the term performative prediction (Perdomo et al.,\n2020), which implies the phenomenon where predictions influence the outcomes. The goal\nof this paper is to design an efficient algorithmic framework for minimizing the performative\nrisk which models stochastic optimization under decision-dependent distributions.\n1.1 Existing Methods for Performative Risk Minimization\nUnlike the standard stochastic optimization problem, the decision θmay affect the under-\nlying distribution D(θ). Hence, a natural starting point to minimize the performative risk\nis to consider the following iterative algorithm, referred to as repeated risk minimization\n(RRM). Given an initial solution θ0∈Θ where Θ is the domain, we apply\nθt+1∈argmin\nθ∈ΘEz∼D(θt)[f(θ, z)] (RRM)\nfort≥0. Here, computing the next iterate θt+1requires solving a stochastic optimization\ninstance where the underlying distribution is fixed with D(θt). Another approach is a\ngradient-based method such as\nθt+1=θt−ηtEz∼D(θt)[∇f(θt, z)] (RGD)\nwhere ηtis a step size, and we refer to this procedure as repeated gradient descent (RGD).\nNote that running RGD, as well as RRM, is based on access to the distribution D(θt) for\nevery iteration t≥0, which may not be feasible in practice. A more sample-efficient method\nis to apply the standard stochastic gradient descent (SGD) update, given by\nθt+1=θt−ηt∇f(θt, zt) where zt∼ D(θt). (SGD)\nDrusvyatskiy and Xiao (2023) analyzed variants of SGD such as stochastic proximal gradi-\nent, proximal point, clipped gradient, and accelerated gradient methods.\nConvergence of these iterative methods has been established; (Perdomo et al., 2020) for\nRRM, (Perdomo et al., 2020; Mendler-D¨ unner et al., 2020) for RGD, and (Mendler-D¨ unner\net al., 2020) for SGD. They showed convergence to a performatively stable solution, under\nsome strong convexity and smoothness assumptions on the loss function f. Here, we say\nthat a solution θPSis performatively stable if it satisfies\nθPS∈argmin\nθ∈ΘEz∼D(θPS)[f(θ, z)]\n2Parameter-Free Performative Regret Minimization\nIn particular, θPSis a fixed point of RRM. However, let alone the validity of the structural\nassumptions on the loss functions, the performatively stable solution θPSis in general not\na minimizer of the performative risk (Perdomo et al., 2020; Miller et al., 2021). Let θPO\ndenote a minimizer of the performative risk, i.e.,\nθPO∈argmin\nθ∈ΘEz∼D(θ)[f(θ, z)].\nIt turns out that PR( θPS) can be arbitrarily large compared to PR( θPO) (Miller et al.,\n2021).\nDerivative-free zeroth-order optimization methods have been proposed to minimize the\nperformative risk directly (Izzo et al., 2021; Miller et al., 2021; Izzo et al., 2022; Ray et al.,\n2022). For the derivative-free methods to minimize the performative risk PR( θ), the require-\nment, however, is that PR( θ) is a convex function of the decision θ. Miller et al. (2021)\nprovided some sufficient conditions on the distribution map to guarantee the convexity of\nthe performative risk. They argued that if the distribution map satisfies a certain stochas-\ntic dominance condition, which is related to stochastic orders (Shaked and Shanthikumar,\n2007), then convexity of the loss function fleads to a convex performative risk. Neverthe-\nless, as noted by Perdomo et al. (2020), the performative risk is non-convex in general even\nif the loss function is convex.\nTo tackle the general case of non-convex performative risk, Jagadeesan et al. (2022) de-\nveloped a bandit optimization-based algorithm. The problem of minimizing the performa-\ntive risk is indeed a bandit optimization problem because until the decision θis deployed it is\ndifficult to estimate the distribution D(θ) and thus the performative risk PR( θ). That said,\nthe framework of Jagadeesan et al. (2022) is inspired by the zooming algorithm ( Zooming )\nfor Lipschitz bandits due to Kleinberg et al. (2008). The core idea is to adaptively discretize\nthe solution space Θ thereby narrowing down the location of the optimal decision θPO. In\nfact, if the loss function fisLθ-Lipschitz continuous in θandLz-Lipschitz continuous in z,\ntheε-sensitivity of the distribution map (defined formally in Section 3) implies that PR( θ)\nis (Lθ+Lzε)-Lipschitz continuous (Jagadeesan et al., 2022). Here, the ε-sensitivity mea-\nsures how much the distributions D(θ) andD(θ′) can differ for two distinct decisions θand\nθ′. Then applying Zooming directly on PR( θ) would guarantee a sublinear regret.\nAlthough this lays down a good starting point, direct application of Zooming fails to\nutilize the fact that the feedback obtained after deploying decision θisD(θ), based on which\nthe learner can evaluate PR( θ) but also infer the distribution D(θ′) of other solutions θ′\nusing the ε-sensitivity. Jagadeesan et al. (2022) referred to this as performative feedback .\nBuilding on this idea, they developed a variant of Zooming , and they provided a regret\nupper bound that is parameterized by not Lθ+LzεbutLzε. Here, note that Lzεvanishes\nasε→0 while Lθ+Lzεdoes not. Moreover, Lzεdoes not depend on Lθ, so the algorithm\nworks even when Lθis not bounded. The two main components of their algorithm are\nadaptive discretization and sequential elimination based on performative confidence bounds\nwhich we explain in Section 3.\nThe algorithm of Jagadeesan et al. (2022) solves performative risk minimization, but\nseveral issues hinder its practical implementation. First, to implement the adaptive dis-\ncretization procedure, we need to know the Rademacher complexity C∗(f) of learning the\nperformative risk PR( θ) under the loss function fbased on data samples from distribution\n3Parameter-Free Performative Regret Minimization\nD(θ). The Rademacher complexity parameter C∗(f) can be very high depending on the\nstructure of f. Second, to build a performative confidence bound, we need the Lipschitz\nconstant Lzand the sensitivity parameter ε. One may argue that there is a way of esti-\nmating the Lipschitz constant Lzfor a known class of loss functions, but the sensitivity\nparameter εdetermines the global landscape of the distribution map, which means that it\nwould be difficult to measure εin advance. Third, one iteration of the algorithm is compu-\ntationally expensive. This is because each time a decision θis deployed, we need to compute\nthe performative confidence bound for every solution θ′∈Θ remaining in the search space.\nSuch an issue is inherent in Lipschitz bandit-based methods. Although Jagadeesan et al.\n(2022) did not demonstrate an implementation of their algorithm, our numerical results in\nSection 6 show that the algorithm is not efficient and incurs a high regret in practice.\nThe aforementioned limitations of the existing method due to Jagadeesan et al. (2022)\nfor performative risk minimization motivate the following question.\nCan we design a practical algorithm for performative risk minimization that\nrelies on minimal knowledge about the problem parameters?\nIn this paper, we devise efficient parameter-free algorithms for performative risk minimiza-\ntion. We not only demonstrate strong theoretical performance guarantees but also show\nexperimental results to highlight their numerical effectiveness.\n1.2 Our Contributions\nAs in (Jagadeesan et al., 2022), we study the problem of minimizing the performative risk\nwith performative feedback. The algorithm of Jagadeesan et al. (2022) is an adaptation\nofZooming by Kleinberg et al. (2008), and as a result, it requires knowledge of problem\nparameters such as the Rademacher complexity C∗(f), the Lipschitz constant Lz, and the\nsensitivity parameter ε.\nOur main contribution is to design practical algorithms that do not assume knowledge\nof the problem parameters. To develop such parameter-free algorithms, we build upon the\nidea of optimistic optimization methods that may adapt to unknown smoothness of the\nobjective function. Here, parameter-free optimistic optimization methods originate from\nthe simultaneous optimistic optimization ( SOO) algorithm (Munos, 2011) and the stochastic\nextension of SOO(StoSOO ) algorithm (Valko et al., 2013), and they are devised to optimize\nblack-box objective functions that are possibly non-convex.\nOur algorithms are inspired by two more recent optimistic optimization-based parameter-\nfree methods due to Bartlett et al. (2019), SequOOL for the deterministic evaluation case\nand StroquOOL for the noisy case. We start by considering the conceptual setting where\nthe distribution D(θ) associated with the deployed decision θcan be fully observed. We\ncall this case the full-feedback setting. Next, we study the more practically relevant setting\nwhere we obtain a few samples from D(θ) after deploying decision θ, and we refer to this\ncase as the data-driven setting. We develop our algorithms for the full-feedback setting and\nthe data-driven setting based on SequOOL andStroquOOL , respectively.\nTo highlight our results early in this paper, let us provide an informal summary of our\nmain theorems. The following states a performance guarantee for the full-feedback case.\n4Parameter-Free Performative Regret Minimization\nTheorem 1 (Full-Feedback Case, Informal) Suppose that the distribution map satis-\nfies the ε-sensitivity condition and the loss function f(θ, z)isLz-Lipschitz continuous in z\nfor any θ. Let ddenote the Lzε-near-optimality dimension. For the full-feedback setting,\nAlgorithm 1 after Tdecision deployments for a sufficiently large Tfinds a solution θwith\nPR(θ)−PR(θPO) =\n\nLzε·2O\u0010\n−T\nlogT\u0011\n,ifd= 0,\n˜O\u0010\nLzε·T−1\nd\u0011\n,ifd >0.\nHere, the optimality gap bounds hide dependence on the ambient dimension Dof the de-\ncision domain Θ. The notion of near-optimality dimension was first introduced by Bubeck\net al. (2011a), and they argued that the near-optimality dimension and the zooming dimen-\nsion due to Kleinberg et al. (2008) are closely related. In this paper, we use a more refined\ndefinition of the near-optimality dimension due to Grill et al. (2015).\nNote that the optimality gap bounds as well as the near-optimality dimension depend on\nparameters Lzandεbut not on Lθ. In fact, direct application of SequOOL to minimize the\nperformative risk PR( θ) would result in dependence on Lθ+Lzεas the Lipschitz constant of\nPR(θ) isLθ+Lzε(Jagadeesan et al., 2022). More precisely, we would need the ( Lθ+Lzε)-\nnear-optimality dimension, and the resulting bounds would be ( Lθ+Lzε)2O(−T/logT)and\n˜O((Lθ+Lzε)T−1/d). The important distinction is that our optimality gap bounds vanish as ε\nbecomes arbitrarily small, which setting corresponds to the standard stochastic optimization\nproblem with decision-agnostic distributions. Moreover, the Lzε-near-optimality dimension\nis always less than or equal to the ( Lθ+Lzε)-near-optimality dimension. Another aspect\nto highlight in Theorem 6 is that when d= 0, we show that the optimality gap decays at\nan exponentially fast rate, which was not discovered by Jagadeesan et al. (2022).\nNext, we state our performance guarantee for the data-driven setting where we receive a\nfinite number of samples from D(θ) after deploying decision θ. The optimality gap bounds\non Algorithm 2 hide dependence on the dimension D, the number of samples received after\neach decision deployment, the Rademacher complexity of learning the performative risk.\nTheorem 2 (Data-Driven Case, Informal) Assume the same conditions on the distri-\nbution map and the loss function. Let ddenote the Lzε-near-optimality dimension. For\nthe data-driven setting, Algorithm 2 after Tdecision deployments for a sufficiently large T\nfinds a solution θsuch that with high probability,\nPR(θ)−PR(θPO) =\n\nLzε·2O\u0010\n−T\nlogT\u0011\n, low-noise regime with d= 0,\n˜O\u0010\nLzε·T−1\nd\u0011\n, low-noise regime with d >0,\n˜O\u0010\nT−1\n2+ (Lzε)d\nd+2·T−1\nd+2\u0011\n,high-noise regime.\nThelow-noise andhigh-noise regimes are defined in Section 5. In particular, for the high-\nnoise regime, the bound is ˜O(T−1/2+LzεT−1/d) which incurs the additional term T−1/2\ndue to errors in estimating the performative risk through noisy feedback. In particular, the\ncase of ε= 0 is under the high-noise regime, in which case the bound reduces to ˜O(T−1/2).\nLastly, we test the numerical performance of our framework on instances in which the\nassociated performative risk is non-convex. The experimental results show that our al-\ngorithms outperform the existing methods that include not only the sequential zooming\n5Parameter-Free Performative Regret Minimization\nalgorithm of Jagadeesan et al. (2022) but also SOO(Munos, 2011), StoSOO (Valko et al.,\n2013), SequOOL , and StroquOOL (Bartlett et al., 2019) applied directly to the performative\nrisk as a black-box function without utilizing the performative feedback.\n2 Related Work\nThis section summarizes prior work on performative prediction and optimistic optimization.\n2.1 Performative Prediction and Performative Risk Minimization\nPrevious work on performative prediction has mainly focused on first-order and zeroth-\norder gradient-based optimization methods (Perdomo et al., 2020; Mendler-D¨ unner et al.,\n2020; Drusvyatskiy and Xiao, 2023; Brown et al., 2022; Miller et al., 2021; Izzo et al., 2021;\nMaheshwari et al., 2022; Li and Wai, 2022; Ray et al., 2022; Dong et al., 2023; Izzo et al.,\n2022). Convergence of these gradient-based methods to performatively stable solutions is\nstudied, and Miller et al. (2021); Izzo et al. (2021) discovered some convexity conditions\nunder which some gradient-based methods converge to a performative optimal solution. Al-\nthough a performatively stable solution provides a good proxy for a performatively optimal\nsolution, its performance can be arbitrarily worse than the optimum. Moreover, in general,\nthe performative risk is non-convex and does not satisfy the convexity conditions. For the\ngeneral case, Jagadeesan et al. (2022) developed a variant of Zooming for minimizing the\nperformative risk. Mofakhami et al. (2023) studied RRM for training non-convex neural\nnetworks, but they considered a different setting in terms of defining the ε-sensitivity of\nthe distribution map. For a comprehensive survey on performative prediction, we refer the\nreader to Hardt and Mendler-D¨ unner (2023) and references therein.\nOne of the most closely related application domains is strategic classification (Dalvi\net al., 2004; Br¨ uckner et al., 2012; Hardt et al., 2016), which models a game between\nan institution deploying a classifier and an agent who adapts its features to increase its\nlikelihood of being positively labeled. Recent work in this area includes (Dong et al., 2018;\nChen et al., 2020; Milli et al., 2019; Bechavod et al., 2021; Zrnic et al., 2021).\n2.2 Optimistic Optimization\nBlack-box optimization andcontinuum-armed bandits aim to optimize an objective function\nunder minimal knowledge about the function. Some early work provides algorithms that\nassume some weak or local smoothness conditions around a global optimal solution, such\nasZooming (Kleinberg et al., 2008), HOO(Bubeck et al., 2011a), DOO(Munos, 2011), HCT(azar\net al., 2014). Here, HOO,DOO, and HCTare optimistic optimization-based methods, which\nmeans that these algorithms use some optimistic estimates of the black-box objective func-\ntion when running a global search of the solution space. However, Zooming ,HOO,DOO, and\nHCTrequire the knowledge of the local smoothness parameter. Then Munos (2011) presented\nSOOthat works even when the local smoothness parameter is unknown. Valko et al. (2013)\ndeveloped StoSOO which extends SOOfor the case of stochastic function evaluation, but its\nconvergence guarantee holds for the limited case of the near-optimality dimension being 0.\nPOOdue to Grill et al. (2015) and GPO,PCTdeveloped by Shang et al. (2019) work for more\ngeneral families of objective functions. Later, Bartlett et al. (2019) presented SequOOL for\n6Parameter-Free Performative Regret Minimization\nthe deterministic function evaluation case and StroquOOL for the stochastic case, which\nwork for general families of functions and exhibit state-of-the-art numerical performance.\nRecently, Li et al. (2023b) provided VHCT, which does not require the budget on the\nnumber of decision deployments beforehand but needs the knowledge of the smoothness\nparameter. There exist more algorithms that work under more specific assumptions on\nsmoothness. For example, DiRect (Jones et al., 1993) and methods for continuum-armed\nbandits due to (Slivkins, 2011; Bubeck et al., 2011b; Malherbe and Vayatis, 2017) can take\nLipschitz-continuous objective functions.\n3 Preliminaries: Optimization with Performative Feedback\nIn this section, we introduce the basics of performative prediction. Then we explain how\nto make use of performative feedback for performative risk minimization as established\nby Jagadeesan et al. (2022). In addition, we elaborate briefly on some limitations of the\nperformative confidence bound-based zooming algorithm by Jagadeesan et al. (2022).\nAs mentioned in the introduction, the ε-sensitivity measures how much the distribution\nD(θ) can change with changes in decision θ. Formally, we assume that the distribution map\nsatisfies the following. Recall that Θ denotes the decision domain.\nAssumption 1 ( ε-sensitivity) A distribution map D(·)isε-sensitive with α > 0if for\nanyθ, θ′∈Θwe have\nW(D(θ),D(θ′))≤ε∥θ−θ′∥α,\nwhere Wdenotes the 1-Wasserstein distance.\nThe original definition due to Perdomo et al. (2020) considers the case α= 1, while\nour framework allows arbitrary positive values of α. We remark that our framework is\nparameter-free in that we do not require knowledge of the parameters εandαin advance.\nIn theory, as we build upon optimistic optimization methods, we may take any semi-metric\nℓ, satisfying ℓ(θ, θ′) =ℓ(θ′, θ) and ℓ(θ, θ′) = 0 if and only if θ=θ′forθ, θ′∈Θ. That\nbeing said, we may run our algorithms regardless of the sensitivity structure of the distri-\nbution map, but we derive the theoretical performance guarantees based on the sensitivity\nstructure given in Assumption 1.\nNext, we define the notion of performative feedback used to infer the distribution D(θ)\nas well as the performative risk PR( θ) after deploying decision θ.\nAssumption 2 (performative feedback) Deploying decision θonce, we receive feedback\nabout the distribution as follows.\n•(Full-Feedback Setting) distribution D(θ)itself.\n•(Data-Driven Setting) m0i.i.d. samples z(1)\nθ, . . . , z(m0)\nθfrom distribution D(θ).\nFor the data-driven setting, we may deploy the same decision θmultiple times, say n.\nThen we may construct the empirical distribution bD(θ) with nm0i.i.d. samples from D(θ).\nUsing the performative feedback for θ, which provides D(θ) orbD(θ), we may compute the\nperformative risk PR( θ) or its empirical estimate\ncPR(θ) =Ez∼bD(θ)[f(θ, z)].\n7Parameter-Free Performative Regret Minimization\nMoreover, based on the performative feedback for θ, we may infer the performative risk of\nother decisions θ′. To be specific, we use the notion of decoupled performative risk (Perdomo\net al., 2020) defined as follows.\nDPR( θ, θ′) =Ez∼D(θ)\u0002\nf(θ′, z)\u0003\nand[DPR( θ, θ′) =Ez∼bD(θ)\u0002\nf(θ′, z)\u0003\nfor any θ, θ′∈Θ where DPR( θ, θ′) is the decoupled performative risk of decision θ′under\ndistribution D(θ) and[DPR( θ, θ′) is its empirical estimate. The decoupled performative risk\noffers a good approximation of the performative risk, which we elaborate on below.\nAssumption 3 There is some Lz>0such that f(θ,·)for any fixed θ∈ΘisLz-Lipschitz\ncontinuous.\nUnder Assumptions 1 and 3, the Kantorovich-Rubinstein duality theorem (Kantorovich and\nRubinstein, 1958; Villani, 2008) implies the following statement (Jagadeesan et al., 2022).\nLemma 3 Under Assumptions 1 and 3, for θ, θ′∈Θ,\n\f\fPR(θ′)−DPR( θ, θ′)\f\f≤Lzε∥θ−θ′∥α.\nTherefore, as long as decisions θandθ′are close, the decoupled performative risk DPR( θ, θ′)\ndeduced based on the performative feedback D(θ) for θwould be a good proxy for the\nperformative risk PR( θ′) of decision θ′. By Lemma 3,\nDPR( θ, θ′)−Lzε∥θ−θ′∥α≤PR(θ′)≤DPR( θ, θ′) +Lzε∥θ−θ′∥α\nis a valid confidence interval for the performative risk of θ′∈Θ. Note that the confidence\ninterval is tighter than the interval\nPR(θ)−Lθ∥θ−θ′∥ −Lzε∥θ−θ′∥α≤PR(θ′)≤PR(θ) +Lθ∥θ−θ′∥+Lzε∥θ−θ′∥α\nwhich holds under the assumption that f(·, z) isLθ-Lipschitz continuous for any fixed z∈ Z\nwhere Zdenotes the domain of the stochastic parameter z(Jagadeesan et al., 2022). Here,\nthe latter interval can be deduced by a black-box evaluation of PR( θ) while we derived the\nformer using the performative feedback.\nWhen we have a set Sof multiple decisions θwith known D(θ), then for any θ′∈Θ,\nmax\nθ∈S\b\nDPR( θ, θ′)−Lzε∥θ−θ′∥α\t\n≤PR(θ′)≤min\nθ∈S\b\nDPR( θ, θ′) +Lzε∥θ−θ′∥α\t\nis also valid, and we refer to the bounds as performative confidence bounds . The zooming\nalgorithm of Jagadeesan et al. (2022) updates the performative confidence bounds whenever\na new decision is deployed, based on which suboptimal decisions are sequentially deleted.\nThis approach, however, has two key limitations. First, we need to know Lzandεto derive\nperformative confidence bounds. Second, the computational complexity of computing the\nperformative confidence bounds associated with SisO(|S| · | Θ|), which is an expensive\nper-time cost. Later, our experimental results reveal that the algorithm turns out to be not\nnumerically efficient.\n8Parameter-Free Performative Regret Minimization\n4 Optimistic Optimization-Based Parameter-Free Framework\nMotivated by the challenges of the existing method, our goal is to design an efficient\nparameter-free framework for performative risk minimization. For simple presentation, we\nassume that the loss function and the decision domain.\nAssumption 4 (bounded domain and objective) Θ⊆[0,1]Dwhere Dis the ambient\ndimension. Moreover, f(θ, z)∈[0,1]for all θ∈Θandz∈ Z.\nLet us explain the basic setup of our optimistic optimization-based framework as follows.\nWe assume that a hierarchical partitioning (Bubeck et al., 2011a; Munos, 2011) of the\ndecision domain is given. Basically, a hierarchical partitioning Pof Θ is given by {Ph,i:\n0≤h≤hmax, i∈[Ih]}where hmaxis the deepest depth, Ihis the width at depth hwith\nI1= 1,{Ph,i:i∈[Ih]}is a partition of Θ, and Ph,iis partitioned into {Ph+1,j:j∈J}\nfor some J⊆[Ih+1]. Throughout the paper, we refer to Ph,ias a cell of depth h. The\nhierarchical partitioning naturally corresponds to a tree structure. When a cell Ph,iis\npartitioned into {Ph+1,j:j∈J},Ph,iis the parent cell of its child cells Ph+1,jforj∈J.\nMoreover, we assume that the partition at each depth level consists of cells of uniform size.\nAssumption 5 (uniform partition) sup{∥θ−θ′∥:θ, θ′∈ P h,i} ≤√\nD2−hfor0≤h≤\nhmaxand1≤i≤Ih. Moreover, Ih≤2Dh.\nThere exists a hierarchical partitioning that satisfies Assumption 5 as we may take 2D\nsubsets of box [0 ,1]Dby dividing each coordinate direction equally and repeat the process\nfor each subset.\nOur framework adopts the tree-search mechanism as done for many optimistic optimiza-\ntion algorithms such as SOO,StoSOO ,SequOOL , and StroquOOL . These algorithms select an\narbitrary decision θh,ifor each cell Ph,ias its representative in advance, and evaluating\ncellPh,imeans deploying decision θh,i. Iff(·, z) isLθ-Lipschitz continuous for any z, then\nAssumption 5 implies that for any θh,i∈ Ph,i,\nPR(θh,i)−inf\nθ∈Ph,iPR(θ)≤Lθ√\nD2−h+LzεDα/22−αh.\nHowever, this bound may be too weak for our setting because the term Lθ√\nD2−hcan be\nmuch larger than the other term LzεDα/22−αhwhen the sensitivity parameter εis small. In\ncontrast, based on performative feedback, we use a specific rule for choosing a representative\ndecision θh,igiven by\nθh,i∈argmin\nθ∈Ph,iDPR( θh−1,j, θ).\nHere, θh−1,jis the representative of the parent cell Ph−1,jof depth h−1 containing Ph,i.\nBased on the performative feedback about decision θh−1,j, we may compute the decoupled\nperformative risk DPR( θh−1,j, θ). Note that the procedure of choosing θh,iis much cheaper\nthan computing performative confidence bounds because the former requires evaluating\ndecisions in a local cell Ph,iwhereas the latter considers the entire domain Θ.\nExplaining the important components of our framework, we present Algorithm 1 for the\nfull-feedback setting. logTdenotes the T-th harmonic number, that is, logT=PT\nt=11/t.\n9Parameter-Free Performative Regret Minimization\nAlgorithm 1 Deterministic Optimistic Optimization with Performative Feedback (DOOP)\nInput: test budget T, hierarchical partitioning P,hmax=\u0004\nT/2DlogT\u0005\nSetL0← {P 0,1}and initialize L1← ∅\nTake a solution θ0,1∈ P0,1\nRun Open( P0,1)\nforh= 1tohmaxdo\nInitialize Lh+1← ∅\nTake⌊hmax/h⌋cells with the ⌊hmax/h⌋smallest values in {PR(θh,i) :Ph,i∈ Lh}\nforeachPh,iof the ⌊hmax/h⌋cellsdo\nRun Open( Ph,i)\nend for\nend for\nTake ( h, i)∈argmin(h,i){PR(θh,i) :h∈[0 :hmax+ 1],Ph,i∈ Lh}\nReturn θT←θh,i\nSubroutine Open( Ph,i)\nInput: cellPh,i\nforeach child cell Ph+1,jofPh,ido\nTake a solution θh+1,j∈argminθ∈Ph+1,jDPR( θh,i, θ) and deploy it\nReceive D(θh+1,j) to compute DPR( θh+1,j, θ) for θ∈ Ph+1,j\nUpdate Lh+1← L h+1∪ {P h+1,j}\nend for\nWe use notation [ a:b] to denote the set {a, a+ 1, . . . , b}for integers a, bwith a < b . We\nadopt SequOOL by Bartlett et al. (2019) as the backbone of Algorithm 1. As SequOOL , our\nalgorithm explores the depth sequentially by testing multiple cells at the same depth level\nbefore going down to the next level. As going deeper, fewer cells are tested, thus focusing on\na narrower area. This can be viewed as an exploration-exploitation procedure. Moreover,\nopening a cell Ph,iof depth hmeans considering its child cells {Ph+1,j:j∈J}ath+ 1 by\ndeploying their representative decisions θh+1,j. In Algorithm 1, Lhdenotes the set of cells\nPh,iof depth hwhose representative decision θh,ihas been deployed. Then L0, . . . ,Lhmax+1\nnaturally form a tree whose vertices correspond to cells.\nTo analyze the performance of Algorithm 1, we use the notion of near-optimality dimen-\nsion, as mentioned in the introduction. Its definition has been refined, and we adopt the\nversion considered by Grill et al. (2015); Bartlett et al. (2019), that is, the near-optimality\ndimension associated with a given hierarchical partitioning.\nDefinition 4 (near-optimality dimension) For any ν >0,C≥1, and ρ∈(0,1), the\n(ν, ρ, C )-near-optimality dimension , denoted d(ν, ρ, C ), offwith respect to the hierarchical\npartitioning Pis defined as\nd(ν, ρ, C ) = infn\nd∈R+:Nh(6νρh)≤Cρ−dh∀h≥0o\nwhere Nh(ϵ)is the number of cells Ph,iof depth hsuch that infθ∈Ph,iPR(θ)≤PR(θPO) +ϵ.\n10Parameter-Free Performative Regret Minimization\nIn particular, we will use the ((2√\nD)αLzε,2−α,1)-near-optimality dimension. It gets large\nas the sensitivity parameter εincreases. Note that by Assumption 5, the number of cells\nof depth his at most 2Dh. This gives rise to a global upper bound on d(ν,2−α,1) that\nholds for any ν >0, that is, d(ν,2−α,1)≤D/α. Hence, when εis small and PR( ·) has\nsufficient curvature around the performative-optimal solution θPO, the ((2√\nD)αLzε,2−α,1)-\nnear-optimality dimension is supposed to be much smaller than D/α. When the ambient\ndimension Dis fixed, one may regard the factor (2√\nD)αas a fixed constant and hide it by\nreplacing Nh(6νρh) with Nh(6(2√\nD)ανρh) in the definition of d(ν, ρ, C ).\nThe following lemma is the key to analyzing the performance of Algorithm 1. Follow-\ning Bartlett et al. (2019), we define ⊥has the depth of the deepest cell containing θPO\nopened until Algorithm 1 finishes opening cells of depth h.\nLemma 5 Letddenote the ((2√\nD)αLzε,2−α,1)-near-optimality dimension. Then θTre-\nturned by Algorithm 1 satisfies the following bound.\nPR(θT)−PR(θPO)≤2(2√\nD)αLzε2−α(⊥hmax+1).\nNote that the bound on the optimality gap scales with Lzε, not Lθ+Lzε. Based on this, we\nprove the following theorem which provides a theoretical guarantee on the performance of\nAlgorithm 1. As in the analysis of SequOOL by Bartlett et al. (2019), we use the Lambert W\nfunction . The function is to describe the solution hto the equation x=h·ehash=W(x).\nTheorem 6 Letddenote the ((2√\nD)αLzε,2−α,1)-near-optimality dimension. For the full-\nfeedback setting, Algorithm 1 after Tdecision deployments finds a solution θwith\nPR(θ)−PR(θPO)≤(\n2(2√\nD)αLzε2−αhmax, ifd= 0,\n2(2√\nD)αLzεe−(1/d)W(hmaxαdlog 2),ifd >0\nwhere hmax=\u0004\nT/2DlogT\u0005\n. Moreover, if d >0andhmaxαdlog 2≥e, then θsatisfies\nPR(θ)−PR(θPO)≤2(2√\nD)αLzε\u0012hmaxαdlog 2\nlog(hmaxαdlog 2)\u0013−1\nd\n.\nAshmax= Ω(T/logT) and hmax=O(T/logT), we deduce from Theorem 6 with α= 1 the\noptimality gap bounds in Theorem 1. We provide the proof of the theorem in Appendix A.\nWe follow the proof outline of Bartlett et al. (2019) for SequOOL , but we need to adapt\nthe analysis to our specific design of the procedure of opening a cell based on performative\nfeedback.\nThe last remark is that the optimality gap PR( θ)−PR(θPO) is the simple regret whereas\nJagadeesan et al. (2022) studies the cumulative regret incurred by their algorithm. Although\nTheorem 6 characterizes an upper bound on the simple regret only, we later report our\nnumerical results on the cumulative regret of Algorithm 1.\n5 Data-Driven Setting\nFor the data-driven setting, we receive a few data samples as performative feedback, which\nprovides an estimation of the distribution. Through the data samples, we obtain the em-\npirical distribution bD(θ) after deploying decision θ. Then we may compute the estimator\n11Parameter-Free Performative Regret Minimization\n[DPR( θ, θ′) of the decoupled performative risk DPR( θ, θ′) for other decisions θ′. Here, con-\ntrolling the estimation error |DPR( θ, θ′)−[DPR( θ, θ′)|is crucial to achieve a better perfor-\nmance. To reduce the error, we evaluate a cell multiple times to obtain enough data samples\nfrom the distribution of the representative decision. Following StroquOOL by Bartlett et al.\n(2019), Algorithm 2 implements this idea, extending Algorithm 1 to the data-driven setting.\nAs Algorithm 1, Algorithm 2 takes fewer cells at deeper depth levels, thereby imple-\nmenting the exploration-exploitation trade-off principle. On top of this, the algorithm keeps\ntrack of the number of times each cell has been evaluated. When exploring cells at a certain\ndepth, the algorithm distributes the evaluation budget over cells based on how many times\nthey have been evaluated. Among the cells that have been evaluated many times, we focus\non a few that have a low performative risk. For the cells that have not been evaluated many\ntimes, we distribute the evaluation budget over more cells, among which we encourage ex-\nploration. Furthermore, as in StroquOOL , Algorithm 2 has the cross-validation phase, in\nwhich we focus on cells whose representative decision has a low estimated performative risk.\nLetnopen\nh,idenote the number of times cell Ph,iis opened, and let ndeploy\nh,idenote the\nnumber of times its representative decision θh,iis deployed. Note that if Ph+1,jis a child cell\nofPh,i, then we have nopen\nh,i=ndeploy\nh+1,j. Recall that [ a:b] denotes the set {a, a+ 1, . . . , b}for\nintegers a, bwith a < b . For a positive integer a, let [ a] denote the set {1, . . . , a }. Moreover,\nas in Algorithm 1, L0, . . . ,Lhmax+1represent the tree search structure of Algorithm 2.\nIn what follows, we analyze the performance of Algorithm 2. Note that we compute\n[DPR( θ, θ′) for many pairs of θandθ′, and at the same time, we need the estimation error\n|DPR( θ, θ′)−[DPR( θ, θ′)|uniformly bounded for all pairs. To achieve this, we introduce the\nRademacher complexity associated with the loss function f.\nDefinition 7 (Rademacher complexity) Given an objective function f, the Rademacher\ncomplexity C∗(f)is defined as\nC∗(f) = sup\nθ∈Θsup\nn∈N√n·Eϵ,zθ\nsup\nθ′∈Θ\f\f\f\f\f\f1\nnnX\nj=1ϵjf(θ′, zθ\nj)\f\f\f\f\f\f\n,\nwhere ϵj∼Rademacher andzθ\nj∼ D(θ)forj∈[n], which are all independent of each other.\nGiven the Rademacher complexity of the loss function, we may provide a uniform upper\nbound on the estimation error. Let us define the clean event under which the estimation\nerror|DPR( θ, θ′)−[DPR( θ, θ′)|is uniformly bounded over all pairs.\nDefinition 8 (Clean event) We define the clean event, denoted Eclean ,δ, as the event that\nsup\nθ∈Ph,i\f\f\f[DPR( θh,i, θ)−DPR( θh,i, θ)\f\f\f≤2C∗(f) + 2p\nlog(T/δ)q\nndeploy\nh,im0∀Ph,i∈ Lh,∀h∈[hmax+ 1]\nand\f\f\fcPR(θT(p))−PR(θT(p))\f\f\f≤2C∗(f) + 2p\nlog(T/δ)√hmaxm0∀p∈[0 :pmax].\nWe may prove that the clean event holds with high probability, parameterized by δ.\n12Parameter-Free Performative Regret Minimization\nAlgorithm 2 Stochastic Optimistic Optimization with Performative Feedback (SOOP)\nInput: test budget T, hierarchical partitioning P,\nhmax=\u0016T\n2D+1(log2T+ 1)2\u0017\n, p max=⌊log2hmax⌋\n◀Initialization Phase ▶\nSetL0← {P 0,1}and initialize L1← ∅\nTake a solution θ0,1∈ P0,1, deploy it hmaxtimes, and set ndeploy\n0,1←hmax\n◀Exploration Phase ▶\nRun Open( P0,1, hmax)\nforh= 1tohmaxdo\nInitialize Lh+1← ∅\nforp=⌊log2(hmax/h)⌋down to 0do\nTake ⌊hmax/h2p⌋cells that correspond to the ⌊hmax/h2p⌋smallest values inn\ncPR(θh,i) :Ph,i∈ Lh, nopen\nh,i= 0, ndeploy\nh,i≥2po\nforeachPh,iof the ⌊hmax/h2p⌋cellsdo\nRun Open( Ph,i,2p)\nend for\nend for\nend for\n◀Cross-validation Phase ▶\nforp= 0topmaxdo\nTake ( h, i)∈argmin(h,i)n\ncPR(θh,i) :h∈[0 :hmax+ 1],Ph,i∈ Lh, ndeploy\nh,i≥2po\nSetθT(p)←θh,i\nDeploy hmaxtimes solution θT(p) to form cPR(θT(p))\nend for\nReturn θT←θT(p) with p∈argminp∈[0:pmax]cPR(θT(p))\nSubroutine Open( Ph,i, n)\nInput: cellPh,i, number n\nSetnopen\nh,i←n\nforeach child cell Ph+1,jofPh,ido\nTake θh+1,j∈argminθ∈Ph+1,j[DPR( θh,i, θ), deploy it ntimes, and set ndeploy\nh+1,j←n\nForm[DPR( θh+1,j, θ) for θ∈ Ph+1,j\nInitialize nopen\nh+1,j←0\nUpdate Lh+1← L h+1∪ {P h+1,j}\nend for\nLemma 9 The clean event holds with probability at least 1−δ, i.e.,P[Eclean ,δ]≥1−δ.\n13Parameter-Free Performative Regret Minimization\nNext, we present the key lemma for our analysis. Following Bartlett et al. (2019), we\ndefine ⊥h,pas the depth of the deepest cell containing the performative optimal solution θPO\nopened for at least 2ptimes until Algorithm 2 finishes opening cells of depth h.\nLemma 10 Assume that the clean event Eclean ,δholds for some δ∈(0,1), and let ddenote\nthe((2√\nD)αLzε,2−α,1)-near-optimality dimension d((2√\nD)αLzε,2−α,1). Then for any\np∈[0 :pmax], the following bound on the regret holds.\nPR(θT)−PR(θPO)\n≤2(2√\nD)αLzε2−α(⊥hmax,p+1)+8C∗(f) + 8p\nlog(T/δ)√2pm0+4C∗(f) + 4p\nlog(T/δ)√hmaxm0.\nTherefore, to provide an upper bound on the simple regret, it is sufficient to provide upper\nbounds on the three terms on the right-hand side. In particular, the second term did\nnot appear in the analysis of StroquOOL by Bartlett et al. (2019). Nevertheless, we show\nthat under Algorithm 2, the three terms are controlled, thereby leading to the desired\nperformance guarantees.\nWe saw that the simple regret of Algorithm 1 behaves differently depending on whether\nthe near-optimality dimension dsatisfies d= 0 or d > 0. Similarly, the simple regret\nof Algorithm 2 varies depending on problem parameters. To illustrate, let us define the\nlow-noise andhigh-noise regimes. For simplicity, we use notations νandBdefined as\nν= (2√\nD)αLzε and B=2√\n2\u0010\nC∗(f) +p\nlog(T/δ)\u0011\n√m0.\nHere, νcaptures the term Lzε, and Bis related to the estimation error. Intuitively, if Bis\nhigh, then the estimation error is large. We define ˜has\n˜h=1\nα(d+ 2) log 2W\u0012hmaxν2α(d+ 2) log 2\nB2\u0013\n=1\nα(d+ 2) log 2\u0012\nlog\u0012hmaxν2α(d+ 2) log 2\nB2\u0013\n−log log\u0012hmaxν2α(d+ 2) log 2\nB2\u0013\u0013\n+o(1)\n= Ω\u0012\nlog\u0012\nL2\nzε2T\nlogT\u0013\u0013\nWe refer to the case B < L zε·2−α˜has the low-noise regime and the case B≥Lzε·2−α˜has\nthe high-noise regime.\nTheorem 11 Letddenote the ((2√\nD)αLzε,2−α,1)-near-optimality dimension. For the\ndata-driven setting, Algorithm 2 after Tdecision deployments finds a solution θthat satisfies\nthe following with probability at least 1−δ. Under the low-noise regime with d= 0,\nPR(θ)−PR(θPO)≤(2 + 2√\n2)(2√\nD)αLzε2−αhmax+4C∗(f) + 4p\nlog(T/δ)√hmaxm0\nwhere hmax=⌊T/2D+1(log2T+ 1)2⌋. Under the low-noise regime with d >0,\nPR(θ)−PR(θPO)≤(2 + 2√\n2)(2√\nD)αLzεe−(1/d)W(hmaxαdlog 2)+4C∗(f) + 4p\nlog(T/δ)√hmaxm0.\n14Parameter-Free Performative Regret Minimization\n(a) Ackley Function A(x1, x2)\n (b) Rastrigin Function R(x1, x2)\nFigure 1: Contour plots of the Ackley and Rastrigin functions on [ −5.12,5.12]2\nUnder the high-noise regime,\nPR(θ)−PR(θPO)≤6(2√\nD)αLzε2−α˜h+4C∗(f) + 4p\nlog(T/δ)√hmaxm0.\nMoreover, if hmax≥max{1, e/αd log 2, B2e/ν2α(d+ 2) log 2 }, then under the low-noise\nregime,\nPR(θ)−PR(θPO)≤\n\n(2 + 3√\n2)(2√\nD)αLzε2−αhmax, ifd= 0,\n(2 + 3√\n2)(2√\nD)αLzε\u0010\nhmaxαdlog 2\nlog(hmaxαdlog 2)\u0011−1/d\n,ifd >0.\nLastly, if hmax≥B2e/ν2α(d+ 2) log 2 , then under the high-noise regime,\nPR(θ)−PR(θPO)\n≤6(2√\nD)αLzε\u0012hmaxν2α(d+ 2) log 2 /B2\nlog(hmaxν2α(d+ 2) log 2 /B2)\u0013−1\nd+2\n+4C∗(f) + 4p\nlog(T/δ)√hmaxm0.\nAshmax= Ω( T/logT),hmax=O(T/logT), and ˜h= Ω(log( L2\nzε2T/logT)), we deduce\nfrom Theorem 11 with α= 1 the simple regret bounds in Theorem 2. We provide the proof\nof the theorem in Appendix B.\n6 Experiments\nIn this section, we empirically demonstrate how our algorithms, DOOP for the full-feedback\ncase and SOOP for the data-driven setting, numerically perform for performative regret\nminimization. We compare DOOP and the existing methods SOO,SequOOL , and SZooming\nfor the full-feedback case, and we test SOOP against StoSOO ,StroquOOL , and SZooming\nfor the data-driven setting. Here, SZooming indicates the variant of the zooming algorithm\n15Parameter-Free Performative Regret Minimization\n(a)f(θ, z) =A(θ) +zwith z∼Exp (1 /R(θ))\n (b)f(θ, z) =R(θ) +zwith z∼Exp (1 /A(θ))\nFigure 2: Cumulative regret comparison in the full-feedback case\nby Jagadeesan et al. (2022). For SOO,StoSOO ,SequOOL , and StroquOOL , we used the package\ndeveloped by Li et al. (2023a).\nWe tested the algorithms for synthetic objectives on a bounded two-dimensional domain\nfor optimization. In our experiments, we used two multi-modal functions as shown in\nFigure 1 to express our loss function f(θ, z) and the distribution map D(θ); the first is the\nAckley function given by\nA(x1, x2) =−20·exp\u0014\n−0.2q\n0.5(x2\n1+x2\n2)\u0015\n−exp [0 .5 (cos(2 πx1) + cos(2 πx2))]\nand the second is the Rastrigin function given by\nR(x1, x2) = 20 +\u0000\nx2\n1−10 cos(2 πx1)\u0001\n+\u0000\nx2\n2−10 cos(2 πx2)\u0001\n.\nNote that both functions have a global minimum at A(0,0) = R(0,0) = 0, and their domains\nare both [ −5.12,5.12]2. With the Ackley and Rastrigin functions, we define two types of\nthe loss function.\n•f(θ, z) =A(θ) +zwith z∼ D(θ) = Exp (1 /R(θ)) and θ∈[−5.12,5.12]2,\n•f(θ, z) =R(θ) +zwith z∼ D(θ) = Exp (1 /A(θ)) and θ∈[−5.12,5.12]2\nwhere Exp(1 /λ) denotes the exponential distribution with mean λ. In both cases, we have\nPR(θ) =Ez∼D(θ)[f(θ, z)] =A(θ) +R(θ).\nFor the full-feedback case, we test DOOP with SOO,SequOOL , and SZooming , and the\nperformative risk is constructed based on combining the Ackley function and the Rastrigin\nfunction . Recall that the tree search-based algorithms, not including SZooming , require a\nhierarchical partitioning, and for them, we used the binary partitioning. For the tree search-\nbased algorithms, we used the same maximum level of depth hmax. For SZooming , the\ndecision domain Θ is set to be a finite set of 3,025 discrete points on domain [ −5.12,5.12]2.\n16Parameter-Free Performative Regret Minimization\n(a)f(θ, z) =A(θ) +zwith z∼Exp (1 /R(θ))\n (b)f(θ, z) =R(θ) +zwith z∼Exp (1 /A(θ))\nFigure 3: Cumulative regret comparison in the data-driven setting\nIn addition, the sensitivity parameter ϵand the Lipschitz constant Lzare chosen according\nto the shape of the objective functions on the decision domain Θ; both ϵandLzforz∼\nExp(1 /f(θ)) is calculated to be sup ( |f(θ1)−f(θ2)|/∥θ1−θ2∥) for any θ1, θ2∈Θ.\nFor the data-driven setting, we test SOOP with StoSOO ,StroquOOL , and SZooming . The\nperformative feedback consists of m0= 10 samples drawn randomly from D(θ). While\nboth A(θ) and R(θ) are multi-modal on the given domain, each has different sets of local\noptima and range. In particular, R(θ) yields values in a broader range, thus the associated\ndistribution Exp (1 /R(θ)) has a larger variance since the variance of Exp(1 /λ) isλ2. The\nother components of the experimental setup are the same as those for the full-feedback case.\nFigures 2 and 3 summarize our experimental results. As shown in Figures 2 and 3, DOOP\nand SOOP outperform the other methods in terms of cumulative regret. We have noticed\nthat SZooming incurs a very high cumulative regret in the first phase, and this is because\nthere exist high estimation errors in the first phase of SZooming and it turns out that the\nmajority of exploration of SZooming occurs during the first phase. Moreover, SZooming\nis not computationally efficient, as it takes a huge amount of time to find an optimal\ndecision. For the full-feedback case, SZooming takes 73691 seconds for f(θ, z) =A(θ) +z\nwith z∼ D(θ) = Exp (1 /R(θ)) and 3737.0 seconds for f(θ, z) =R(θ) +zwith z∼ D(θ) =\nExp (1 /A(θ)). In contrast, DOOP takes 4.2390 seconds and 2.3609 seconds, respectively. For\nthe data-driven case, SZooming takes 73112 seconds for f(θ, z) =A(θ) +zwith z∼ D(θ) =\nExp (1 /R(θ)) and 3999.4 seconds for f(θ, z) =R(θ) +zwith z∼ D(θ) = Exp (1 /A(θ)). In\ncontrast, SOOP takes 2.5896 seconds and 1.4877 seconds, respectively.\nAcknowledgments and Disclosure of Funding\nThis research is supported, in part, by KAIST Starting Fund (KAIST-G04220016), FOUR\nBrain Korea 21 Program (NRF-5199990113928), and National Research Foundation of Ko-\nrea (NRF-2022M3J6A1063021).\n17Parameter-Free Performative Regret Minimization\nAppendix A. Regret Analysis of DOOP\nA.1 Approximation Bounds under the Hierarchical Partitioning Scheme\nIn this section, we prove two lemmas that are related to the quality of the representative\ndecision of each cell.\nLemma 12 Forθ∈ Ph,iandPh,i∈ Lh, we have\n|DPR( θh,i, θ)−PR(θ)| ≤(√\nD)αLzε2−αh.\nProof Note that\nDPR( θh,i, θ)≥PR(θ)−Lzε∥θh,i−θ∥α≥PR(θ)−(√\nD)αLzε2−αh\nwhere the first inequality is from Lemma 3 and the second inequality follows from Assump-\ntion 5 with θh,i, θ∈ P h,i. Similarly, we can argue that DPR( θh,i, θ)≤PR(θ) +Lzε∥θh,i−\nθ∥α≤PR(θ) + (√\nD)αLzε2−αh,as required.\nLemma 13 LetPh,i∈ Lh. Then\nPR(θh,i)≤inf\nθ∈Ph,iPR(θ) + 2(2√\nD)αLzε2−αh.\nProof Letθ⋆\nh,i∈argminθ∈Ph,iPR(θ), and let Ph−1,jbe the parent cell of Ph,i. Note that\nPR(θh,i)≤DPR( θh−1,j, θh,i) + (2√\nD)αLzε2−αh≤DPR( θh−1,j, θ⋆\nh,i) + (2√\nD)αLzε2−αh\nwhere the first inequality follows from Lemma 12 and the second inequality holds due to\nour choice of θh,iminimizing DPR( θh−1,j, θ) over θ∈ Ph,i. Lastly, by Lemma 12, we have\nDPR( θh−1,j, θ⋆\nh,i)≤PR(θ⋆\nh,i) + (2√\nD)αLzε2−αh.\nConsequently, it follows that PR( θh,i)≤PR(θ⋆\nh,i) + 2(2√\nD)αLzε2−αh,as required.\nA.2 Proof of Theorem 6\nRecall that ⊥his defined on the depth of the deepest cell containing θPOopened until\nAlgorithm 1 finishes opening cells of depth h.\nLemma 14 Letddenote the ((2√\nD)αLzε,2−α,1)-near-optimality dimension. For any h∈\n[hmax], ifhmax/h≥2αhd, then ⊥h=hwith⊥0= 0.\nProof Leth∈[hmax], and assume that hsatisfies the condition of the lemma. Then we\nwill argue by induction that ⊥h′=h′for all h′∈[h], thereby proving that ⊥h=h.\nNote that P0,1= Θ contains θPOandP0,1is opened, so ⊥0= 0. Next, we assume that\n⊥h′−1=h′−1 for some h′∈[h]. Then it is sufficient to show that ⊥h′=h′. Let i⋆\nh′−1\n18Parameter-Free Performative Regret Minimization\ndenote the index such that Ph′−1,i⋆\nh′−1is the cell containing θPOat depth h′−1. Algorithm 1\nopens ⌊hmax/h′⌋cells from depth h′cells. Suppose for a contradiction that cell Ph′,i⋆\nh′is not\none of them. This implies that for each solution θh′,iof the ⌊hmax/h′⌋cells from depth h′,\nwe have PR( θh′,i)≤PR(θh′,i⋆\nh′).Consequently, it follows that\nPR(θh′,i)≤PR(θh′,i⋆\nh′)≤PR(θPO) + 2(2√\nD)αLzε2−αh′\nwhere the second inequality follows from Lemma 13 as θPOis contained in cell Ph′,i⋆\nh′. This\nimplies that\nNh(6(2√\nD)αLzε2−αh′)≥\u0016hmax\nh′\u0017\n+ 1≥\u0016hmax\nh\u0017\n+ 1≥2αhd+ 1≥2αh′d+ 1\nwhere ⌊hmax/h′⌋comes from cells Ph′,iand 1 is due to cell Ph′,i⋆\nh′in the first inequality,\nthe second and the fourth inequalities hold because h′≤h, and the third inequality comes\nfrom the condition of the lemma. This in turn implies that Nh(6(2√\nD)αLzε2−αh′)>2αh′d,\na contradiction. Therefore, it follows that ⊥h′=h′. Then the induction argument shows\nthat⊥h=h, as required.\nBased on Lemma 14, we prove Lemma 5 that shows\nPR(θT)−PR(θPO)≤2(2√\nD)αLzε2−α(⊥hmax+1).\nProof [Proof of Lemma 5 ] LetP⊥hmax+1,i⋆be the cell at depth ⊥hmax+1 containing θPO.\nNote that\nPR(θT)≤PR(θ⊥hmax+1,i⋆)≤PR(θPO) + 2(2√\nD)αLzε2−α(⊥hmax+1)\nwhere the first inequality holds due to the choice of θTand the second inequality follows\nfrom Lemma 13.\nFor simplicity, we introduce notations ρandνdefined as\nρ= 2−αand ν= (2√\nD)αLzε.\nMoreover, we define ¯has the number satisfying\nhmax\n¯h=ρ−d¯h.\nNote that if d= 0, then ¯h=hmax. Ifd >0, then\n¯h=1\ndlog(1/ρ)W(hmaxdlog(1/ρ))\nwhere W(·) denotes the Lambert Wfunction.\nLemma 15 (Bartlett et al. (2019)) Letddenote the (ν, ρ,1)-near-optimality dimension.\nThen⊥hmax+ 1≥¯h.\n19Parameter-Free Performative Regret Minimization\nCombining Lemmas 5 and 15, we are ready to provide the desired regret bounds on Algo-\nrithm 1.\nProof [Proof of Theorem 6 ] As¯h=hmaxwhen d= 0 and ¯h=W(hmaxαdlog 2) /αdlog 2,\nit follows directly from Lemmas 5 and 15 that\nPR(θ)−PR(θPO)≤(\n2(2√\nD)αLzε2−αhmax, ifd= 0,\n2(2√\nD)αLzεe−(1/d)W(hmaxαdlog 2),ifd >0.\nLastly, Hoorfar and Hassani (2008) showed that if x≥e, then W(x)≥log(x/log(x)).\nHence, if d >0 and hmaxαdlog 2≥e, then θsatisfies\nPR(θ)−PR(θPO)≤2(2√\nD)αLzε\u0012hmaxαdlog 2\nlog(hmaxαdlog 2)\u0013−1\nd\n,\nas required.\nAppendix B. Regret Analysis of SOOP\nB.1 Total Number of Solution Deployments\nRecall that\nhmax=\u0016T\n2D+1(log2T+ 1)2\u0017\nand pmax=⌊log2hmax⌋.\nLemma 16 The total number of solution deployments before the cross-validation phase is\nat most 3T/4, and in the cross-validation phase, the total number of solution deployments\nis at most T/4.\nProof Note that as T≥2, we have log2T+1≥2, in which case hmax≤T/2D+3.Hence, we\ndeploy solution θ0,1at most T/2D+3≤T/8 times. Moreover, we open P0,1at most T/2D+3\ntimes, and since P0,1has 2Dchild cells, it corresponds to at most T/8 solution deployments.\nNext, during the exploration phase, we makePhmax\nh=1Ppmax\np=0⌊hmax/h2p⌋2popenings. Here,\nhmaxX\nh=1pmaxX\np=0\u0016hmax\nh2p\u0017\n·2p≤hmaxX\nh=1pmaxX\np=0hmax\nh= (pmax+ 1)hmaxhmaxX\nh=11\nh≤hmax(pmax+ 1)2≤T\n2D+1\nwhere the last inequality holds due to pmax≤log2T. Since each opening requires 2Dsolution\ndeployments, it incurs T/2 solution deployments. In total, before the cross-validation phase,\nwe make T/8 +T/8 +T/2 = 3 T/4 solution deployments.\nIn the cross-validation phase, the number of solution deployments is given by\nhmax(pmax+ 1)≤T\n2D+1(log2T+ 1)≤T\n2D+2≤T\n4,\nas required.\n20Parameter-Free Performative Regret Minimization\nB.2 Rademacher Complexity-Based Concentration Bounds for Estimating the\nPerformative Risk\nIn this section, we prove Lemma 9 which shows that the clean event holds with probability\nat least 1 −δ, i.e.,P[Eclean ,δ]≥1−δ.\nProof [Proof of Lemma 9 ] By Lemma 16, Algorithm 2 deploys at most Tsolutions. Let\nndistinct denote the number of distinct solutions deployed by Algorithm 2. As new solutions\nare deployed during the exploration phase, we have ndistinct ≤3T/4 by Lemma 16. Among\nthendistinct solutions, we use notation ( h(s), i(s)) to indicate the cell Ph(s),i(s)containing\nthesth deployed solution for 1 ≤s≤ndistinct . As hmaxis fixed, ndistinct ,h(s), and ndeploy\nh(s),i(s)\nare all deterministic functions of s. In particular, we use notation ndeploy\ns :=ndeploy\nh(s),i(s)to\nemphasize that ndeploy\nh(s),i(s)is deterministic in s. Then we maintain a virtual tape of samples\nfor each solution θ. Basically, for each solution θ, we maintain {zθ\nj:j= 1, . . . , Tm 0}, and if\nθbecomes the sth solution deployed, then we use ndeploy\ns m0samples in {zθ\nj:j= (ndeploy\n1 +\n···+ndeploy\ns−1)m0+ 1, . . . , (ndeploy\n1 +···+ndeploy\ns )m0}to estimate bD(θ). For 1 ≤s≤ndistinct ,\nlet us define Es\nclean ,δas the event that\nsup\nθ∈Ps\f\f\f[DPR( θs, θ)−DPR( θs, θ)\f\f\f≤2C∗(f) + 2p\nlog(T/δ)q\nndeploy\ns m0\nwhere θsdenotes the sth solution deployed θh(s),i(s),Psdenotes Ph(s),i(s)containing the sth\nsolution deployed, and ndeploy\ns denotes the number of times solution θh(s),i(s)is deployed.\nMoreover, for p∈[0 :pmax], let us define ET,p\nclean ,δas the event that\n\f\f\fcPR(θT(p))−PR(θT(p))\f\f\f≤2C∗(f) + 2p\nlog(T/δ)√hmaxm0.\nThen we know that\nP[Eclean ,δ] =P[E1\nclean ,δ∩ ··· ∩ Endistinct\nclean ,δ∩ET,0\nclean ,δ∩ ··· ∩ ET,pmax\nclean ,δ]\n≥1−ndistinctX\ns=1P[¬Es\nclean ,δ]−pmaxX\np=0P[¬ET,p\nclean ,δ]\nwhere the inequality is the union bound. For simplicity, let Jdenote J={(ndeploy\n1 +···+\nndeploy\ns−1)m0+ 1, . . . , (ndeploy\n1 +···+ndeploy\ns )m0}. Note that\nP[¬Es\nclean ,δ] =P\nsup\nθ∈Ps\f\f\f[DPR( θs, θ)−DPR( θs, θ)\f\f\f>2C∗(f) + 2p\nlog(T/δ)q\nndeploy\ns m0\n\n=P\nsup\nθ∈Ps\f\f\f\f\f\f1\nndeploy\ns m0X\nj∈Jf(θ, zθs\nj)−DPR( θs, θ)\f\f\f\f\f\f>2C∗(f) + 2p\nlog(T/δ)q\nndeploy\ns m0\n\n≤P\nsup\nθ∈Θ\f\f\f\f\f\f1\nndeploy\ns m0X\nj∈Jf(θ, zθs\nj)−DPR( θs, θ)\f\f\f\f\f\f>2C∗(f) + 2p\nlog(T/δ)q\nndeploy\ns m0\n\n21Parameter-Free Performative Regret Minimization\nwhere the inequality holds because Ps⊆Θ. Here, the right-most side of this inequality is\nequal to\nE¯θ∼θs\nP\nsup\nθ∈Θ\f\f\f\f\f\f1\nndeploy\ns m0X\nj∈Jf(θ, z¯θ\nj)−DPR( ¯θ, θ)\f\f\f\f\f\f>2C∗(f) + 2p\nlog(T/δ)q\nndeploy\ns m0|θs=¯θ\n\n.\nTherefore, to provide an upper bound on P[¬Es\nclean ,δ], it suffices to provide an upper bound\non\nP\nsup\nθ∈Θ\f\f\f\f\f\f1\nndeploy\ns m0X\nj∈Jf(θ, z¯θ\nj)−DPR( ¯θ, θ)\f\f\f\f\f\f>2C∗(f) + 2p\nlog(T/δ)q\nndeploy\ns m0|θs=¯θ\n (1)\nfor every ¯θ∈Θ. Note that data samples in {z¯θ\nj:j= (ndeploy\n1 +···+ndeploy\ns−1)m0+\n1, . . . , (ndeploy\n1 +···+ndeploy\ns )m0}are independent of the event that θs=¯θbecause the\nsamples are obtained after the sth solution for deployment is chosen. Therefore, the prob-\nability term (1) is equal to\nP\nsup\nθ∈Θ\f\f\f\f\f\f1\nndeploy\ns m0X\nj∈Jf(θ, z¯θ\nj)−DPR( ¯θ, θ)\f\f\f\f\f\f>2C∗(f) + 2p\nlog(T/δ)q\nndeploy\ns m0\n. (2)\nWhat remains is to bound this probability term for every ¯θ∈Θ. By the bounded differences\ninequality and Assumption 4, with probability at least 1 −(δ/T), we have\nsup\nθ∈Θ\f\f\f\f\f\f1\nndeploy\ns m0X\nj∈Jf(θ, z¯θ\nj)−DPR( ¯θ, θ)\f\f\f\f\f\f\n≤E\nsup\nθ∈Θ\f\f\f\f\f\f1\nndeploy\ns m0X\nj∈Jf(θ, z¯θ\nj)−DPR( ¯θ, θ)\f\f\f\f\f\f\n+s\n2 log( T/δ)\nndeploy\ns m0.(3)\nLetϵjdenote i.i.d. Rademacher random variables. Then by a symmetrization argument,\nthe right-hand side of (3) is at most\nE\nsup\nθ∈Θ\f\f\f\f\f\f1\nndeploy\ns m0X\nj∈Jf(θ, z¯θ\nj)−DPR( ¯θ, θ)\f\f\f\f\f\f\n+s\n2 log( T/δ)\nndeploy\ns m0\n≤2·E\nsup\nθ∈Θ\f\f\f\f\f\f1\nndeploy\ns m0X\nj∈Jf(θ, z¯θ\nj)·ϵj\f\f\f\f\f\f\n+s\n2 log( T/δ)\nndeploy\ns m0\n≤2q\nndeploy\ns m0·sup\nn≥1√n·E\nsup\nθ∈Θ\f\f\f\f\f\f1\nnnX\nj=1f(θ, z¯θ\nj)·ϵj\f\f\f\f\f\f\n+s\n2 log( T/δ)\nndeploy\ns m0\n≤2C∗(f) + 2p\nlog(T/δ)q\nndeploy\ns m0(4)\n22Parameter-Free Performative Regret Minimization\nwhere {zθ\nj}j∈Ndenotes an infinite sequence of samples from D(θ) for θ∈Θ. By (3)\nand (4), the probability term (2) as well as (1) is at most δ/T. Therefore, it follows\nthatP[¬Es\nclean ,δ]≤δ/T.\nNext, we consider P[¬ET,p\nclean ,δ]. Note that the total number of solution deployments\nduring the exploration phase, denoted ntotalis deterministic. Note that θT(p) is deployed\nforhmaxtimes and obtain samples {zθT(p)\nj :j∈J′}where J′={(ntotal+phmax)m0+\n1, . . . , (ntotal+ (p+ 1)hmax)m0}. Then we have\nP[¬ET,p\nclean ,δ]\n=P\"\f\f\fcPR(θT(p))−PR(θT(p))\f\f\f>2C∗(f) + 2p\nlog(T/δ)√hmaxm0#\n=P\n\f\f\f\f\f\f1\nhmaxm0X\nj∈J′f(θT(p), zθT(p)\nj)−DPR( θT(p), θT(p))\f\f\f\f\f\f>2C∗(f) + 2p\nlog(T/δ)√hmaxm0\n\n≤P\nsup\nθ∈Θ\f\f\f\f\f\f1\nhmaxm0X\nj∈J′f(θ, zθT(p)\nj)−DPR( θT(p), θ)\f\f\f\f\f\f>2C∗(f) + 2p\nlog(T/δ)√hmaxm0\n.\nAs before, we can argue that P[¬ET,p\nclean ,δ]≤δ/T. Since ndistinct + (pmax+ 1)hmax≤T\nby Lemma 16, it follows that P[Eclean ,δ]≥1−δasndistinct ≤T.\nB.3 Approximation Bounds under the data-driven setting\nIn this section, we prove the following lemma analyzing the quality of the representative\ndecision of each cell under the data-driven setting.\nLemma 17 Assume that Eclean ,δholds for some δ∈(0,1). LetPh,i∈ L h, and let Ph−1,j\nbe the parent cell of Ph,i. Then\nPR(θh,i)≤inf\nθ∈Ph,iPR(θ) + 2(2√\nD)αLzε2−αh+4C∗(f) + 4p\nlog(T/δ)q\nndeploy\nh−1,jm0.\nProof Letθ⋆\nh,i∈argminθ∈Ph,iPR(θ). By Lemma 12,\nPR(θh,i)≤DPR( θh−1,j, θh,i) + (2√\nD)αLzε2−αh.\nAsEclean ,δholds, we have\nDPR( θh−1,j, θh,i)≤[DPR( θh−1,j, θh,i) +2C∗(f) + 2p\nlog(T/δ)q\nndeploy\nh−1,jm0.\nMoreover,\n[DPR( θh−1,j, θh,i)≤[DPR( θh−1,j, θ⋆\nh,i)≤DPR( θh−1,j, θ⋆\nh,i) +2C∗(f) + 2p\nlog(T/δ)q\nndeploy\nh−1,jm0\n23Parameter-Free Performative Regret Minimization\nwhere the first inequality is due to our choice of θh,iminimizing [DPR( θh−1,j, θ) over θ∈ Ph,i\nand the second inequality holds because Eclean ,δholds. Lastly, by Lemma 12,\nDPR( θh−1,j, θ⋆\nh,i)≤PR(θ⋆\nh,i) + (2√\nD)αLzε2−αh.\nConsequently, it follows that\nPR(θh,i)≤PR(θ⋆\nh,i) + 2(2√\nD)αLzε2−αh+4C∗(f) + 4p\nlog(T/δ)q\nndeploy\nh−1,jm0,\nas required.\nB.4 Proof of Theorem 11\nRecall that ⊥h,pis defined as the depth of the deepest cell containing the performative\noptimal solution θPOopened for at least 2ptimes until Algorithm 2 finishes opening cells\nof depth h.\nLemma 18 Assume that the clean event Eclean ,δholds for some δ∈(0,1), and let ddenote\nthe((2√\nD)αLzε,2−α,1)-near-optimality dimension d((2√\nD)αLzε,2−α,1). For any h∈\n[⌊hmax/2p⌋]andp∈[0 :⌊log2(hmax/h)⌋], if the following condition holds, then ⊥h,p=h\nwith⊥0,p= 0.\n2C∗(f) + 2p\nlog(T/δ)√2pm0≤(2√\nD)αLzε2−αhandhmax\nh2p≥2αhd.\nProof Let (h, p) with h∈[⌊hmax/2p⌋] and p∈[0 :⌊log2(hmax/h)⌋] satisfy the condition of\nthe lemma. Then we will argue by induction that ⊥h′,p=h′for all h′∈[h], thereby proving\nthat⊥h,p=h.\nNote that P0,1= Θ contains θPOandP0,1is opened hmaxtimes with hmax≥2pmax, so\n⊥0,p= 0. Next, we assume that ⊥h′−1,p=h′−1 for some h′∈[h]. Then it is sufficient to\nshow that ⊥h′,p=h′. Let i⋆\nh′−1denote the index such that Ph′−1,i⋆\nh′−1is the cell containing\nθPOat depth h′−1. By the induction hypothesis, cell Ph′−1,i⋆\nh′−1is opened at least 2p\ntimes, i.e., nopen\nh′−1,i⋆\nh′−1≥2p. This implies that nopen\nh′−1,i⋆\nh′−1≥2pbecause ndeploy\nh′−1,i⋆\nh′−1≥2p′=\nnopen\nh′−1,i⋆\nh′−1for some p′according to the design of Algorithm 2. Let i⋆\nh′denote the index such\nthatPh′,i⋆\nh′is the cell containing θPOat depth h′. This means that Ph′,i⋆\nh′is a child cell of\nPh′−1,i⋆\nh′−1andndeploy\nh′,i⋆\nh′=nopen\nh′−1,i⋆\nh′−1≥2p.\nWe open ⌊hmax/h′2p⌋cells from depth h′cells with at least 2pdeployments. Suppose for\na contradiction that cell Ph′,i⋆\nh′is not one of them. This implies that for each solution θh′,i\nof the ⌊hmax/h′2p⌋cells with 2pdeployments from depth h′, we have cPR(θh′,i)≤cPR(θh′,i⋆\nh′).\nMoreover, such θh′,isatisfies the following.\nPR(θh′,i)−(2√\nD)αLzε2−αh′≤PR(θh′,i)−(2√\nD)αLzε2−αh\n≤PR(θh′,i)−2C∗(f) + 2p\nlog(T/δ)√2pm0(5)\n24Parameter-Free Performative Regret Minimization\nwhere the first inequality holds because h′≤hand the second inequality holds due to the\ncondition of the lemma. Furthermore,\nPR(θh′,i)−2C∗(f) + 2p\nlog(T/δ)√2pm0≤PR(θh′,i)−2C∗(f) + 2p\nlog(T/δ)q\nndeploy\nh′,im0\n≤cPR(θh′,i)\n≤cPR(θh′,i⋆\nh′)(6)\nwhere the first inequality holds because ndeploy\nh′,i≥2pand the second inequality holds due to\nthe assumption that Eclean ,δholds. Combining (5) and (6), we deduce that\nPR(θh′,i)−(2√\nD)αLzε2−αh′≤cPR(θh′,i⋆\nh′).\nSimilarly, we can argue that\nPR(θh′,i⋆\nh′) + (2√\nD)αLzε2−αh′≥cPR(θh′,i⋆\nh′).\nConsequently, it follows that\nPR(θh′,i)≤PR(θh′,i⋆\nh′) + 2(2√\nD)αLzε2−αh′\n≤inf\nθ∈ΘPR(θ) + 4(2√\nD)αLzε2−αh′+4C∗(f) + 4p\nlog(T/δ)√2pm0\nwhere the second inequality follows from Lemma 17, ndeploy\nh′−1,i⋆\nh′−1≥2p, and θPOis contained\nin cell Ph′,i⋆\nh′. Furthermore, by the condition of this lemma, it follows that\nPR(θh′,i)≤PR(θPO) + 6(2√\nD)αLzε2−αh′.\nIn addition, since θPOis contained in cell Ph′,i⋆\nh′, Lemma 17 implies that\nPR(θh′,i∗)≤PR(θPO) + 4(2√\nD)αLzε2−αh′.\nThis implies that\nNh(6(2√\nD)αLzε2−αh′)≥\u0016hmax\nh′2p\u0017\n+ 1≥\u0016hmax\nh2p\u0017\n+ 1≥2αhd+ 1≥2αh′d+ 1\nwhere ⌊hmax/h′2p⌋comes from cells Ph′,iand 1 is due to cell Ph′,i⋆\nh′in the first inequality,\nthe second and the fourth inequalities hold because h′≤h, and the third ineuality comes\nfrom the condition of the lemma. This in turn implies that Nh(6(2√\nD)αLzε2−αh′)>2αh′d,\na contradiction. Therefore, it follows that ⊥h′,p=h′. Then the induction argument shows\nthat⊥h,p=h, as required.\n25Parameter-Free Performative Regret Minimization\nNext, we prove Lemma 10 which shows that\nPR(θT)−PR(θPO)\n≤2(2√\nD)αLzε2−α(⊥hmax,p+1)+8C∗(f) + 8p\nlog(T/δ)√2pm0+4C∗(f) + 4p\nlog(T/δ)√hmaxm0.\nProof [Proof of Lemma 10 ] Let p∈[0 :pmax], and let\n(h, i)∈argmin\n(h,i)n\ncPR(θh,i) :h∈[hmax+ 1],Ph,i∈ Lh, ndeploy\nh,i≥2po\n.\nRecall that θT(p) is set to θh,iand that we obtain hmaxm0new samples from D(θT(p))\nfrom which we construct cPR(θT(p)). Moreover, cPR(θT)≤cPR(θT(p)). As Eclean ,δholds, it\nfollows that\nPR(θT)−2C∗(f) + 2p\nlog(T/δ)√hmaxm0≤cPR(θT)\n≤cPR(θT(p))\n≤PR(θT(p)) +2C∗(f) + 2p\nlog(T/δ)√hmaxm0.(7)\nAgain, as Eclean ,δholds and θT(p) =θh,i,\nPR(θT(p))≤cPR(θh,i) +2C∗(f) + 2p\nlog(T/δ)q\nndeploy\nh,im0≤cPR(θh,i) +2C∗(f) + 2p\nlog(T/δ)√2pm0.(8)\nRecall that ⊥hmax,pis the depth of the deepest cell containing θPOopened for at least 2p\ntimes until Algorithm 2 finishes opening cells of depth hmax. Let ( ⊥hmax,p+ 1, i⋆) denote\nthe deepest cell containing θPOand a solution deployed at least 2ptimes. By the choice of\n(h, i), we have\ncPR(θh,i)≤cPR(θ⊥hmax,p+1,i⋆)≤PR(θ⊥hmax,p+1,i⋆) +2C∗(f) + 2p\nlog(T/δ)√2pm0(9)\nwhere the second inequality holds because Eclean ,δholds and ndeploy\n⊥hmax,p+1,i⋆≥2p. Moreover,\nsince the parent cell of P⊥hmax,p+1,i⋆is opened at least 2ptimes, it means that the parent\ncell contains a solution deployed at least 2ptimes. Then by Lemma 17, it follows that\nPR(θ⊥hmax,p+1,i⋆)≤PR(θPO) + 2(2√\nD)αLzε2−α(⊥hmax,p+1)+4C∗(f) + 4p\nlog(T/δ)√2pm0.(10)\nIn summary, we deduce from (7) – (10) that\nPR(θT)−PR(θPO)\n≤2(2√\nD)αLzε2−α(⊥hmax,p+1)+8C∗(f) + 8p\nlog(T/δ)√2pm0+4C∗(f) + 4p\nlog(T/δ)√hmaxm0,\n26Parameter-Free Performative Regret Minimization\nas required.\nNote that the regret bound given by Lemma 10 holds for any p∈[0 :pmax]. Hence, to\nprove an upper bound on the regret PR( θT)−PR(θPO), we need to choose an appropriate\npthat achieves a small value of\n2(2√\nD)αLzε2−α(⊥hmax,p+1)+8C∗(f) + 8p\nlog(T/δ)√2pm0.\nAs in Bartlett et al. (2019), the strategy is to choose punder which there is a strong lower\nbound on ⊥hmax,p+ 1. In our case, however, we have the additional term ˜O(1/√\n2p). In\nfact, we will argue that the choice of p, under which ⊥hmax,p+ 1 is large, also makes the\nadditional term small.\nFor simplicity, we use notations ρ,ν, and Bdefined as\nρ= 2−α, ν = (2√\nD)αLzε, B =2√\n2\u0010\nC∗(f) +p\nlog(T/δ)\u0011\n√m0.\nWith these notations, Lemma 18 can be restated as follows.\nLemma 19 Assume that Eclean ,δholds for some δ∈(0,1). Let ddenote the (ν, ρ,1)-near-\noptimality dimension d(ν, ρ,1). For any h∈[⌊hmax/2p⌋]andp∈[0 :⌊log2(hmax/h)⌋], if the\nfollowing condition holds, then ⊥h,p=h.\nB√\n2p+1≤νρhandhmax\nh2p≥ρ−dh.\nNext, we define ˜hand ˜pas the numbers satisfying the following condition.\nhmaxν2ρ2˜h\n˜hB2=ρ−˜hdandB√\n2˜p=νρ˜h.\nThen by definition of the Lambert Wfunction, we have\n˜h=1\n(d+ 2) log(1 /ρ)W\u0012hmaxν2(d+ 2) log(1 /ρ)\nB2\u0013\n.\nHere, B≥Lzε·2−α˜hholds if and only if 2˜p≥1. Hence, the case when 2˜p≥1 corresponds\nto the high-noise regime and the setting where 2˜p<1 corresponds to the low-noise regime.\nNext, as in Bartlett et al. (2019), we define ¨hand ¨pas follows.\n•(High-noise regime) Set ¨h=˜hand ¨p= ˜p.\n•(Low-noise regime) Set ¨has¨h=¯hthat satisfies hmax/¯h=ρ−d¯hand ¨p= 0.\nNote that for this choice of ¨hand ¨p, we have hmax/¨h2¨p=ρ−d¨h. under both regimes.\nMoreover, with Lemma 19, we may argue that the following statement holds.\n27Parameter-Free Performative Regret Minimization\nLemma 20 (Bartlett et al. (2019)) Assume that Eclean ,δholds for some δ∈(0,1). Let\nddenote the (ν, ρ,1)-near-optimality dimension d(ν, ρ,1). Then ¨h≤˜hand\n⊥hmax,⌊¨p⌋+ 1≥ ⌊¨h⌋+ 1≥¨h\nunder both the high-noise and low-noise regimes.\nNow we are ready to complete the proof of Theorem 11.\nProof [Proof of Theorem 11 ] Under Eclean ,δ, Lemmas 10 and 20 imply that\nPR(θT)−PR(θPO)≤2(2√\nD)αLzε2−α¨h+8C∗(f) + 8p\nlog(T/δ)p\n2⌊¨p⌋m0+4C∗(f) + 4p\nlog(T/δ)√hmaxm0\nholds.\nLet us first consider the low-noise regime. Since 2˜p<1, we know that B < νρ˜h. By\nLemma 20, we have ¨h≤˜h, which implies that B < νρ˜h≤νρ¨h. Then as ¨ p= 0 under the\nlow-noise regime, it follows that\nPR(θT)−PR(θPO)≤(2 + 2√\n2)(2√\nD)αLzε2−α¨h+4C∗(f) + 4p\nlog(T/δ)√hmaxm0.\nMoreover, if hmax≥1, then as B≤νρ¨h,\nPR(θT)−PR(θPO)≤(2 + 3√\n2)(2√\nD)αLzε2−α¨h.\nWhen d= 0, we have ¨h=hmax. When d >0, we have\n¨h=1\nαdlog 2W(hmaxαdlog 2) .\nFor the high-noise regime, we have ¨h=˜hand\n8C∗(f) + 8p\nlog(T/δ)p\n2⌊¨p⌋m0=4B√\n2⌊¨p⌋+1≤4B√\n2¨p= 4νρ˜h.\nTherefore, under the high-noise regime, we have\nPR(θT)−PR(θPO)≤6(2√\nD)αLzε2−α˜h+4C∗(f) + 4p\nlog(T/δ)√hmaxm0.\nRecall that ˜his given by\n˜h=1\nα(d+ 2) log 2W \n(4D)αα(d+ 2) log 2\n8(C∗(f) + 4p\nlog(T/δ))2L2\nzε2m0hmax!\n.\nLastly, Hoorfar and Hassani (2008) showed that if x≥e, then W(x)≥log(x/log(x)).\nHence, if d >0 and hmaxαdlog 2≥eunder the low-noise regime, then θsatisfies\nPR(θ)−PR(θPO)≤(2 + 3√\n2)(2√\nD)αLzε\u0012hmaxαdlog 2\nlog(hmaxαdlog 2)\u0013−1\nd\n.\n28Parameter-Free Performative Regret Minimization\nMoreover, if B2hmaxν2α(d+ 2) log 2 ≥e, then\nPR(θ)−PR(θPO)\n≤6(2√\nD)αLzε\u0012hmaxν2α(d+ 2) log 2 /B2\nlog(hmaxν2α(d+ 2) log 2 /B2)\u0013−1\nd+2\n+4C∗(f) + 4p\nlog(T/δ)√hmaxm0,\nas required.\nReferences\nM. G. azar, A. Lazaric, and E. Brunskill. Online stochastic optimization under corre-\nlated bandit feedback. 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URL https://proceedings.neurips.cc/paper_\nfiles/paper/2021/file/812214fb8e7066bfa6e32c626c2c688b-Paper.pdf .\n33"    },    {        "title": "2402.15251v1.Magnons_and_fundamental_magnetic_interactions_in_a_ferromagnetic_monolayer__The_case_of_Ni_monolayer.pdf",        "content": "Magnons and fundamental magnetic interactions in a ferromagnetic monolayer: The\ncase of Ni monolayer\nKhalil Zakeri,1,∗Albrecht von Faber,1and Arthur Ernst2, 3\n1Heisenberg Spin-dynamics Group, Physikalisches Institut,\nKarlsruhe Institute of Technology, Wolfgang-Gaede-Str. 1, D-76131 Karlsruhe, Germany\n2Institute for Theoretical Physics, Johannes Kepler University, Altenberger Strasse 69, A-4040 Linz, Austria\n3Max-Planck-Institut f¨ ur Mikrostrukturphysik, Weinberg 2, D-06120 Halle, Germany\nThe experimental investigations of the mag-\nnetic interactions in an atomically thin magnetic\nlayer are essential to understand the physics of\nlow-dimensional magnets. The full spectrum of\ncollective magnetic excitations (magnons) would\nprovide an access to these fundamental interac-\ntions on the atomic scale. Here in order to be\nable to excite the magnons by means of spin-\npolarized electrons we couple a Ni monolayer to\none and two atomic layers of Co and probe the\nfull experimental magnon dispersion relation up\nto the Brillouin zone boundary. Comparing to\nthe results of ab initio calculations we quantify\nthe complex pattern of the magnetic exchange\ninteraction in the Ni monolayer. We show that\nalthough the magnons in this system are rather\nstiff, the Heisenberg exchange coupling between\nthe Ni spins is weak. We unravel the origin of the\nobserved large magnon stiffness constant being a\nconsequence of the small spin density of the Ni\natoms.\nThe collective excitations of magnetic solids can be de-\nscribed by their representative quasi-particles, called\nmagnons. Of particular interest are the high wavevec-\ntor magnons, since they are governed by the magnetic\nexchange interaction between neighboring spins [1]. This\ninteraction, also known as Heisenberg exchnage interac-\ntion (HEI), is indispensable for understanding many phe-\nnomena in magnetism [2–4]. The Heisenberg spin Hamil-\ntonian describing this fundamental interaction is usually\ngiven by HHEI=−P\ni̸=jJijSi·Sj, where the exchange\nparameter Jij, describes the interaction between atomic\nspins SiandSjsitting on sites iandj. The pattern\nof HEI in low-dimensional itinerant magnets can be very\ncomplex [5–9]. Fortunately, probing the full magnon dis-\npersion relation provides a direct and unambiguous way\nof resolving the complex pattern of HEI [9–11].\nSeveral experimental techniques based on the inelastic\nscattering of neutrons, photons and electrons have been\nsuccessfully implemented to probe magnons in bulk and\nultrathin films of Fe and Co down to the monolayer and\neven submonolayer regime. However, so far it has been\nchallenging to probe the full magnon dispersion relation\nin Ni, in particular a Ni monolayer. This may have sev-\neral reasons. (i) The inelastic scattering cross-section of\nneutrons as bulk probes by magnons scales with the mag-netic form factor which, in turn, is directly related to the\nspin density of the unit cell. Owing to its rather small\nmagnetic moment and its itinerant magnetism character,\nNi has not been in the favor of the neutron scattering\nexperiments [12]. The magnon spectrum of bulk Ni has\nonly been measured over a very small range of momentum\n[13–15]. (ii) Likewise, resonant inelastic x-ray scattering\nexperiments have been limited by effects associated with\nthe x-ray fluorescence [16]. In the most recent exper-\niments only a small fraction of the Brillouin zone (BZ)\ncould be covered [17]. (iii) Theoretical calculations of the\ninelastic electron scattering cross-section have revealed\nthat due to the small exchange splitting and the pres-\nence of Stoner excitations at low energies it is practically\nnot feasible to measure the magnon dispersion relation in\nNi thin films and at Ni surfaces when using electrons as\nprobing particles [18]. All these together have made the\nfull magnon dispersion relation in Ni inaccessible. Given\nthe fact that the neutron and x-ray scattering techniques\ndo not have the monolayer sensitivity the experimental\nmagnon spectrum as well as the magnetic exchange pa-\nrameters in a Ni monolayer have remained hitherto fully\nunknown. Meanwhile, however, the theory of magnetic\nexcitations has been well advanced [2, 3, 19–28]. In or-\nder to verify the validity of the theoretical approaches,\nthe authors had to compare their results to the only one\navailable set of the experimental data, which covers a\nvery small fraction of the Brillouin zone [13–15].\nThe magnons and the fundamental magnetic interac-\ntions in the Ni monolayer are also of great interest in the\ncontext of unconventional topological superconductivity\nin the Bi/Ni bilayer, where the interfacial Ni magnons\nare proposed to be responsible for the superconducting\npairing mechanism being of dxy±idx2+y2character [29].\nBesides their importance for the fundamental un-\nderstanding of the physics of low-dimensional solids,\nmagnons are also of great interest to the field of magnon-\nics, where the idea is to use the magnons as information\ncarriers [30–33].\nIn this Letter by performing spin-polarized high resolu-\ntion electron energy-loss spectroscopy (SPHREELS) ex-\nperiments on specifically designing multilayer structures\nwe probe, for the first time, the full dispersion relation of\nthe magnon mode, which is partially localized in the Ni\nmonolayer. We show that the Ni magnons are rather stiff\nand show a notable dispersion relation up to the surfacearXiv:2402.15251v1  [cond-mat.str-el]  23 Feb 20242\n0.20.40.60.81.0\n0100 200 300 400 50002468\nElectron Energy Loss (meV)Intensity (kCounts/s) I\n I\n II\n Difference (kCounts/s)\nIr(001)𝐸𝑖, 𝐤𝐢\n𝐸𝑓, 𝐤𝐟\n𝑖𝑓\nM\nNi\n0 100 200 300 4000.25 Å-10.40 Å-10.50 Å-10.55 Å-10.60 Å-10.65 Å-10.70 Å-10.75 Å-10.80 Å-1Difference Intensity (arbitrary units)\nElectron Energy Loss (meV)|q|||= 0.90 Å-1(a) (b)\nFIG. 1. (a) Typical SPHREELS spectra recorded on the\nCo/Ni/Ir(001) structure at |q∥|= 0.7˚A−1. The spectra\nwere recorded at the incident energy of Ei= 10 eV and at\nroom temperature. The red and blue spectra, denoted by\nI↓andI↑, were recorded with the spin polarization vector\nof the incident electron beam being parallel and antiparallel\nto the magnetization M, respectively. The difference spec-\ntrum I↓−I↑is shown by the green color. The scattering\ngeometry is schematically illustrated in the inset. The energy\nand wavevector of the incident (scattered) beam are shown\nbyEiandki(Efandkf), respectively. (b) A series of differ-\nence spectra recorded for different magnon wavevectors |q∥|\nranging from 0.25 to 0.90 ˚A−1. The excitation energy (peak\nposition) is marked by the black circles.\nBZ boundary. This is in sharp contrast to the results of\nthe Fe monolayer on the same substrate or on W(110) and\nPd(001), where the magnons are found to be rather soft.\nComparing the results to those of first-principles calcu-\nlations we shall comment on the observed large magnon\nstiffness constant. Moreover, we will provide the complex\npattern of HEI in the Ni monolayer.\nWe have shown earlier that in multilayer structures the\nspatial localization of the magnons depends on the pat-\ntern of Jij[9, 34]. Under some circumstances one may\nselectively excite magnons which are localized either pre-\ndominantly at the surface or interface, or almost equally\nat both. Moreover, previous investigations have revealed\nthat the excitation cross-section of magnons with elec-\ntrons is the highest when the sample surface is composed\nof Co atoms [34–40]. Hence, in order to substantially en-\nhance the magnon excitation cross-section we cover the\nNi monolayer with one and two atomic layers of Co and\ndesigned the following multilayers: Co/Ni/Ir(001) and\n2Co/Ni/Ir(001). A similar idea has recently been used\nto probe the standing spin waves with a parallel momen-\ntumq∥<0.3˚A−1in Co/Ni multiyers composed of several\natomic layers [41, 42].\nAll the sample preparation and magnon spectroscopy\nexperiments were performed under ultrahigh vacuum\nconditions. We first examine the Co/Ni/Ir(001) epitax-\nial system. The surface of Ir(001) was cleaned using the\n0.00.10.20.30.40.5Magnon Excitation Energy (eV)തΓഥXഥM തΓതΓഥXഥM\nതΓഥXഥM തΓ(a) (b)\nCo\nNi\nIr\nCo\nCo\nNi\nIrFIG. 2. The magnon dispersion relation of (a) Co/Ni/Ir(001)\nand (b) 2Co/Ni/Ir(001). The calculated magnon dispersion\nrelation is shown for two cases: without (light-blue) and with\n(dark-blue) considering the reconstruction of the Ir surface.\nThe lower insets show a schematic representation of the struc-\ntures. The upper inset in (a) shows the surface BZ.\nstandard cleaning procedure, described in details in Refs.\n[34, 43, 44]. The procedure leads to a well-ordered (1 ×5)\nreconstructed surface. The Ni and Co monolayers were\nepitaxially grown by molecular beam epitaxy at room\ntemperature. Figure 1(a) shows typical SPHREELS\nspectra recorded on the Co/Ni/Ir(001) structure at a\nwavevector of |q∥|=|∆k∥|=|ki∥−kf∥|= 0.7˚A−1,\nwhere ki∥andkf∥denote the parallel momentum of the\nincoming and outgoing beam, respectively [see the inset\nof Fig. 1(a)]. The spectra were recorded for the two pos-\nsible spin orientations of the incoming electron beam, i.e.,\nparallel ( I↓, red color) and antiparallel ( I↑, blue) to the\nsample magnetization M. Due to the conservation of the\ntotal angular momentum during the scattering event, the\nmagnons are only excited by incidence of minority elec-\ntrons (electrons with their spin parallel to M). Hence,\nthe difference spectrum I↓−I↑includes all the infor-\nmation regarding the magnons excitation energy (or fre-\nquency) [45, 46] and lifetime [47–49]. In this experiment\nMwas parallel to the [ 110]-direction and the magnon\nwavevector q∥was along the [110]-direction. This corre-\nsponds to the ¯Γ–¯X of the surface BZ. The magnon disper-\nsion relation was probed along this symmetry direction.\nDifferent magnon wavevectors were achieved by chang-\ning the scattering angles θiandθf. A series of differ-\nence spectra recorded for different values of |q∥|ranging\nfrom 0.25 to 0.90 ˚A−1is shown in Fig. 1(b). The re-\nsults clearly indicate that the measured magnon mode\nexhibits a rather stiff dispersion, similar to the stand-\ning spin waves in thick ferromagnetic layers [50]. The\nresulting magnon dispersion relation is summarized in\nFig. 2(a), representing the the acoustic magnon mode of\nthe system. In order to experimentally verify the par-\ntial localization of this mode in the Ni monolayer and to3\nensure that the dispersion relation is governed by the ex-\nchange parameters in both Ni and Co layer, we added an-\nother Co monolayer on top of the structure and probed,\nonce more, the magnon dispersion relation. The results\nof the 2Co/Ni/Ir(001) multilayer are shown in Fig. 2(b).\nThe very similar dispersion relation for the two systems is\nan indication that the probed magnon mode is partially\nlocalized in the Ni monolayer. However, comparing the\nresults to those of Fe monolayer on W(110) [7, 51–53] and\nPd(001) [54, 55], or those of buried Fe monolayer on the\nsame substrate [34, 56], one realizes that in the present\ncase the magnon dispersion relation is very stiff. This is\nsurprising, since it has been found theoretically that both\nHEI and the Curie-temperature of Ni are much smaller\nthan those of Fe and Co [2, 3, 19, 27].\nIn order to quantify the strength of HEI and to unravel\nthe origin of the probed magnon mode we resort to the\nfirst-principles calculations of the magnetic exchange pa-\nrameters in both Co/Ni/Ir(001) and 2Co/Ni/Ir(001) sys-\ntems. Our first-principles calculations are based on den-\nsity functional theory and the fully relativistic Korringa-\nKohn-Rostoker electronic structure method. In this ap-\nproach the values of Jijare computed within the frame-\nwork of the magnetic force theorem [57]. The experimen-\ntal lattice parameters were used as the input of the first-\nprinciples calculations. The calculations provide most\nof the magnetic parameters of the systems including the\nmatrix of Jij. The magnon dispersion relation was then\ncomputed based on the Jij-matrix. The results of calcu-\nlations for the two systems are summarized in Fig. 2. The\ncalculated magnon dispersion relation agrees well with\nthe experimental results. Moreover, similar to the exper-\niment the calculations indicate that the acoustic magnon\nmode of the two systems is very similar. To unravel the\norigin of different magnon modes we also calculated the\nmagnon Bloch spectral function (BSF) and the magnon\ndensity of states (DOS). When these quantities are pro-\njected onto different layers they would provide an access\nto the spatial localization of different magnon modes. In\nFig. 3 we provide the magnon BSF and DOS for the\nCo/Ni/Ir(001) structure (the results of 2Co/Ni/I(001)\nare presented in Supplemental Figure S1 [58]). The pro-\njected BSF of the two magnon modes, shown in Fig. 3,\nindicates that the spectral weight of the acoustic magnon\nmode near the high symmetry ¯X and ¯M points as well as\nalong the ¯X–¯M path is the highest when magnon BSF\nis projected onto the Ni monolayer. This means that in\nthis part of BZ this mode is almost entirely localized in\nthe Ni layer. Likewise, the partial magnon DOS of the Ni\nmonolayer [Fig.3(b)] is maximum in this range of energy.\nWhile above a magnon energy of about 250 meV both\nthe magnon BSF and DOS are zero, the low energy part\nof the magnon spectrum shows a finite spectral weight\nin the Ni monolayer. The projected magnon BSF and\nDOS onto the Co layer indicate that except for the high-\nsymmetry ¯X and ¯M points and along the ¯X–¯M direction,\n0.00.10.20.30.4Magnon Excitation Energy (eV)\n MDOS\nതΓഥXഥMതΓ(a) (b)\nതΓ ഥXഥM തΓ(c)1\n0\n MDOS(d)\n2FIG. 3. The magnon Bloch spectral function (BSF) and the\nmagnon density of states (DOS) for the Co/Ni/Ir(001) sys-\ntem, projected onto the Ni monolayer [(a) and (b)] and onto\nthe Co layer [(c) and (d)]. For the sake of clarity, the data in\n(b) are multiplied by a factor of 2.\nthe magnons have a considerable spectral weight in the\nCo layer. The main conclusion of the results presented in\nFig. 3 is that the experimentally probed magnon mode\ndescribes magnons, which have a finite spectral weight in\nboth layers. The larger spectral weight of this mode in\nthe Co layer makes it easily accessible to the electrons as\nprobing particles. Hence, this mode is very efficiently ex-\ncited in this system. A similar conclusion can be drawn\nfor the 2Co/Ni/Ir(001) system [58].\nOur first-principles calculations indicate that the near-\nest neighbor intralayer and interlayer interaction with\n(without) considering the surface reconstruction are only\nJN∥= 1.03 (1.24) meV and JN⊥= 5.71 (6.48) meV,\nrespectively. Likewise, the interaction between spins lo-\ncated at larger distances are also rather weak. The sec-\nond nearest neighbor intralayer and interlayer interaction\nwith (without) considering the surface reconstruction are\nonly JNN∥= 0.164 (0.29) meV and JNN⊥= 0.24 (0.2)\nmeV, respectively. The values of Jijand the magnetic\nmoments are provided in Tab. I. The fact that the HEI\nin Ni is weak is in agreement with the calculations for\nbulk fcc Ni [19, 21–23, 25–27]. Note that due to the lack\nof the full magnon spectrum, the exchnage parameters\nin Ni are not known experimentally. The main conclu-\nsion of the results presented in Tab. I is that the ex-\nchange parameters in the Ni monolayer are rather small.\nMoreover, the pattern of the exchange interaction is com-\nplex and includes both ferromagnetic (positive) and an-\ntiferromagnetic (negative) exchange constants. Owing to\nthe weak HEI in the Ni monolayer one would expect a\nlarger magnon BSF and DOS of the acoustic magnon\nmode in this layer. However, it is important to consider\nthat the magnetic moment of Ni is by a factor of about\n3.5 smaller than that of the Co. This means that the dy-\nnamic component of the magnetization (which somewhat\nscales with the static magnetic moment) is smaller in the\nNi layer. Hence, one observes a larger magnon BSF and\nDOS when they are projected onto the Co layer. Note4\nTABLE I. The magnetic exchange parameters (in meV) and the magnetic moments (in µB) as calculated by our first-principles\ncalculations. Co 1and Co 2refer to the first and second Co atomic layer on top of the Ni monolayer. For the sake of simplicity\nonly the HEI in the Ni monolayer up to the third nearest neighbors (NNN) are presented.\nMultilayer system JNiNi\nN∥JNiCo 1\nN⊥ JNiNi\nNN∥JNiNi\nNNN∥JNiCo 1\nNN⊥JNiCo 1\nNNN⊥JNiCo 2\nN⊥ JNiCo 2\nNN⊥JNiCo 2\nNNN⊥µNiµCo1µCo2\nCo/Ni/Ir(001)–1 ×1 1.24 6.48 0.29 -0.047 0.2 -0.22 – – – 0.52 1.98 –\nCo/Ni/Ir(001)–1 ×5 1.03 5.71 0.164 0.002 0.24 -0.25 – – – 0.55 1.95 –\n2Co/Ni/Ir(001)–1 ×1 2.75 6.28 -0.001 0.275 0.6 -0.21 -1.13 0.11 -1.06 0.57 1.84 1.87\n2Co/Ni/Ir(001)–1 ×5 1.84 4.93 0.008 0.178 0.55 -0.21 -0.69 0.24 -0.79 0.56 1.81 1.88\nthat the magnon BSF and DOS exhibit finite values in\nthe Ni monolayer. The large magnon BSF and DOS in\nthe Co layer helps to easily excite this magnon mode.\nNear the ¯Γ–point the magnon dispersion relation may\nbe approximated by E(q∥)≃Dq2\n∥, where Ddenotes the\nmagnon exchange stiffness constant. Fitting the data for\nq∥<0.4˚A−1, we find a value of 356 ±5 meV ˚A2, in\nreasonable agreement with the experimental [13–15] and\ntheoretical [19, 21–23, 25–27] values of the bulk Ni. Due\nto the weak HEI in the Ni monolayer, this is somewhat\nsurprising. In the following we shed light on the origin\nof the stiff magnon mode in the Ni monolayer.\nWithin the adiabatic formalism (linear spin-wave the-\nory) the magnon dispersion relation is given by E(q) =\n2gµB\nµiΣj̸=0J0,j[1−expi(q·R0j)], where g= 2 is the g-\nfactor, µBis the Bohr magneton, µiis the magnetic mo-\nment of the origin site, R0jrepresents the displacement\nvector of site jwith respect to the origin and qdenotes\nthe three dimensional vector of the magnon momentum.\nA careful investigation of the electronic DOS reveals that\nthe Ni atoms possess a low spin density. A low spin den-\nsity leads to both a weak exchange interaction as well as\na small magnetic moment [0.55 (0.52) µBfor the recon-\nstructed (unreconstructed) Ir(001)]. The fact that the\nmagnetic moment of Ni is much smaller than that of the\nCo has experimentally been verified by probing both bulk\nsamples [59] as well Co/Ni multilayers [60–62]. Since the\nexchnage stiffens scales inversely with µi, a weak HEI in\nNi can lead to a large magnon exchange stiffness, com-\nparable to that of the Co and Fe. Hence, assuming the\nsame exchange constants for Ni and Co is not valid, even\nthough for the small magnon momentum the dispersion\nrelation of the two systems might be very similar [41, 42].\nRecently, the emergence of topological superconductiv-\nity in the Bi/Ni bilayer is attributed to the magnetic ex-\ncitations at the interface and their coupling to the surface\nstate electrons [29]. We anticipate that our quantitative\nresults on the values of HEI in the Ni monolayer would\ncontribute to a quantitative understanding of supercon-\nducting mechanism in this system. Moreover, they would\nprovide guidelines for tuning superconductivity in similar\nbilayer structures in which the magnons play a decisive\nrole in the pairing mechanism [63, 64].\nIn conclusion, we prepared atomically architectured\nmultilayers composed of Ni and Co epitaxial monolay-ers on Ir(001). Owing to the large magnetic moment of\nthe surface Co monolayer and the fact that the excita-\ntion cross-section at Co surfaces is high, the magnons\nin such designed multilayers can be very efficiently ex-\ncited by spin-polarized electron scattering experiments,\ne.g., SPHREELS. The acoustic magnon mode of these\nsystems was found to be localized in both the Co as well\nas in the Ni monolayers. The full experimental acous-\ntic magnon dispersion relation, probed up to the sur-\nface BZ, enabled us to quantitatively resolve the com-\nplex pattern of HEI in the Ni monolayer. We observed a\nrather weak exchange coupling within the Ni monolayer,\neven though the magnon exchange stiffness is rather large\n(much larger than that of an Fe monolayer on various\nsubstrates). The large magnon stiffness constant is a con-\nsequence of the low spin density of Ni atoms, as confirmed\nby our first-principles calculations. In addition to the\nfact that our results resolve the long-standing question\nregarding the quantitative values of HEI in the Ni mono-\nlayer, they are also of the interest to the field of topolog-\nical superconductivity in magnetic/topological materials\nheterostructures.\nACKNOWLEDGMENTS\nFinancial support by the Deutsche Forschungsgemein-\nschaft (DFG) through the DFG Grants No. ZA 902/7-1\nand No. ZA 902/8-1 is acknowledged. 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In GM in particular, the use of reverse-time diffusion\nprocesses depending on the log-densities of Ornstein-Uhlenbeck forward processes are a popular\nsampling tool. In [5] the authors point out that these log-densities can be obtained by solution of a\nHamilton-Jacobi-Bellman (HJB) equation known from stochastic optimal control. While this HJB\nequation is usually treated with indirect methods such as policy iteration and unsupervised training\nof black-box architectures like Neural Networks, we propose instead to solve the HJB equation by\ndirect time integration, using compressed polynomials represented in the Tensor Train (TT) format\nfor spatial discretization. Crucially, this method is sample-free, agnostic to normalization constants\nand can avoid the curse of dimensionality due to the TT compression. We provide a complete deriva-\ntion of the HJB equation’s action on Tensor Train polynomials and demonstrate the performance of\nthe proposed time-step-, rank- and degree-adaptive integration method on a nonlinear sampling task\nin 20 dimensions.\n1 Introduction and related work\nConsider the problem of sampling from a probability measure µ∗onRd,d∈N, with Lebesgue-density\nπ∗(y) =1\nZexp(−Φ(y)), (1.1)\nwhere Φ:Rd→Ris a sufficiently regular function called the potential andZ∈(0,∞)is a normalization constant\nsuch thatR\nRdπ∗(y)dy= 1. Throughout this manuscript, we assume that Φis known and can be evaluated, while the\nnormalization constant Zis unknown and difficult or even impossible to compute. Over time, a myriad of different\nsampling methods have been devised, including Markov Chain Monte Carlo (MCMC) methods [35, 6, 34], methods\nbased on Stein variational gradient descent [24], Langevin dynamics [36, 14, 15, 32, 7, 12], or Langevin dynamics\n∗equal contributionarXiv:2402.15285v1  [stat.ML]  23 Feb 2024GM with TT approximations of HJB equations A P REPRINT\npreconditioned with measure transport [41] to name just a few. In the last few years, interacting particle systems have\nreceived a lot of attention [16, 14, 15, 32, 7, 12]. An important application where one aims to sample from densities\nof the form (1.1) stems from solutions of inverse problems via Bayesian inference [40].\nSince our approach is linked to (interacting particle-) Langevin samplers, we take a moment to review these methods\nin more detail. All methods proposed in [14, 15, 32, 7] work with an It ˆo diffusion process of the form\ndXt=f(Xt)dt+g(Xt)dWt, (1.2)\nwhere Wis a standard Brownian motion with appropriate dimension, fis the drift and gis the diffusion. Under\ncertain assumptions on the potential, e.g. (strong) convexity, convexity at infinity, regularity and growth conditions,\nthis process is ergodic [42, 15] and admits either µ∗(in the case of a single particle process) or ⊗B\ni=1µ∗(in the case of\na system of B∈Ninteracting particles) as an invariant measure.\nSamples from µ∗are obtained by propagating an initial batch of arbitrarily distributed samples through the process\n(1.2) for infinite time. In the classical overdamped Langevin dynamics, the drift term fis given by the negative gradient\n−∇Φof the potential. In state-of-the-art interacting particle methods like the Affine Invariant Langevin Dynamics\n(ALDI) [14], this drift is modified by a reversible perturbation of the underlying process (see e.g. [41, Equation 2.4]\nfor a general definition of reversible perturbations and [15, Definition 3.1] for the specific perturbation of ALDI).\nReversible perturbations can increase convergence speed [33], while ensuring that the perturbed SDE maintains the\nsame invariant measure as the unperturbed system and is still time-reversible. Even if the resulting system is time-\nreversible, the reverse-time process is not considered in those works, since the forward process (1.2) admits µ∗as\ninvariant measure.\nWhile the time-homogeneous drift term of (1.2) makes these methods conceptually simple, it comes with a potential\ndownside with regard to the class of measures µ∗that can be approximated. ALDI comes with theoretical convergence\nguarantees only in the case of a potential with Gaussian tails outside of a compact set [14]. In [12], the authors propose\nusing a time-inhomogeneous process\ndXt=f(t, Xt)dt+g(Xt)dWt, (1.3)\nwhere f(t,·)is defined by gradients of log-densities of intermediate measures defined upon time dependent interpo-\nlation, e.g. a convex combination of the target potential Φand a simpler auxiliary potential. By the choice of the\nauxiliary measure, the flow towards the target distribution is fixed. While this so called homotopy -approach can sub-\nstantially increase convergence speed in practice, e.g. to sample from multimodal target distributions, the choice of\nauxiliary measures allowing for optimal flows remains an open question.\nContrary, reverse-time diffusion processes offer a principled way of defining a process of the form (1.3), which can\nbe used to sample from µ∗. The key observation, dating back to [1], is that the reverse-time process corresponding to\n(1.3) defines again a diffusion process of the form (1.3). For some years, this property has been used in what is now\ncalled Diffusion Generative Modelling [37, 18, 39]. In contrast to Bayesian inference, where µ∗is known but difficult\nto sample from, the goal here is to generate new samples from some completely unknown data distribution of which\na finite number of samples {xi}D\ni=1,D∈N, are available. The central idea is to use an Ornstein-Uhlenbeck process\nmapping any distribution to a standard-normal distribution N(0, Id)fort→ ∞ and then, by using the available\nsamples {xi}D\ni=1, learning the drift of the reverse-time process, mapping N(0, Id)back to the data distribution [39].\nMore specifically, the gradient-log-density or score of the Ornstein-Uhlenbeck process is learned by minimizing a\nscore-matching objective function [22, 38], which is essentially a weighted time-average of mean-squared errors (see\ne.g. [39, Equation 7]). The score determines the reverse process. Once the score is known, new samples from the data\ndistribution can be obtained by sampling from the standard-normal distribution and propagating the samples through\nthe reverse process. However, classical score-matching relies on the samples {xi}D\ni=1of the data distribution, which\nare usually not available in a Bayesian setting. Hence, we consider an alternative approach.\nThe authors of [5] point out that the negative log-density of a reverse-time diffusion process satisfies a Hamilton-\nJacobi-Bellman (HJB) equation. Since the score is invariant under additive constants to the log-density, it suffices\nto solve this HJB equation up to an additive constant to obtain the correct score. In particular, the normalization\nconstant of the target density need not be known. Hence, solving the corresponding HJB equation is a viable method\nof obtaining the score in a Bayesian setting.\nTensor Trains [29] have been used successfully in several works on approximations of HJB equations for nonlinear\noptimal control, see e.g. [31, 11] and references therein. In [31] the solution of the deterministic finite horizon HJB\nequation is obtained by a combination of Monte-Carlo (MC) sampling and policy iteration. While this approach\nis appealing due to its model-free nature, the policy iteration requires the solution of multiple nonlinear optimization\nproblems at each time step. Furthermore, MC sampling may lead to slow convergence. In [11] a spectral discretization\nis used, circumventing the slow convergence rate of MC sampling and achieving algebraic convergence for a class of\ndeterministic infinite horizon optimal control problems. In contrast to these works, we propose a method not reliant on\n2GM with TT approximations of HJB equations A P REPRINT\npolicy iteration. Furthermore, no nonlinear optimization has to be performed except at the initial time point. Instead,\nthe HJB right-hand side is discretized by orthogonal projection onto polynomial space, resulting in an ODE in tensor\nspace. Subsequently, this ODE is integrated using methods for time-integration of Tensor Trains.\n1.1 Contribution and Outline\nThe main contribution of this work lies in providing an interpretable solver based on compressed polynomials for the\nreverse-time HJB equation as it appears in the context of Generative Modelling and Bayesian Inference. Specifically,\nwe integrate the HJB equation using orthogonal projections of the right-hand side and rank-retractions onto a smooth\nmanifold within polynomial space defined by Tensor Trains of a fixed rank. The solver adaptively chooses its stepsize\nbased on current projection- and retraction-errors as well as the local stiffness, which is estimated by local lineariza-\ntions of the HJB. This approach is sample-free and agnostic to normalization constants and can therefore be used in a\nBayesian setting. We demonstrate the performance of the solver on a nonlinear test case in d= 20 dimensions.\nThe outline of the rest of the paper is as follows.\n• Section 2 covers the relevant theory of diffusion processes necessary to construct a process of the form (1.3),\nwhich can be used to sample from µ∗. In particular, Remark 2.1 offers one such form as a reverse-time\nOrnstein-Uhlenbek process. The corresponding reverse-time HJB equation determining the score of this\nprocess is given in (2.5).\n• In Section 3 we introduce our approximation class for the log-densities, namely functional Tensor Trains with\northogonal polynomial ansatz functions. A motivation for this ansatz class can be found in Appendix C.This\nsection further introduces all algebraic operations on tensor space necessary to solve a projected version of\nthe HJB equation.\n• Section 4 is the main part of the paper, where we are concerned with the solution of the HJB. We state\nthe equivalence of the HJB projected onto polynomial space of fixed degree with an ODE in tensor space\n(Theorem 4.1). Furthermore, we give a precise version of the proposed solution algorithm (Algorithm 2).\n• Finally, the performance of the solver is demonstrated on a Gaussian test case as well as a 20-dimensional\nnonlinear potential in Section 5.\n2 Reverse-time diffusion processes and HJB equation\nLet the terminal time T >0and a d-dimensional Ornstein-Uhlenbeck process (Xt)t∈[0,T]be defined by\ndXt=−Xtdt+√\n2dWt, X 0∼µ∗, (2.1)\nwhere Wtdenotes standard d-dimensional Brownian motion. The probability density function πtof this process\nsatisfies the Fokker-Planck equation\n∂tπt= ∆πt+x· ∇πt+dπt, π 0=π∗, (2.2)\nfort∈[0, T]. Since the (standard normal) invariant measure of (2.1) satisfies a log-Soboloev inequality, the cor-\nresponding law µXtof (2.1) converges exponentially in Kullback-Leibler divergence (KL) to the standard normal\ndistribution N(0, Id)onRd[27], i.e.\nKL(µXt||N(0, Id))≤e−2tKL(µ∗||N(0, Id)). (2.3)\nHence, for sufficiently large T, the measure µXTwill be close to a standard normal distribution in KL divergence. The\nfollowing remark provides a reverse-time process (Yt)t∈[0,T]withY0∼µXTandYT∼µ∗.\nRemark 2.1 (Reverse-time Ornstein-Uhlenbeck process) .Let(Xt)t∈[0,T]be defined by (2.1) . Then, for any λ∈[0,1]\nthe process (Yt)t∈[0,T]defined by\ndYt= [Yt+ (2−λ)∇logπT−t(Yt)] dt+»\n2(1−λ)dWt, Y 0∼µXT (2.4)\nsatisfies µYt=µXT−tand in particular µYT=µ∗. This result is an immediate consequence of [21, Appendix G],\nwhich covers a much wider range of diffusion processes. The most common choices for λareλ= 0, used for the\nreverse process e.g. in [39], and λ= 1, which leads to a reverse ODE known as probability flow ODE [39].\nTo formulate the reverse process (Yt)t∈[0,T]we need the score ∇logπt. If a sufficient number of samples of µ∗are\navailable, we can apply score matching techniques (see [37, 39] and references therein). Lacking these samples, we\n3GM with TT approximations of HJB equations A P REPRINT\ncould try to obtain πtby solving (2.2), but the fact that πtneeds to be a density for every tmakes this approach\ncumbersome for approximation methods. Instead, we apply a Hopf-Cole transformation vt:=−logπtto (2.2). A\nshort calculation by product and chain rule (see Appendix A) yields that vtsatisfies the PDE\n∂tvt= ∆vt+x· ∇vt− ∥∇ vt∥2\n2−d, v 0=−logπ∗, (2.5)\nfort∈[0, T]. This nonlinear PDE is the time-reverse of a HJB equation appearing in finite-horizon stochastic\noptimal control. As [5] pointed out, we can now apply techniques from optimal control to approximate the score. A\nstraightforward way is to approximately solve the HJB equation (2.5) with some suitable class of functions such as\nNeural Networks [44, 4, 28]. Instead of this black-box approach, we propose solving (2.5) by means of compressed\npolynomials represented by a low-rank tensor format, the details of which are provided in the next section. In contrast\nto Neural Networks, this approach is highly interpretable and utilizes the structure of the HJB equation. In particular,\nwe make frequent use of the fact that the right-hand side F(v):= ∆v+x· ∇v− ∥∇ v∥2\n2−dof (2.5) can be split into\na constant, linear and nonlinear contribution, given by\nConst( v) =d, (2.6)\nLin(v) = ∆ v+x· ∇v, (2.7)\nNonLin( v) =−∥∇v∥2. (2.8)\nBefore going into the detail about the polynomial approximation in the following section, we briefly sketch some of\nthe core ideas.\nFirst, we note the constant term (2.6) can be dropped from (2.5) since the score is agnostic to constant shifts of the\nlog-density. More precisely, vtis a solution to (2.5) if and only if vt:=vt+tdis a solution to ∂tvt= ∆vt+x·∇vt−\n∥∇vt∥2\n2,v0=−logπ∗,t∈[0, T]. The two solutions vtandvtdiffer only by a constant shift for every t∈[0, t],\nhence the score satisfies ∇logπt=−∇vt=−∇vt. By the same reasoning, an arbitrary constant can be added to\nthe initial condition of (2.5) without affecting the score. By choosing this constant equal to −log(Z), we achieve\n−logπ∗−log(Z) = Φ . Thus, from now on we consider the equation\n∂tvt= Lin( vt) + NonLin( vt), v 0= Φ. (2.9)\nMorevover, if vtis a polynomial of fixed degree N∈Nfor any t, then Lin(vt)is also a polynomial of degree N. This\nmeans that if v0is a polynomial of a fixed degree, integrating only the linear part of (2.9) would yield a polynomial of\nsame degree for all t∈[0, T]. For the nonlinear part NonLin( vt)this is only true for N= 2. In this quadratic case,\n(2.9) can be solved to arbitrary accuracy. In particular, if µ∗is a zero mean Gaussian with density\nπ∗(x) =1p\n(2π)d|Σ|e−x⊺Σ−1x(2.10)\nfor positive definite Σ∈Rd,d, then (2.9) corresponds to the HJB equation of a linear-quadratic optimal control problem\nwith solution given by vt(x) =x⊺Ptx, where Pt∈Rd,d,t∈[0, T], solves a Ricatti matrix differential equation(see\nAppendix B).\nSolving (2.9) e.g. with an explicit Euler method for time discretization leads to a steady increase of the degree over\ntime for all initial degrees larger than N= 2. This is due to the nonlinear term: if vtfor some t∈[0, T]is a\npolynomial of degree N, then NonLin( vt)is (in general) a polynomial of degree ≤2N. To prevent this degree\nincrease, we project the nonlinear part of the right-hand side back onto the space spanned by polynomials of degree N\nbefore performing the time integration step. Furthermore, since the linear space of polynomials suffers from the curse\nof dimensionality, we use a compression orretraction after every time step, finding a best approximation of the new\niterate in a low dimensional manifold. In the case of an explicit Euler method, the resulting integration scheme can be\nwritten as\nvt+τt= Compression [ vt+τt(Lin( vt) + Projection [NonLin( vt)])], (2.11)\nwhere τt>0is the current adaptively chosen stepsize. The precise definition of all terms involved is the subject of\nthe next section.\n3 Functional Tensor Trains (FTT) and Tensor Trains (TT)\nIn this section we introduce the approximation class used as a spatial discretization for the HJB equation. Let K⊂Rd\nbe a compact hypercube defined by ai, bi∈Rwithai< bifori= 1, . . . , d andK=×d\ni=1[ai, bi]. A function\n4GM with TT approximations of HJB equations A P REPRINT\nn∈Nd\n0 dimension array n= (n1, . . . , n d)\nkn+l (kn1+l, . . . , kn d+l)fork, l∈N0\n[n] indexing [n] =×d\ni=1{0, . . . , n i}\nn1≥n2,n≥kcomponent wise comparison n,n1,n2∈Nd,k∈N\nα,β,γ multiindex in Nd\n0, note that we always index starting from 0\nRntensor space Rn1,...,n d\nA,B,C tensor elements in Rn\nr rankr= (r1, . . . , r d−1)inNd−1\nr1r2multiplication r1r2= (r1\n1r2\n1, . . . , r1\nd−1r2\nd−1)inNd−1\nki, li rank enumeration indices in {1, . . . , r i}\nAi, Bi, Ci component order 3tensor in Rri−1,ni+1,riwith entries indexed by [ki−1, αi, ki]\nAi[αi] matrix extraction Ai[αi] =Ai[ :, αi,: ]∈Rri−1,riof component tensor Ai\nAi[ki−1,:, ki] vector extraction in Rni+1for each rank enumeration ki−1, ki\nA[α] tensor indexing A[α1, . . . , α d]forA∈Rn,α∈[n],n∈Nd\n0\nTable 1: List of compact notations used in this work.\nf:K→Ris said to have functional Tensor Train (FTT) [30] rank r= (r1, . . . , rd−1)∈Nd−1with the convention\nr0=rd= 1, if it can be written as\nf(x1, . . . , x d) =F1(x1)F2(x2)···Fd(xd) (3.1)\nwith matrix valued functions Fi(xi)∈Rri−1,ri,xi∈[ai, bi]fori= 1, . . . , d . For discussions regarding the approxi-\nmation of functions of mixed regularity or compositional structures we refer to [3, 17, 2].\nIn order to obtain a discrete approximation class, for each i= 1, . . . , d andα∈N0letpi\nαdenote the α-th orthonormal\nLegendre polynomial with respect to the standard L2inner product on [ai, bi]. Forn∈Nd\n0, we define the discrete set\nof orthonormal polynomials of degree nby\nΠn:={pα:=dO\ni=1pi\nαi|α∈[n]}, (3.2)\nwhere [n]is defined as in Table 1. For fwith FTT rank r, we then may approximate\nf(x1, . . . , x d)≈X\nα∈[n]C[α]pα(x1, . . . , x d), (3.3)\nwith a tensor array C∈Rn+1with Tensor Train (TT) rank r= (r1, . . . , r d−1)⊺∈Nd−1bounded by the FTT rank r.\nIn particular we have the decomposition into a Tensor Train (or Matrix Product State) format\nC[α] =C1[α1]C2[α2]···Cd[αd], (3.4)\nwith matrices Ci[αi]∈Rri−1,riand the convention that r0=rd= 1. Note that the relation of randrdepends\non the relation of Fiand the polynomials in i-th direction. In particular it holds r=rif for all i= 1, . . . , d and\nα= 0, . . . , n iit holds\nbiZ\naiFi(xi)pi\nα(xi)dxi̸=0∈Rri−1,ri.\nProvided that the ranks can be bounded, the TT format exhibits a storage complexity of\nO(max( n1, . . . , n d)dmax( r1, . . . , r d−1)2), which scales only linearly in the dimension d, hence avoiding the curse\nof dimensionality. The set of such Tensor Trains of fixed rank rdefines a manifold Mr⊂Rn+1, see e.g. [20].\nAs a first step of our HJB solver, we propose to approximate V0=−logπ∗in a functional Tensor Train format based\non orthogonal polynomial space discretization as in (3.3) for some TT rank r∈Nd−1. A motivation for this type\n5GM with TT approximations of HJB equations A P REPRINT\nof approximation for Bayesian posteriors can be found in Appendix C. In what follows we discuss the actions of the\nlinear and nonlinear operators defined in (2.7) and (2.8) on functions given in that format. To that end, we define for\nany tensor A∈Rn+1the associated polynomial vA∈span Π nby\nvA=X\nα∈[n]A[α]pα. (3.5)\n3.1 The linear part\nThis section is concerned with the operator Linfrom (2.7), appearing in the right-hand side of the HJB in (2.9).\nLet the differential operator D:C2(R)→ C(R)be defined as Dv=∂2\nxv+x∂xvforv∈ C2(R)and let I:C2(R)→\nC2(R)denote the identity operator. Then, it holds\nLin = D ⊗ I ⊗ . . .⊗ I+I ⊗ D ⊗ I ⊗ . . .⊗ I+. . .+I ⊗. . .⊗ I ⊗ D . (3.6)\nAs a first result we discuss the effect of the operator on functions vgiven in FTT format.\nLemma 3.1. Letf∈ C2(K)have FTT-rank r∈Nd−1. Then, Lin(f)has FTT-rank at most 2r.\nProof. The assertion follows immediately since Lin(f)defines a Laplace-like sum of FTTs, meaning that each sum-\nmand only modifies a single component of the FTT. More precisely, we have\nLin(f)(x) = [DF1(x1)F1(x1)]ïF2(x2) 0\nDF2(x2)F2(x2)ò\n···ïFd−1(xd−1) 0\nDFd−1(xd−1)Fd−1(xd−1)òïFd(xd)\nDFd(xd)ò\n,(3.7)\nwhich defines a product of matrix valued functions as in (3.1). The rank bound follows immediately from the block\nstructure of (3.7) and the dimensions of Fi,DFifori= 1, . . . , d .\nWhen applied to polynomials, the linear operator can be expressed in terms of its action on the polynomial’s coeffi-\ncients. More precisely, the discretization of Linon the finite set Πnfor some n= (n1, . . . , n d)⊺∈Nd\n0implies a\nlinear operator L:Rn+1→Rn+1given as\nL:=dX\ni=1Li,Li:=Ñ\ni−1O\nj=1Inj+1é\n⊗Di⊗Ñ\ndO\nj=i+1Inj+1é\n, (3.8)\nwith identity matrix In∈Rn,n. For the structure of the matrix Di∈Rni+1,ni+1we refer to Appendix E.1, specifically\nequation (E.5).For the moment it suffices to note that Digoverns the action of the differential operator Don the\ncoefficients of the polynomials in dimension i. It can be shown that the action of Linon a polynomial corresponds to\nalgebraic manipulation of the coefficient tensor with respect to L, which is the result of the following lemma.\nLemma 3.2. Letn∈Nd\n0,LinandLfrom (3.6) and(3.8) . Then, for A∈Rn+1we have\nLinvA=vLA. (3.9)\nProof. LetLi[β,α] :=ÄNi−1\nj=1Inj+1[βj, αj]ä\n⊗Dni[βi, αi]⊗ÄNd\nj=i+1Inj+1[βj, αj]ä\n. Then, the action of Lion\nAdefines a tensor Bigiven as Bi[β] =P\nα∈[n]Li[β,α]A[α].Moreover, LA=dP\ni=1Bi. Hence,\nLinvA=dX\ni=1X\nα∈[n]Ñ\ni−1O\nj=1Ié\n⊗ D ⊗Ñ\ndO\nj=i+1Ié\nA[α]p1\nα1⊗ ··· ⊗ pd\nαd\n=dX\ni=1X\nβ∈[n]X\nα∈[n]Li[β,α]A[α]p1\nα1⊗ ··· ⊗ pd\nαd\n=dX\ni=1X\nβ∈[n]Bi[β]p1\nβ1⊗ ··· ⊗ pd\nβd\n=vdP\ni=1Bi\n6GM with TT approximations of HJB equations A P REPRINT\nThe contraction LAis cumbersome for full tensors A. However, it is easy to implement if A∈ M ris a Tensor Train\nof fixed rank r∈Nd−1, such that A[α] =A1[α1]A2[α2]···Ad[αd]withAi[αi]∈Rri−1,rifori= 1, . . . , d . In this\ncase, let Di,Ai[βi] :=niP\nαi=0Di[βi, αi]Ai[αi]fori= 1, . . . , d . Then,\n(LA) [β] =dX\ni=1X\nα∈[n]Li[β, α]A[α]\n=dX\ni=1X\nα∈[n]Ñ\ni−1O\nj=1Inj+1[βj, αj]Aj[αj]é\n⊗Di[βi, αi]Ai[αi]⊗Ñ\ndO\nj=i+1Inj+1[βj, αj]Aj[αj]é\n=dX\ni=1Ñ\ni−1O\nj=1Aj[βj]é\n⊗Di,Ai[βi]⊗Ñ\ndO\nj=i+1Aj[βj]é\n= [D1,A1[β1]⊺A1[β1]⊺]ïA2[β2]⊺0\nD2,A2[β2]⊺A2[β2]⊺ò\n···ïAd−1[βd−1]⊺0\nDd−1,Ad−1[βd−1]⊺Ad−1[βd−1]⊺òïAd[βd]\nDd,Ad[βd]ò\n.\nHence, LA is given in TT format through a Laplace-like sum with TT rank bounded by 2r, which is consistent with\nLemma 3.1. In particular, this formula involves only contractions of the matrices Di∈Rni+1,ni+1with the order\nthree tensors Ai∈Rri−1,ni+1,rifori= 1, . . . , d . No contraction with the full tensor Ais required.\n3.2 The nonlinear part\nThis section is concerned with the operator NonLin( ·) =∥∇ · ∥2from (2.8), appearing in the right-hand side of the\nHJB equation in (2.9), in case the arguments are functions given in FTT format. This operator is a combination of\npartial derivatives, squares and a summation. We split the results into two Lemmas. First, we derive a more general\nbound on the FTT-rank of a product of functions with bounded FTT-rank.\nLemma 3.3. Letg,fhave FTT-rank rfandrg, respectively. Then g·fhas FTT-rank at most rgrf.\nProof. We write f(x) =F1(x1)·. . .·Fd(xd)andg(x) =G1(x1)·. . .·Gd(xd)withFi(xi)∈Rrf\ni−1,rf\ni,Gi(xi)∈\nRrg\ni−1,rg\nifori= 1, . . . , d . Let⊗kdenote the standard matrix-Kronecker product with the convention that for two\nscalar values a, b∈Rwe set a⊗kb=a·b. Then, we have\nf(x)g(x) =F1(x1)·. . .·Fd(xd)·G1(x1)·. . .·Gd(xd) =F1(x1)⊗kG1(x1)·. . .·Fd(xd)⊗kGd(xd),\nwhere Fi(xi)⊗kGi(xi)∈Rrf\ni−1rg\ni−1,rf\nirg\ni.\nSecond, the rank bound of the nonlinear right hand side is provided.\nLemma 3.4. Letfhave FTT-rank r, then NonLin( f) =∥∇f∥2=dP\ni=1Ä∂f\n∂xiä2has FTT-rank at most 2r2.\nProof. Note that∂f\n∂xi=F1(x1). . . ∂ xiFi(xi). . . F d(xd)has FTT-rank ≤rfor all i. By Lemma 3.3, (∂f\n∂xi)2has\nFTT-rank at most r2. To bound the FTT-rank ofPd\ni=1Ä∂f\n∂xiä2, the derivation is the same as in the proof of Lemma\n3.1, only that the operator Dis replaced by an operator mapping C1(R)→ C(R)andv7→(∂xv)2.\nWe now turn our view on the discretization of the corresponding operator with respect to Πn.\n3.2.1 The operator in Tensor Train format\nFor a practical algorithm, we need a discretization of NonLin on the finite set Πnsuch as (3.8) for the linear part.\nHere, we refrain from deriving a formula in the general setting of a full coefficient tensor and directly examine the\ncase of a TT with fixed rank. We consider the square operation first. Let n∈Ndand the multiplication operation\nMg:f7→g·f, where g, fare given by\nf(x) =vA(x) =X\nαA[α]dY\ni=1pi\nαi(xi), g(x) =vB(x) =X\nβ∈[n]B[β]dY\nj=1pj\nβj(xj) (3.10)\n7GM with TT approximations of HJB equations A P REPRINT\nwith tensors A,B∈Rn+1both given in Tensor Train format with TT-rank r= (r1, . . . , r d−1)∈Nd−1and\nA[α] =A1[α1]···Ad[αd],B[β] =B1[β1]···Bd[βd]. (3.11)\nWe aim to define a Tensor Train operator MB:Rn+1→R2n+1such that\nMg(f) =vMB(A)=X\nγ∈[2n]MB(A)[γ]dY\ni=1pi\nγi(xi). (3.12)\nForn∈N0letTi,n∈Rn+1,n+1denote the transformation matrix mapping the coefficients of Legendre polynomials\nup to degree non[ai, bi]to the corresponding coefficients of standard monomials 1, x, x2, . . .up to degree n. Let\nˆA[α] := ˆA1[α1]···ˆAd[αd],ˆAi[αi] =niX\nα′\ni=0Ti,ni[αi, α′\ni]Ai[α′\ni], (3.13)\nˆB[j] :=ˆB1[β1]···ˆBd[βd],ˆBi[βi] :=niX\nβ′\ni=0Ti,ni[βi, β′\ni]Bi[β′\ni], (3.14)\ndefine the coefficient tensors of fandgwith respect to monomials. Now, for i= 1. . . , d andαi= 0, . . . , n idefine\nthe matrix Di,αiby\nDi,αi:=\n0αi,ni+1\nIni+1,ni+1\n0ni+1−αi,ni+1\n∈R2ni+1,ni+1, (3.15)\nwhere 0m,n∈Rm,nis a matrix with all entries equal to 0, which we define to be empty if mornequal\nzero. Furthermore, for i= 1, . . . , d ,ki−1, ℓi−1∈ {1, . . . , r i−1},ki, ℓi∈ {1, . . . , r i}we define the vector\nˆCi[ki−1, ℓi−1, ki, ℓi]∈R2ni+1as\nˆCi[ki−1, ℓi−1, ki, ℓi] =niX\nαi=0Di,αiˆBi[ki−1,:, ki]ˆAi[ℓi−1, αi, ℓi]. (3.16)\nNote that Di,αiˆBi[ki−1,:, ki]denotes a matrix-vector multiplication, whereas ˆAi[ℓi−1, αi, ℓi]is scalar valued.\nWith slight abuse of notation, we denote the γi-th entry of the vector ˆCi[ki−1, ℓi−1, ki, ℓi]byˆCi[ki−1, ℓi−1, γi, ki, ℓi]∈\nR, which defines an order 5tensor ˆCi∈Rri−1,ri−1,2ni+1,ri,ri. For convenience, we reshape ˆCito an order 3tensor by\nflattening together the first two and last two dimensions, again overloading notation with ˆCi∈Rr2\ni−1,2ni+1,r2\ni. Now\nwe revert to the Legendre polynomial system and define the coefficient tensor C∈R2n+1given in TT format by\nC[γ] :=C1[γ1]···Cd[γd], C i[γi] =2niX\nγ′\ni=0T−1\ni,2ni[γi, γ′\ni]ˆCi[γ′\ni]. (3.17)\nThis construction yields the following result.\nLemma 3.5. Letfandghave FTT-rank rand given as in (3.10) . Then, fghas FTT-rank at most r2, in particular\ng(x)f(x) =X\nγ∈[2n]C[γ]dY\ni=1pi\nγi(xi), (3.18)\nwith coefficient tensor Cwith TT-rank at most r2given by (3.17) .\nBy this Lemma, we have\nMB(A) =C (3.19)\nwithCfrom (3.17). For ease of notation, we further define the square operation S:Rn+1→R2n+1,A7→MA(A).\nFinally, note that the partial derivative ∂xidefines a linear operator that, analogous to Section 3.1, implies a linear\noperator Lxi:Rn+1→Rn+1based on the polynomial discretization such that ∂xivA=vLxiA. This operator has\n8GM with TT approximations of HJB equations A P REPRINT\nthe form Lxi=I ⊗. . .⊗ I ⊗ Dxi⊗ I ⊗ . . .⊗ I withDxi∈Rni+1,ni+1given in Appendix E.2.Putting all of the\nabove together, we see that\n⟨∇vB,∇vA⟩=dX\ni=1(∂xivA)(∂xivB) =dX\ni=1vLxiAvLxiB=dX\ni=1vMLxiB(LxiA)=vdP\ni=1MLxiB(LxiA)(3.20)\nThis leads to a Tensor Train operator representing the nonlinear part (2.8). In particular, for A,B∈Rn+1let\nN L(A):=−dX\ni=1S(LxiA), (3.21)\nN LB(A):=−dX\ni=1MLxiB(LxiA). (3.22)\nThen, by (3.20), we have NonLin( vA) =vN L(A)and−⟨∇vB,∇vA⟩=vN LB(A). This concludes the derivation of\nthe nonlinear part.\n3.3 Projection and retraction\nThe discussion so far shows that linear and nonlinear operations on the polynomial discretization with Tensor Trains\nmay increase the rank as well as the underlying polynomial degree. Therefore, we shall discuss operations that keep a\nfixed polynomial degree and a fixed TT-rank with possible error control, namely projection andretraction . Regarding\nthe projection, let n,m∈N0,n≤mand define Pm,n: span Π m→span Π nby\nPm,n(·):=n1,...,n dX\nα1,...,α d=0dO\ni=1pi\nαi*dO\ni=1pi\nαi,·+\n(3.23)\nDue to the orthonormality of the pi\nαi, the projection is simply obtained by truncating the coefficients, as the following\nresult states.\nLemma 3.6. Forn≤mandA∈Rm+1we have Pm,nVA=VPm,nA, where Pm,n:Rm+1→Rn+1is defined by\n(Pm,nA)[α1, . . . , α d] =A[α1, . . . , α d] (3.24)\nfor all A∈Rm+1andα∈Nn\n0.\nNote that by Parseval’s identity, the projection error in L2-norm can be computed by simply adding the squares of the\nelements that are eliminated by the projection, i.e. with assumptions of Lemma 3.6, we have\n∥Pm,nvA−vA∥2\nL2(K)=m1,...,m dX\nα1=n1+1,...,α d=nd+1A[α1, . . . , α d]2. (3.25)\nA possible realization of a retraction operator\nRr:[\nˆr≥rMˆr→ M r, (3.26)\nfor given fixed rank r∈Nd−1, is obtained by using the TT rounding scheme first presented in [29, Algorithm 2],\nwhich is based on efficient high-order singular value decomposition exploiting the structure of TTs. The operators\nin (3.24) and (3.26) provide us with the necessary tensor operations to fix the degree as well as the rank of the HJB\nsolution, concluding this section.\n4 A direct low-rank HJB solver\nIn this section, we consider polynomial potentials Φ∈span Π nfor some n∈Nd\n0. If the potential is not available in\npolynomial form, we can obtain a suitable polynomial approximation e.g. by the Alternating Linear Scheme (ALS)\n[19] as was done in [31] for the purpose of approximating value functions. Crucially, the ALS yields an approximation\nin a chosen low rank TT format. For Φ∈span Π n, we consider a projected version of the modified HJB equation\n(2.9) restricted to the hypercube Kdefined by\nß∂tvt=P2n,n[Lin(vt) + NonLin( vt)],\nv0= Φ ,inK, (4.1)\n9GM with TT approximations of HJB equations A P REPRINT\nfort∈[0, T]and some T >0large enough. Note, that the projection only acts on the nonlinear part, as the linear part\ndoes not increase the polynomial degree.\nWith the work from the previous section, we can show that this PDE is equivalent to an ODE on a tensor space. Let\nL,N L,N LBfor any B∈Rn+1andP≡P2n+1,n+1be given by (3.8), (3.21), (3.22) and (3.24), respectively. Then\nthe following theorem holds true.\nTheorem 4.1 (Projected HJB equation is equivalent to tensor-valued ODE) .LetA(t)∈Rn+1be a solution of the\ntensor-valued ODE\n˙A(t) =LA(t) +PN L (A(t)),A(0) = A0, (4.2)\nfort∈[0, T]. Then vt:=vA(t)solves (4.1) . Conversely, if vt∈span Π nsolves (4.1) , then there exists a unique\nA(t)∈Rn+1such that vt=vA(t)andA(t)solves (4.2) .\nProof. LetA(t)∈Rn+1solve (4.2). Then, ˙vA(t)=v˙A(t)=vLA(t)+PN L (A(t))=vLA(t)+vPN L (A(t))=\nLin(vA(t)) +P2n+1,n+1\u0002NonLin( vA(t))\u0003andvA(0)=vA0= Φ, showing the first part of the claim. Conversely, if\nvt∈span Π nsolves (4.1), then there exists a unique A(t)withvt=vA(t)andv˙A(t)=∂tvt=vLA(t)+PN L (A(t)).\nSince the mapping A7→vAis injective, this yields the second part of the claim.\nThe solution algorithm for (4.2) which will be presented in the following relies on local linearizations of the HJB for\nstiffnes based stepsize control. Hence, we state the following result on the form of such local linearizations.\nLemma 4.1 (Local linearization) .LetB∈Rn+1. Then, the linearization of (4.2) atBis given by\n˙A(t) = (L+ 2PN L B)A(t)−PN L (B). (4.3)\nProof. Note that the linearization of NonLin( v) =−∥∇v∥2around a fixed v0∈span Π nis given by\nNonLin v0(v) =−2⟨∇v0,∇v⟩+∥∇v0∥2=−2⟨∇v0,∇v⟩ −NonLin( v0). (4.4)\nNow, for A,B∈Rn+1we have\nNonLin vB(vA) =−2⟨∇vB,∇vA⟩ −NonLin( vB)\n= 2vN LBA−vN L(B)\n=v2N LBA−N L(B).\nSince the other operators appearing on the right hand side of (4.1) are linear, (4.3) follows.\nBy Theorem 4.1, it suffices to solve (4.2) for A(t)since this solution defines the solution of (4.1) via vt=vA(t).\nIn the rest of this section, we present principled ways of computing approximate solutions to (4.2) on the low rank\nmanifold Mr. Two methods are investigated:\n1. A simple explicit Euler scheme with adaptive step sizes and retraction after every step, see Section 4.1.\n2. A dynamical low rank integrator designed for time integration of Tensor Trains [26], see Section 4.2.\n4.1 Time adaptive explicit Euler scheme\nPreliminaries. In the following, we define a number of time points N∈N, a sequence of times 0 =t0< t1<\n. . . < t N=T, a TT-rank function t7→rt∈Nd−1assigning to every time a Tensor Train rank and discrete\napproximations Mrtn∋Ytn≈A(tn),n= 0, . . . , N , to the solution A(t)of (4.2). Throughout this section, let\nτmax, δproj, δrank, δcontr >0and a reduction parameter ρ∈(0,1). Denote the potential of the standard normal\ndistribution by v∞(x) =∥x∥2/2and note that this function has FTT rank (2, . . . , 2)[30, Theorem 2]. In practice\nwe choose rtto be bounded by TT-rank( v0)andTT-rank( v∞)with adaptive rank reduction based on TT-rounding\nerror induced by the retraction from (3.26).\nTime adaptive explicit Euler step. Starting with n= 0, we have Ytn∈ M rtn. By Section 3, the right-hand side\nof (4.2) applied to Ytn, i.e. the tensor LYtn+PN L (Ytn)has TT-rank at most 2r+ 2r2and so the addition\nYtn+τn=Ytn+τn(LYtn+PN L (Ytn)) (4.5)\nhas TT-rank at most 3r+ 2r2for any τn>0.\n10GM with TT approximations of HJB equations A P REPRINT\nSince we require the next iterate to be a Tensor Train of rank rtn+τn, we retract to the appropriate manifold, setting\nYtn+τn=Rrtn+τn(Ytn+τn), (4.6)\nwhere Rrdenotes the TT rounding procedure based on higher order singular value decomposition and mapping to\nMr, which was presented in [29, Algorithm 2]. Note that (4.6) corresponds to (2.11) with the Compression given by\nthe retraction operator, i.e. by the higher order singular value decomposition. Up to now the choice of the step size τn\nwas arbitrary. In what follows we set constraints on the step size τnbased on three stability criteria.\nCriterion 1: local stiffness. At each iteration, we restrict the stepsize dependent on the local stiffness of the ODE.\nWe use a heuristic based on local linearizations of (4.2) to determine suitable upper bounds for the stepsize. By Lemma\n4.1, the local stiffness at the current iterate Ytnis governed by the linear operator\nHYtn:=L+ 2PN L Ytn. (4.7)\nIf the current iterate Ytndefines a zero mean Gaussian with diagonal covariance diag( aii, i= 1, . . . , d ), the eigen-\nvalues of HYtncan be bounded by 2Pd\ni=1|1−2aii|(the details of the calculation can be found in Appendix E.3).In\ngeneral, HYtndefines a non-symmetric TT operator. To the knowledge of the authors, estimation of the largest ab-\nsolute eigenvalue of general non-symmetric TT operators is an open question. Here, we rely on a simpler idea. In\nparticular as we are dealing with real valued tensors Ytn, we avoid analyzing the operator action on complex space.\nIn contrast, we are interested in the effect of the current operator in the neighborhood of the current iterate. This is\nrealized by estimating the largest absolute real eigenvalue of HYtndenoted by λtnwith corresponding eigenspace\nthat is not orthogonal to the current iterate Ytn, by a power iteration. The resulting scheme is detailed in Algorithm\n1. The current Tensor Train iterate Ytnserves as an initial guess for the eigentensor. The procedure then resem-\nbles a standard power iteration with an additional retraction step in line 6, which reduces computational burden. In\npractice we are only interested in the absolute value of the eigenvalue or a meaningful upper bound λtnand not in\nthe corresponding eigentensor. Note that the eigenvalue usually converges at much higher order than the eigentensor.\nThe aforementioned upper bound then is obtained through a simple rounding up strategy with a specified number of\naccurate non-zero digits, see Algorithm 1. Based on the return λtnof the power iteration, we define a maximal stable\nstepsize τλby\nτλ:=2ρ\n|λtn|. (4.8)\nIn experiments, this stiffness estimation proves essential for the solver to converge.\nAlgorithm 1 Upper bound estimating the principal real eigenvalue λtnofHYtnfrom (4.7) based on power iteration.\nInput:ß\n• current iterate X0=Ytn• maximum allowed TT rank r∈Nd−1\n• application of HYtn• number of correct non-zero digits p∈N\nOutput: upper bound λtn\n1:Letλk\ntndenote the k-th iterate.\n2:Setk= 0.\n3:while p-th non-zero digit of λk\ntnis changing do\n4: LetK∈N,X0∈ M r.\n5: ˆXk=Xk/∥Xk∥F\n6: Xk+1=Rr(HYtnˆXk)\n7: λk\ntn=⟨ˆXk,Xk+1⟩\n8: k=k+ 1\n9:end while\n10:Define position Pof first non zero digit with P=⌈−log10(λk\ntn)⌉.\n11:Define upper bound treshold ϵp= 10−(P+p).\n12:Define λk\ntn=λk\ntn+ϵp.\n11GM with TT approximations of HJB equations A P REPRINT\nCriterion 2: local relative projection error. For stepsize τ > 0consider the iterate Ytn+τdefined by (4.5) for\nτn=τand let Ytn+τ=Ytn+τ(LYtn+N L(Ytn))be an Euler step with the non-projected equation. Let\nτproj:=®\nτmax, if∥PN L (Ytn)−N L(Ytn)∥F= 0,\nδproj\n∥PN L (Ytn)−N L(Ytn)∥F/∥N L(Ytn)∥F,else.(4.9)\nThen, for any τ≤τprojwe get\n∥Ytn+τ−Ytn+τ∥F≤τ∥PN L (Ytn)−N L(Ytn)∥F≤δproj∥N L(Ytn)∥F. (4.10)\nHence, the projection error of the Euler step, normalized with respect to the magnitude of the degree increasing\nnonlinear part N L(Ytn), is bounded from above by δproj.\nCriterion 3: local relative retraction error. Determine maximum τranksuch that\n∥Ytn+τrank−Ytn+τrank∥F\n∥Ytn+τrank∥F≤δrank. (4.11)\nHere, we initially choose τ0\nrank=τn−1and proceed with τk\nrank=1\n2τk−1\nrankuntilτk\nrankfulfils condition (4.11). Then, we\nuse bisection iteration to determine the maximum τrank∈(1\n2τk\nrank, τk\nrank]satisfying (4.11).\nFinal stepsize choice. After these three criteria, the next step size τnin (4.6) and the next time tn+1are defined as\nτn:= min {τmax, τλ, τproj, τrank, T−tn}, (4.12)\ntn+1:=tn+τn, (4.13)\nwhere the term T−tnensures that we end exactly at terminal time T. The single time step (4.6) is repeated for\nn= 0,1, . . .with stepsize (4.12) until tn+1=T, in which case we define N=n+ 1.\nIn addition to the adaptivity in the stepsize, the solver also incorporates adaptivity in the polynomial degree as well as\nthe TT rank, which is detailed in the following.\nAdaptive decrease of polynomial degree Motivated by the fact that vt→v∞∈Π(2,...,2)ast→ ∞ at exponential\nrate, we introduce a simple adaptive choice for the polynomial degree. Assume that the degrees of Ytnat time tnare\ngiven by ntn∈Nd\n0. LetYk\ntndenote the order d−1tensor, which for k= 1, . . . , d is given as\nYk\ntn= (Ytn[α])α∈[ntn],αk=(ntn)k.\nThis is a slice of the full coefficient tensor Ytnfixing αk= (ntn)kwhich is the highest polynomial degree in the k-th\ndimension at time tn. Now, in case of\n∥Yk\ntn∥F≤δcontr, (4.14)\nwe truncate the highest polynomial degree in the k-th direction. Since Ytnis given in TT format with\nYtn[α] =Yt,1[α1]·. . .·Yt,d[αd],\nthis operation is realized by truncation of the component tensor Ykand possibly adapting the TT-ranks.\nAdaptive choice of TT rank Motivated by the conjecture, that the FTT rank of vtis bounded by the FTT rank of v0\nandv∞, i.e.r∞= (2, . . . , 2), we perform two retraction steps with respect to these bounds after the time step at time\ntn. First a retraction with respect to the rank\nˆrtn+τn= max {rtn,r∞} (4.15)\nis performed where the maximum is understood component wise. This serves to ensure that the rank of the solution\nremains bounded by the maximum of the initial rank and the rank of the standard normal potential. Furthermore, a\nrounding procedure [29, Algorithm 2] with respect to δcontris performed to potentially further decrease the rank and\nthus define rtn+τn. In practice both retraction steps can be performed efficiently in a single operation, which leads to\n(4.6).\nThe proposed time adaptive explicit Euler scheme is summarized in Algorithm 2.\n12GM with TT approximations of HJB equations A P REPRINT\nAlgorithm 2 Time adaptive explicit Euler Scheme to solve HJB equation based on Tensor Trains\nInput:\n\n•v0given in TT format ,\n•T >0maximum finite time horizon ,\n•τmax>0, bound for the stepsize\n• reduction stiffness parameter ρ∈(0,1),\n• step size proposal hyperparameter δproj, δrank,\n• degree of freedom contribution tolerance δcontr>0.\nOutput: Discrete sequence (vtn)ndefined on subsequently determined adaptive time points tn∈[0, T].\n1:Sett= 0.\n2:while t≤Tdo\n3: Determine next time step :\n4: Compute maximal stable stepsize τλ. ▷see (4.8)\n5: Compute step size proposal τprojbased on projection error. ▷see (4.9)\n6: Compute step size proposal τrankbased on relative retraction error. ▷see (4.11)\n7: Determine final step size τ= min {τmax, τλ, τproj, τrank, T−t}.\n8: Perform a single Euler step\n9: Sett=t+τ.\n10: Approximate vtvia algebraic manipulation of the underlying TT format. ▷see (4.5)\n11: Perform a retraction step of the resulting coefficient in TT format. ▷see (4.6)\n12: (Re-)compression\n13: Check for potential polynomial degree decrease using δcontr. ▷see (4.14)\n14: Check for potential rank reduction using δcontr. ▷see (4.15)\n15:end while\n4.2 Dynamical low rank approximation\nWhile the time adaptive explicit Euler scheme presented in the previous section offers a conceptually simple inte-\ngration method, Dynamical low rank appromxation (DLRA) [23, 25, 26] methods offer another principled way of\napproximately integrating tensor valued ODEs of the form (4.2).\nHere, the idea is to formulate an approximation of a tensor valued ODE\n˙A(t) =F(t,A(t)),A(0) = A0,\nwhere n∈Nd,A(t)∈RnandF: [0,∞]×Rn→Rnon a fixed rank manifold Mr. This is done via projection of\nthe right-hand side onto the tangent space of Mr. More precisely, for a fixed r∈Nd−1, the approximation is defined\nas\n˙Y(t) =PTY(t)F(t,Y(t)),Y(0) = Y0≈A0, (4.16)\nwhere Y0∈ M randPTY(t)denotes the orthogonal projection (in Frobenius norm) onto the tangent space of Mr\ninY(t). Note that due to this projection, a solution of (4.16) satisfies Y(t)∈ M rfor all t. In [13] the authors use\nan explicit Euler discretization of (4.16) for the solution of HJB equations appearing in deterministic optimal control\nbased on spatially discretized parabolic PDEs. However, leveraging the form of the tangent space, the projector on\nthe right hand side can be decomposed into a sum of projectors corresponding to orthogonal subspaces. In [25], the\nauthors propose to use this sum structure for a Lie-Trotter type splitting scheme in the case of a matrix valued ODE,\nwhich is termed the projector-splitting integrator. Consequently, [26] extends the projector splitting integrator to the\ntensor setting. One of the key properties of this integrator is that each discrete step preserves the rank r.\nIn our scheme, using a step with the integrator from [26] instead of the explicit Euler step (4.5) leads to a new\niterate Ytn+τnwith the same rank as Ytn. Hence, the retraction (4.6) becomes a mere rounding procedure and the\nrank of two consecutive iterates is monotonically decreasing. This is a desirable property if the initial rank satisfies\nrt0≥(2, . . . , 2). Forrt0≱(2, . . . , 2), the projector-splitting is unsuited because it restricts the rank from above to\nrtn≤rt0and so rtncan not converge to the correct rank (2, . . . , 2).\nIncorporating more recent state-of-the-art dynamical low rank integrators for matrix valued ODEs such as [8, 9] to the\nTensor Train setting could lead to significant improvements of the proposed method. In particular, the Basis Update\n&Galerkin (BUG) integrator [8] introduces rank adaptivity, while the fully parallel integrator [9] could additionally\ngreatly speed up computations in high dimensions. Their application in our method is a topic of future research.\n13GM with TT approximations of HJB equations A P REPRINT\n4.3 Evaluation of the low-rank model\nAs the result of section 4.1 or 4.2 we have a representation of the value function in the spirit of (3.5) at discrete set of\ntime points the form t∈ {t0, t1, . . . , T }of the form\nvt(x) =X\nα∈[nt]Yt[α]pα(x), (4.17)\nfor some nt∈Nd\n0andYtgiven in tensor train format resulting as the discrete solution of (4.2).\nWe now want to discuss the evaluation of vt(x)at arbitrary time t∈[0, T]andx∈Rd. This is motivated by the\nreverse-time sampling process, which is permitted to be time adaptive and may require evaluation in time points not\nincluded in the set {t0, . . . , t N}.\nFor this we propose a very simple solution. Let t∗∈[0, T]. Let\nt= max {t∈ {t0, . . . , t N}:t≤t∗} (4.18)\nLetτ=t∗−t. Then, we compute the coefficient representation in Tensor train format of vt∗through a single Euler-\nor DLRA step with step size τ. Note that this step size is within the step size bounds implied by the adaptive scheme\nproposed earlier. In particular, τis smaller than the step size implied by local stiffness.\nLastly, we discuss how the evaluation of the model class is performed in practice. Aside from the evaluation of\nthe polynomial basis functions, only matrix- and vector products have to be computed to evaluate vt. This efficient\nevaluation is one of the strengths of the Tensor Train format. For x= (xs, . . . , x d)∈Rdandt∈[0, T], the\napproximation is defined by a TT Ytwith dimensions nt= (nt,1, . . . , n t,d)and ranks rt= (rt,1, . . . , r t,d). To\nevaluate (4.17), one first computes pi\nj(xi)for all i= 1, . . . , d andj= 0, . . . , n t,i. Now the TT format provides a\ndecomposition of the form Yt[α] =Yt,1[α1]Yt,2[α2]···Yt,d[αd], where Yt,i∈Rrt,i−1,nt,i,rt,i,α∈[nt]and hence αi\nruns from 0tont,i. In particular, (4.17) implies\nvt(x) =nt,1X\nα1=0. . .nt,dX\nαd=0Yt,1[α1]Yt,2[α2]···Yt,d[αd]p1\nα1(x1)p2\nα2(x2). . . pd\nαd(xd)\n=:Yx1\nt,1·Yx2\nt,2···Yxd\nt,d,(4.19)\nwhere Yxi\nt,i∈Rrt,i−1,rt,iresults from a simple contraction of Yt,iwith the vector (pi\n1(xi), . . . , pi\nnt,i(xi))over the\nnt,i-dimension. Yx1\nt,1·Yx2\nt,2···Yxd\nt,dis now a simple matrix product. Note since rt,0=rt,d= 1, this product boils down\nto a matrix-vector product, when performed from left to right or vice-versa, yielding a scalar value.\n5 Numerical results\nBased on Remark 2.1, we generate approximate samples from µ∗by means of the discrete process described in\nAlgorithm 3. The algorithm utilizes the reverse-time process from Remark 2.1 with λ= 0 discretized at the time-\npoints tnat which approximate solutions Ytnof the projected HJB (4.1) are available. These approximations define\nour surrogate for the score ∇logπtbased on\nvtn≈ −logπtn, n = 0, . . . , N, (5.1)\nwhere vtn:=vYtnis understood in the sense of (3.5). The inner loop over kin Algorithm 3 consists of additional\nLangevin-postprocessing steps [39] after every step with the reverse process.\nAs a necessary condition for convergence of the computed solutions vtnto the potential v∞(x) =1\n2x⊺Idxof the\nstandard normal distribution, we consider convergence of the coefficients of the quadratic part. More precisely, since\nvtnis a polynomial, we can always write\nvtn(x) =atn+b⊺\ntnx+x⊺Σ−1\ntnx+higher order terms , (5.2)\nwithatn∈R, btn∈Rdand a symmetric Σtn∈Rd×d. In this section, we call covariance error at time tnthe term\nCovErr( tn) =\r\rΣ−1\ntn−Id/2\r\r\nF/∥Id/2∥F, (5.3)\ni.e. the relative error in Frobenius norm between the current precision matrix and the precision matrix of the standard\nnormal distribution.\nWe remark that, in the test cases we considered, the results produced by the dynamical low rank integrator [26] (using\nthe same heuristics for adaptive stepsize determination) are similar to the results produced by an explicit Euler stepping\nwith subsequent retraction. Hence, we only present the results of the latter.\n14GM with TT approximations of HJB equations A P REPRINT\nAlgorithm 3 Sampling from π∗\nInput:\n\n• Initial samples {zi\n0}I\ni=1∼ N(0, Id),\n• Times {tn}N\nn=1and discrete HJB solution {vYtn}N\nn=1defined by Algorithm 2 ,\n• Stepsize τand number of steps K∈Nfor Langevin postprocessing,\n• Parameter λ∈[0,1]for reverse-time process\nOutput: Samples {zi\nN}I\ni=1∼µ∗.\n1:fori= 1, . . . , I in parallel do\n2: Generate time points {ti\nn}N\nn=1.\n3: forn= 0,1, . . . , N −1do\n4: Setτi\nn=ti\nn+1−ti\nn\n5: Sample ξn∼ N(0, Id)ifλ̸= 1.\n6: Setzi\nn+1=zi\nn+\u0002zi\nn+ (2−λ)∇vYT−tn(zi\nn)\u0003τi\nn+p\n2(1−λ)τnξn. ▷Reverse-time process step\n7: forℓ= 0,1, . . . , L do\n8: Sample ξk∼ N(0, Id)\n9: zi\nn+1←−zi\nn+1−τ∇vYT−tn(zi\nn+1) +√\n2τξk. ▷Langevin post-processing step\n10: end for\n11: end for\n12:end for\n5.1 Verification result: Gaussian setting\nProblem definition Letd= 10 ,K= [−5,5]10andΦ(x) =x⊺Σ−1x, where Σis a randomly generated symmetric\npositive definite matrix (we sample entries of a matrix Auniformly on [0,1]and then define Σ−1=A⊺A+ 0.1Id).\nNote that in this setting the polynomial degree of the HJB solution is bounded by n= (2, . . . , 2)asπtremains a\nGaussian density if π0andπ∞are Gaussian.\nParameters For Algorithm 2, we choose T= 12 ,τmax= 0.1,ρ= 0.2,δproj=δrank= 0.01,δcontr= 10−8.\nEvaluation By Lemma D.1, Φhas an FTT-rank given by r= (3,4,5,6,7,6,5,4,3). Since the solution of the HJB is\na strictly quadratic polynomial for all times (meaning that no higher or lower degrees than 2 appear), Lemma D.1also\nyields that the FTT rank of the solution is bounded from above by rfor all tn. In Figure 5.1 the ranks of the solution\nduring integration are displayed. Once the solution reaches a covariance error of ∼10−7, the solver starts to truncate\nthe ranks, meaning that at this point higher ranks give a contribution to the solution which is less than δcontr = 10−8\nin relative Frobenius norm. Finally, all ranks higher than 2are truncated, which is to be expected since the standard\nnormal potential has FTT rank r≡2. Att= 12 , the covariance error has decreased to ∼10−11.\nFigure 5.2 displays the stepsizes chosen by the solution algorithm. Since the polynomial degree does not increase and\nthe ranks are bounded from above by the initial rank, the stiffness estimate (4.8) determines the stepsize.\nAs a further demonstration of the consistency of the method, we consider the solution algorithm for different maximal\nstepsizes τmax∈ {0.1,0.01,0.001}. By the standard theory of the explicit Euler scheme, we expect the time dis-\ncretization error to be in O(τmax). As the second order polynomial ansatz introduces no spatial discretization error in\nthe Gaussian setting, the time discretization is the only source of error in the learned score. Figure 5.3 clearly displays\ntheO(τmax)dependence in both the relative Frobenius-error of the covariance matrix as well as the L2-error of the\nscore.\n5.2 Mixed nonlinear density\nProblem definition Letd= 20 ,K= [−5,5]2×[−2,2]2×[−5,5]2×[−2,2]14. Consider the transport map\nT:R2→R2and matrix Σwith\nT(x, y) = (x, y+x2+ 1),Σ =Å1 0 .9\n0.9 1ã\n. (5.4)\nLetΦ1(x, y) =v∞(Σ−1T(x, y)),Φ2(x, y) =x4+y4−4x2−4y2−0.4x+0.1y+8andΦ3(x, y) =x6+y6+3xy.\nDefine Φ(x) = Φ 1(x1, x2) + Φ 2(x3, x4) + Φ 3(x5, x6) +P20\ni=7x2\ni. The first six dimensions of this potential define a\nbanana -shaped marginal density in the first two dimensions, a nonsymmetric multimodal marginal density in the third\nand forth dimensions, and a bimodal marginal density in the fifth and sixth dimensions (see the right most column in\nFigure 5.4). By construction, this potential has rank r= (3,2,2,2,3,2, . . . , 2).\n15GM with TT approximations of HJB equations A P REPRINT\nFigure 5.1: Development of the solution ranks and the covariance error (5.3) over time in the Gaussian setting. Once\nthe solution is close to convergence (in terms of the covariance error), the ranks decrease to the rank (2, . . . , 2)of the\npotential of the standard normal distribution.\nFigure 5.2: Approximations of the maximal absolute eigenvalues of the linearized right-hand side |λt|determined by\nthe power method (left) and accordingly chosen stepsize 2ρ/|λt|(right) over time in the Gaussian setting. Note that\nthe eigenvalues decrease monotonically, permitting a monotonous increase of the stepsize until the maximal permitted\nstepsize τmax= 0.1is reached.\nParameters We choose n= (4,2,4,4,6,6,2, . . . , 2)∈N20according to the degrees appearing in the potential.\nFor Algorithm 2 we set T= 10 ,τmax= 0.05,δproj=δrank= 0.01,δcontr= 10−8. To account for the high stiffness\nof the equation at small time t≪1, we set the stiffness parameter ρin Algorithm 2 to ρ= 0.001as long as t <10−6\nandρ= 0.5fort≥10−6. Langevin postprocessing (see Algorithm 3) is performed with L= 100 steps and stepsize\nτ= 0.005.\nEvaluation While the rank between independent parts of the density does not increase under the HJB flow, the initial\nranks r1=r5= 3 may increase due to the time stepping scheme and hence incur a truncation error. However, with\nthe specified values for ρwe discover that the stepsize resulting from the stiffness criterion (4.8) satisfies both the\nprojection and the truncation criterion (4.10), (4.11) with the requested tolerance, suggesting that the solver keeps\nthese errors sufficiently small. Figure 5.5 shows these stepsizes with a jump around t= 10−6due to the increase in\nthe stiffness control parameter ρ. Figure 5.6 shows the exponential decay in the covariance error (5.3) between the\nHJB solution and the standard normal distribution. Note that there is an initial spike in the error for small times t. In\nexperimentation, this spike seems to decrease in magnitude when permitting higher polynomial degrees. Hence, we\ncan attribute it to a discretization error. The optimal choice of permitted degrees to balance accuracy and computational\nfeasibility is an open question at this point. We conjecture that it is at this point that future research will prove most\nfruitful: the difficult region close to t= 0, where the true solution of the HJB is far away from the standard normal\npotential. Finally, Figure 5.4 shows the densities corresponding to the HJB solution obtained by Algorithm 2 and the\n16GM with TT approximations of HJB equations A P REPRINT\n0 1 2 3 4 5 6 7 8\nForward time (target ---> std. Gauss)108\n107\n106\n105\n104\n103\n102\nRelative Frobenius-error of the covariancemax=1e1\nmax=1e2\nmax=1e3\n0 1 2 3 4 5 6 7 8\nForward time (target ---> std. Gauss)1010\n109\n108\n107\n106\n105\n104\n103\nRelative L2-error of the scoremax=1e1\nmax=1e2\nmax=1e3\nFigure 5.3: Comparison of the discretized HJB solution and its derived gradient for different maximum time steps\nτmax= 0.1,0.01,0.001to the exact score in the setup of random Gaussian target distribution for d= 10 . One can\nclearly see the O(τmax)dependence of the error in both gradient and covariance matrix of the approximate solution.\nNote that for small times t≪1, the discretizations for maximal stepsizes 0.1and0.01lead to similar L2-error in the\ngradient, which can be attributed to an equally small adaptively chosen stepsize in this early (stiff) regime (compare\nto the right plot in Figure 5.2 which shows the realized step sizes for τmax= 0.1).\nsamples at the corresponding time points in the reverse process. We note that the curvature of the banana potential in\nthe first two dimensions as well as the multimodalities in higher dimensions are recovered by the method. Finally, we\nnote the large number of postprocessing steps used in this example. We observed a drastic decrease in sample quality\nfor less postprocessing steps. This observation motivates the discussion in the next section.\n5.3 Sensitivity of score-based sample generation under perturbations\nIn [39] it was pointed out that using the flow-ODE instead of the reverse SDE, we actually can rely on high-order\nintegration schemes and thus reduce the time discretization error significantly. However, both the reverse SDE and\nthe flow ODE are unstable under perturbations. In fact, while the magnitude of perturbations decreases exponentially\nunder the forward process, it can grow exponentially under the reverse process. This can be easily seen in the linear\ncase. Assume that µ∗=N(0,Σ)forΣ = cI, where c >1. Then for all t≥0, the law of (2.1) satisfies µXt=\nN(0,Σt)forΣt= Σe−2t+ (1−e−2t)I= (1 + ( c−1)e−2t)I. Hence for the flow ODE ( λ= 1) and for the reverse\nSDE ( λ= 0) , the dynamic (respective drift) term has the form\nf(t, Xt) =Xt+ (2−λ)∇logπT−t(Xt) = (I−(2−λ)Σ−1\nT−t)Xt.\nThe eigenvalues of the linear dynamic of the ODE ( λ= 1) are 1−(1 + ( c−1)e−2(T−t))−1>0for all t. Hence,\nperturbations in the initial data will grow exponentially in magnitute.\nIn practice, a good sample quality for the generation process relies on control of the time discretization, the error made\nin the score approximation and the accuracy of the start distribution. While the latter can be reduced by letting samples\ndrawn from N(0, I)be transported to the terminal distribution of the forward process, µXT, where by construction\nµXT≈ N(0, I), the approximation error of the score still has negative effect on the sampling quality.\nIn order to illustrate this, we consider an example independent of the low-rank approximation, that allows for exact\nerror control with respect to an accessible exact reference solution. For this, consider the two dimensional example\nwith potential Φ2(x, y) =x4+y4−4x2−4y2−0.4x+0.1y+8from section 5.2, i.e. dµ∗(x) =1\nZexp(−Φ2(x))dx=\nπ∗(x)dx. Since the forward process is an Ornstein-Uhlenbeck process, the exact density is obtained by convolution,\nin particular\nπt(x) =Z\nR2πt(x|x0)π∗(x0)dx0. (5.5)\nwith the transition density\nπt(x|x0) =1p\n|2πΣ(t)|e−1\n2(x−M(t,x0))TΣ(t)−1(x−M(t,x0))\n17GM with TT approximations of HJB equations A P REPRINT\nFigure 5.4: Development of marginal densities (blue) and the samples produced by the corresponding reverse process\ndefined by Algorithm 3 (red) in the setting of the mixed nonlinear case (Section 5.2). The first row shows the values\nof the densities and samples on the (x1, x2)-plane, which is governed by the the Banana potential. The second row\nconcerns the (x3, x4)-plane, which is governed by the nonsymmetric multimodal potential. The third row shows the\n(x5, x6)-dimension, governed by the bimodal potential. On the level of the HJB solver, the plot should be viewed from\nright to left since the target density (right) is transformed to a standard Gaussian (left). On the level of the reverse\nprocess, the samples (red) move from the standard Gaussian on the left to the target measure on the right. We note\nthat in all cases the sampler is able to reproduce the multimodality and curvature of the corresponding density.\nFigure 5.5: pproximations of the maximal absolute eigenvalues of the linearized HJB right-hand side (left) and ac-\ncordingly chosen time stepsizes (right) as in Figure 5.2 but for the mixed nonlinear potential from Section 5.2. Note\nthe jump in the stepsize at t= 10−6which corresponds to a change in the stiffnes control parameter ρ. Up to small\nperturbations which may be attributed to inaccuracy of the power method the stepsizes are monotonically increasing\nagain.\n18GM with TT approximations of HJB equations A P REPRINT\nFigure 5.6: Decay of the covariance error (5.3) for the mixed nonlinear potential of Section 5.2. Note that after\nan initial spike, which may be attributed to degree and rank increase of the true HJB solution, the error decays\nexponentially.\nwith\nM(t, x0) =e−tIx0=e−tx0,Σ(t) =tZ\n0e(s−t)IσσTe(s−t)Ids= (1−e−2t)I.\nNow consider for Q∈Nthe approximation of πtthrough πQ\ntdefined as\nπQ\nt(x) :=QX\ni,j=1wijπt(x|xij)π∗(xij)\nvia Gaussian quadrature encoded in the weights (wij)and abcissas (xij)on a suitable large tensor domain in R2\ncontaining most of the support of π∗. We then define the approximation vM\ntof the exact value function vtas\nvt(x) =−logπt(x)≈vQ\nt(x) :=−logπQ\nt(x)\nIn the experiment we consider the case of Q= 3 for low-accuracy, Q= 10 for medium accuracy and Q= 50 for\nhigh-accuracy on the domain [−5,5]2. As can be seen in Figure 5.7, the sample quality is very poor in the situation\nof low or medium accuracy. In our experiments the effect of perturbations due to low accuracy could be mitigated by\nLangevin postprocessing.\nMotivated by this observation, we suggest an alternative to the reverse SDE sampling, given by the sampling process\ndXt=−∇logπmax{T−α(t),0}(Xt)dt+√\n2dWt, (5.6)\nwhere π0=π∗, and α: [0,∞)→[0, T]is some suitably chosen time dilation. A piecewise constant αcorresponds\nto the Langevin postprocessing employed in Algorithm 3. For a high number of postprocessing steps, the reverse-\nSDE step arguably becomes numerically negligible, and the reverse sampling process approximately follows (5.6).\nThe SDE in (5.6) is an instance of homotopy based Langevin dynamics developed in [12]. Consider again the linear\nexample µXt=N(0,Σt)from the beginning of this section. Then, the drift term has the form\n−∇logπmax{T−α(t),0}=−Σmax{T−α(t),0}\nwhich uniformly provides negative eigenvalues, hence stability is to be expected. This serves as a motivation for the\nLangevin postprocessing employed in the sampling algorithm.\n19GM with TT approximations of HJB equations A P REPRINT\nlow acc\n medium acc\n high acc\nFigure 5.7: Sample quality illustration for different accuracies for the sample generation through the flow ODE (top\nrow) and the reverse SDE (bottom row).\n6 Conclusion and Outlook\nWe presented an interpretable solver for the HJB equation arising from Hopf-Cole transformation of the Fokker-Planck\nequation in the setting of Bayesian inference and Generative Modelling. The approach uses functional Tensor Trains\nand spatial discretization with Legendre polynomials. A surrogate replacement for the HJB equation, which reduces\nto an ODE on tensor space, was derived. The applicability of the method was demonstrated on linear and nonlinear\ntest cases.\nThere are some obvious avenues for future work.\n• Incorporating more recent state-of-the-art dynamical low rank integrators for matrix valued ODEs such as\n[8, 9] to the Tensor Train setting could lead to substantial performance improvements of the proposed method.\nIn particular, the Basis Update &Galerkin (BUG) integrator [8] introduces rank adaptivity, while the fully\nparallel integrator [9] could additionally greatly speed up computations in high dimensions.\n• Sampling from the reverse process via an Euler-Maruyama discretization usually requires a small step size\nand a high number of time steps. In a recent work [43], a Diffusion Exponential Integrator Sampler (DEIS)\nwas proposed, which utilizes the semilinear structure of the learned diffusion process (2.4) to reduce the\ndiscretization error. This integrator could be applied in our setting. In particular, the combination with recent\ndynamical low rank solvers such as [9] could lead to a greatly reduced number of necessary steps both in\nsolving the HJB as well as in discretizing the reverse process.\n• We provided results for the FTT rank structure of the HJB solution in case of Gaussian distributions (Lemma\nD.1)and distributions with independent components (Lemma D.2)but it is an open question if there are further\nrank structures that are preserved under the HJB flow. As a fist step, Lemma D.2can be generalized to\nindependence between groups of components: Let f(x) =f1(x1, . . . , x n) +f2(xn+1, . . . , x d). Then the\nFTT ranks of both fandLin(f) + NonLin( f)satisfy rn≤2. We conjecture that there are further situations\nin which the solution ranks can be bounded:\nConjecture: The HJB flow FTT ranks rtare (up to a constant) bounded by r0ofv0andr∞≡2forv∞.\nThis analysis is part of investigations in a subsequent work.\n• A rigorous analysis providing error estimates between the solution of the projected equation (4.1) and the\nsolution of the HJB equation (2.9) needs to be carried out. For a Gaussian potential, the solution of (4.1)\ncoincides with that of (2.9). For more general densities the quality of the approxmiation largely depends on\nthe initial condition, the contraction properties of the right-hand side of the HJB equation and the projection\nerror.\n20GM with TT approximations of HJB equations A P REPRINT\n• Finally, in the standard reverse scheme, sample quality depends on three factors: initial measure error, accu-\nracy of the score and the time discretization. While the initial measure error can be treated by methods such\nas predictor-corrector schemes, the other sources of error can only be mitigated by increased computational\neffort. Langevin postprocessing provides a cheaper alternative, improving the quality of the samples (up to a\ncertain level) without requiring higher accuracy of the score. We therefore suggested a homotopy approach\nin (5.6). The sample quality achieved by this method, especially in the case of low accuracy of the score, will\nbe investigated in future work.\n7 Acknowledgements\nDS & ME acknowledge support by the Profit project ReLkat - Reinforcement Learning for complex automation engi-\nneering as well as support by the ANR-DFG project COFNET: Compositional functions networks - adaptive learning\nfor high-dimensional approximation and uncertainty quantification . RG, ME & CS acknowledge support by the DFG\nMATH+ project AA5-5 (was EF1-25) - Wasserstein Gradient Flows for Generalised Transport in Bayesian Inver-\nsion. ME acknowledges partial funding by the DFG priority program SPP 2298 “Theoretical Foundations of Deep\nLearning”. This study does not have any conflicts to disclose.\nReferences\n[1] B. D. Anderson. Reverse-time diffusion equation models. Stochastic Processes and their Applications ,\n12(3):313–326, 1982.\n[2] M. Bachmayr. Low-rank tensor methods for partial differential equations. Acta Numerica , 32:1–121, 2023.\n[3] M. Bachmayr, A. Nouy, and R. Schneider. Approximation by tree tensor networks in high dimensions: Sobolev\nand compositional functions. arXiv preprint arXiv:2112.01474 , 2021.\n[4] J. Berner, M. Dablander, and P. Grohs. Numerically solving parametric families of high-dimensional kolmogorov\npartial differential equations via deep learning. 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Actor-critic method for high dimensional static hamilton–jacobi–bellman partial\ndifferential equations based on neural networks. SIAM Journal on Scientific Computing , 43(6):A4043–A4066,\n2021.\n22GM with TT approximations of HJB equations A P REPRINT\nA Hopf-Cole Transformation\nLetπtsatisfy the Fokker-Planck equation\n∂tπt= ∆πt+x· ∇πt+dπt. (A.1)\nThen, by the chain and product rule, we have the identities\n∂tlogπt=1\nπt∂tπt=1\nπt(∆πt+x· ∇πt+dπt), (A.2)\n∆ log πt=∇ ·Å1\nπt∇πtã\n=1\nπt∆πt−1\nπ2\nt∇πt· ∇πt=1\nπt∆πt− ∥∇ logπt∥2\n2. (A.3)\nPutting these together, we see that\n∂tlogπt= ∆ log πt+∥∇logπt∥2\n2+x· ∇logπt+d (A.4)\nand hence vt=−logπtsatisfies\n∂tvt= ∆vt− ∥∇ vt∥2\n2+x· ∇vt−d. (A.5)\nB Optimal Control Perspective\nWe consider the case where π∗is a zero-mean Gaussian with symmetric positive definite covariance matrix Σ. Hence,\n(2.9) becomes\n∂tvt= Lin( vt) + NonLin( vt), v 0=1\n2x⊺Σ−1x. (B.1)\nThis is a familiar form in stochastic optimal control. Consider an SDE\ndXt= (f(Xt) +g(Xt)ut)dt+σ(Xt)dWt, (B.2)\nX0=x0, (B.3)\nwith initial condition x0, control ut, diffusion σ, free drift part fand controlled drift part g·ut. Associate with this\nSDE a cost functional given by\nJ(t, x, u ) =EñZT\ntλu⊺\ntutdt+E(XT)\f\fXt=xô\n, (B.4)\nwhere λ >0andEis a positive definite terminal cost function. The associated HJB equation for the value function V\nreads\n∂tV+f⊺∇V−1\n4λ∇V⊺gg⊺∇V+σ2\n2∆V= 0, V (T, x) =E(x).\nNow, choose f(x) =x,g(x)≡Id,E(x) =x⊺Σ−1x/2,λ= 1/4andσ=√\n2to arrive at\n∂tV=−∇V⊺x+∥∇V∥2−∆V,\nV(T, x) =1\n2x⊺Σ−1x.(B.5)\nClearly, reversing the time by defining← −V(t, x):=V(T−t, x)and regrouping the terms yields\n∂t← −V= ∆← −V+∇← −V⊺x− ∥∇← −V∥2= Lin(← −V) + NonLin(← −V),\n← −V(0, x) =1\n2x⊺Σ−1x,(B.6)\nand hence vt=← −V(t,·). In total, the log-density of the foward SDE is given as the reverse-time value function of\na linear quadratic optimal control problem. The linear quadratic problem has the property that its solution is the\nsame in the deterministic and stochastic setting. Hence, instead of the stochastic problem, we may also consider the\ndeterministic optimal control problem defined by\n˙x=x+u, x (0) = x0, (B.7)\nJ(t, x, u ) =ZT\nt1\n4u(t)⊺u(t)dt+1\n2X(T)⊺Σ−1X(T). (B.8)\n23GM with TT approximations of HJB equations A P REPRINT\nNow, setting A=Id,B=Id,Q≡0∈Rd×d,R=1\n4IdandQf=1\n2Σ−1\nρthis leads to\n˙x=Ax+Bu, x (0) = x0, (B.9)\nJ(t, x, u ) =ZT\ntx(t)Qx(t) +u(t)⊺Ru(t)dt+X(T)⊺QfX(T). (B.10)\nThe solution of this problem is given by the LQR V(t, x) =x⊺Ptx, where Ptsolves a Riccati differential equation\nwith inputs A, B, Q, R, Q f. It follows that\n∇vT−t(x) =∇V(t, x) = 2 Ptx,\nleading for λ= 0to a reverse-time generative process defined by\ndYs= (Id−4Ps)Ysds+√\n2dWs. (B.11)\nC Motivation for using (Functional) Tensor Trains\nWe give an informal motivation for the use of FTTs and in particular polynomials represented in the Tensor Train\nformat in the setting of Bayesian inference for paramteric PDEs. For more rigorous representations and decay rates in\npolynomial chaos representations of solutions of parametric PDEs we refer e.g. to pioneering work in [10] and follow\nup research. In this setting, the fact that usually only very few data points are available often renders high frequency\ncomponents non-informative. Hence, the higher mode dimensions are often close to the prior information even after\ninference. Assuming that the prior is given as a (standard) Gaussian, it is reasonable to assume that these higher modes\nwill be close to Gaussian. In particular, this motivates a form of potential similar to the nonlinear potential used in\nSection 5.2. The following provides a sketch on how such a form might be obtained. Let M1∈N,M1< d, denote a\nnumber of relevant modes and for maximal polynomial degrees d1≥d2≥. . .≥dM1≥2let\nrelevant =M1×\ni=1{0, . . . , d i},√\nrelevant =M1×\ni=1{0, . . . ,⌊p\ndi⌋}.\nObserve that solutions uof parameteric PDEs with spatial variable xand parameter ycan often be written as\nu(x, y)≈X\nα∈√\nrelevantuα(x)pα(y)\nwhere uαis an element of some function space Vfor every α. Let G(y) =u(·, y)∈Vand for some K∈Nlet\nO:V→RKbe a linear observation operator (e.g. point evaluations in x). Hence:\nO(G(y)) =O(u(·, y)) =X\nα∈√\nrelevantuαpα(y),uα∈RK.\nThen, assuming a Bayesian setting with a zero mean Gaussian prior with covariance matrix Σ, the log posterior density\nhas the form\nlogπ(y) =−1\n2∥O(G(y))−δ∥2\nσId−1\n2∥y∥2\nΣ\nwhere δis an observation and σId,σ > 0, is the covariance of the zero mean Gaussian observational noise. By\nthe form of O(G(y)), it then follows that there are coefficient tensors cpriorandclikelihoodsuch that the potential or\nnegative log posterior density is of the form\nΦ(y) =1\n2σKX\nk=1(O(G(y))−δ)2\nk+X\n|β|=2cprior[β]Pβ(y)\n=X\nβ∈relevantclikelihood[β]Pβ(y) +X\n|β|=2,βi≡0fori>M 1cprior[β]Pβ(y)\n| {z }\nnon-Gaussian component+X\n|β|=2,βi≡0fori≤M1cprior[β]Pβ(y)\n| {z }\nGaussian (uninformed) component,(C.1)\nwith non-Gaussian parts confined to the relevant modes 1, . . . , M 1.\n24GM with TT approximations of HJB equations A P REPRINT\nD Functional Tensor Train rank of HJB solutions\nLemma D.1 (Gaussian distributions) .Letd∈Nandf:Rd→Radmit the form f(x) =x⊺Mx for a symmetric\npositive definite matrix M∈Rd,d. Then fhas finite FTT rank r∈Nd−1. In particular for d≥3,\nr≤r:= 2 +®\u00001,2, . . . ,d\n2, . . . , 2,1\u0001, d even,\u00001,2, . . . ,d−1\n2,d−1\n2, . . . , 2,1\u0001, d odd,\nandr= 2∈Nford= 2.\nProof. The case d= 2 follows since Mis invertible and the TT rank coincides with the matrix rank. Let d≥3and\nwrite M= (mij)andr= (ri)d−1\ni=1,r0=rd= 1. We seek a representation\nf(x) =U1(x1)U2(x2)···Ud(xd), U i(xi)∈Rri−1,ri, i= 1, . . . , d.\nLetIn∈Rn,ndenote the identity matrix and 0k,l∈Rklbe a zero matrix and 0k∈Rkbe a zero vector.\nDefine the matrices ˜Ui(xi)fori= 1, das\n˜U1(x1) =\u00001 2x1m11x2\n1\u0001∈R1,r1, ˜Ud(xd) =\u00001 2xdmddx2\nd\u0001⊺∈Rrd−1,1.\nMoreover, for i= 2, . . . , d −1except for i=d−1\n2+ 1in case hat dis odd let\n˜Ui(xi) =\n2xi miix2\ni\nm1ixi\n...\nmi−1,ixi\n0 1Ii0i−1\n0⊺\ni\n∈Rri−1,ri, i= 2, . . . ,°d−1\n2§\n,\n˜Ui(xi) =\n1\n2xi\nmi,ix2\nimd,ixi···mi+1,ixi102,d−i 02\n0d−i Id−i 0d−i\n∈Rri−1,rii=õd+ 3\n2û\n, . . . , d −1.\nIfdis odd we define the middle square component Ui(xi)fori=d−1\n2+ 1by\n˜Ui(xi) =\nmiix2\nimi,i+1xi···mi,dxi1\nm1,ixi\n...\nmi−1,ixi\n1 01\n2M1:i−1,i+1:d 0i−1\n0⊺\nd−i\n, i=d−1\n2+ 1.\nFordeven we define for i=d\n2+ 1 =\u0004d+3\n2\u0005\nUi(xi) =\n0 1\n1 00⊺\nd−i\n0i−1M1:i−1,i+1:d0i−1\n0⊺\nd−i\n˜U(xi)\nand in any other case set Ui(xi) =˜Ui(xi).\nLemma D.2 (Measures of independent variables) .Letd∈Nandf:Rd→Radmit the form f(x) =Pd\ni=1fi(xi)\nforfi∈ C2(R,R),i= 1, . . . , d . Then both fandLin(f) + NonLin( f)have FTT rank r= (2, . . . , 2)⊺∈Nd−1.\nProof. The result follows immediately from [30, Theorem 2] and the structure of Lin(f) + NonLin( f).\n25GM with TT approximations of HJB equations A P REPRINT\nE Details of HJB solutions\nLetp(mon)\nα forα∈N0denote the α-th monomial, i.e. p(mon)\nα (x) =xα. As in Section 3, let Ti,n∈Rn+1,n+1denote\nthe basis transformation matrix between Legendre polynomials of degree non[ai, bi]and the monomials up to degree\nn.\nE.1 Derivation matrices in the linear operator part\nNote that for every i= 1, . . . , d and for every c∈Rni+1, we have\n∂2\nxniX\nα=0cαp(mon)\nα =niX\nα=0(Mi\nddc)αp(mon)\nα, (E.1)\nwhere\nMi\ndd=\n2\n6\n...\nni(ni−1)\n0\n0\n∈R(ni+1)×(ni+1). (E.2)\nIn a similar way, we get\nx∂xniX\nα=0cαp(mon)\nα =niX\nα=0(Mi\nxdc)αp(mon)\nα, (E.3)\nwhere\nMi\nxd=à0\n1\n2\n...\nnií\n∈R(ni+1)×(ni+1). (E.4)\nWith the basis transformation matrix Ti,niwe can express the action of these operators on the coefficients of the\noriginal basis pi. In particular, we have\n(∂2\nx+x∂x)niX\nα=0cαpi\nα= (∂2\nx+x∂x)niX\nα=0(Ti,nic)αp(mon)\nα\n=niX\nα=0((Mi\ndd+Mi\nxd)Ti,nic)αp(mon)\nα\n=niX\nα=0(T−1\ni,ni(Mi\ndd+Mi\nxd)Ti,nic)αpα=niX\ni=0(Dic)αpi\nα,(E.5)\nwithDi:=T−1\ni,ni(Mi\ndd+Mi\nxd)Ti,ni∈R(ni+1)×(ni+1).\nE.2 Derivation of the nonlinear part\nNote that for every c∈Rni+1, we have\n∂xniX\nα=0cαp(mon)\nα =niX\nα=0(Mi\ndc)αp(mon)\nα, (E.6)\nwhere\nMi\nd=à1\n2\n...\nni\n0í\n∈R(ni+1)×(ni+1), (E.7)\n26GM with TT approximations of HJB equations A P REPRINT\nand hence\n∂xniX\nα=0cαpi\nα=∂xniX\nα=0(Ti,nic)αp(mon)\nα (E.8)\n=niX\nα=0(Mi\ndTi,nic)αp(mon)\nα (E.9)\n=niX\nα=0(T−1\ni,niMi\ndTi,nic)αpi\nα=niX\nα=0(Dxic)αpi\nα, (E.10)\nwithDxi=T−1\ni,niMi\ndTi,ni∈R(ni+1)×(ni+1).\nE.3 Estimating the eigenvalues for Gaussian distributions\nLetn= (2, . . . , 2)⊺∈Ndandg(x) =1\n2x⊺Axfor all x∈Rd, where A∈Rd×d. Note that ∇g(x) =Axand hence\nfor any v∈span Π n, we have NonLin gv=⟨Ax,∇v⟩= (Ax)· ∇v. Hence, the linearized HJB at greads\n˙v=x· ∇v+ ∆v−2⟨Ax,∇v⟩+⟨Ax, Ax ⟩, v (0) = g (E.11)\nIn order to determine the stiffness, we need to determine the effect of this right-hand side on the coefficient tensor of\nv. From now on, let v=vC, where C∈Rn+1. Assume that vC(x) =1\n2x⊺ˆCxfor some ˆC∈Rd×d, i.e.VChas only\nterms with degree 2. We know that x· ∇vC+ ∆vC=vLCand⟨Ax,∇vC⟩=Pd\ni=1Pd\nj=1aijxj∂ivC. Now, note\nthat∂ivC=v(Id⊗...⊗Pi⊗...⊗Id)C, where\nPi=T−1\ni,2 0 1 0\n0 0 2\n0 0 0!\nTi,2, (E.12)\nandxjvC=v(Id⊗...⊗Xj⊗...⊗Id)C, where\nXj=T−1\nj,2 0 0 0\n1 0 0\n0 1 0!\nTj,2. (E.13)\nIn the case of i=jthis leads to xi∂iVC=v(Id⊗...⊗T−1\ni,2Mi\nxdT−1\ni,2⊗...⊗Id)C, where\nMi\nxd= 0 0 0\n0 1 0\n0 0 2!\n. (E.14)\nLet\nM=dX\ni,j=1aijId⊗. . .⊗Id⊗Pi⊗Id⊗. . .⊗Id⊗Xj⊗Id⊗. . .⊗Id, (E.15)\nthen we have ⟨Ax,∇vC⟩=vMC.\nDiagonal covariance. We consider the special case where A= diag( aii, i= 1, . . . , d )is a diagonal matrix. In this\ncase, we have\nM=dX\ni=1aiiId⊗. . . I d⊗T−1\ni,2Mi\nxdTi,2⊗Id⊗. . .⊗Id (E.16)\nand hence the linear operator governing the right-hand side is given by\nL−2M=dX\ni=1Id⊗. . . I d⊗Hi⊗Id⊗. . .⊗Id, (E.17)\nwhere Hi:=Di−2aiiT−1\ni,2Mi\nxdTi,2=T−1\ni,2(Mi\ndd+ (1−2aii)Mi\nxd)Ti,2and\nMi\ndd+ (1−2aii)Mi\nxd= 0 0 2\n0 1−2aii 0\n0 0 2(1 −2aii).!\n(E.18)\n27GM with TT approximations of HJB equations A P REPRINT\nThe point spectrum σ(Hi)ofHiis given by σ(Hi) ={0,1−2aii,2(1−2aii)}. The eigenvector corresponding to\nthe eigenvalue with largest absolute value 2(1−2aii)is given by\nˆvi=T−1\ni,2Å1\n1−2aii,0,1ã⊺\nTi,2. (E.19)\nLetvidenote any eigenvector of Hi. Then, v= (v1⊗. . .⊗vd)is an eigenvector of L−2M. Since this leads to 3d\npossible combinations, the whole spectrum of L−Mis defined by such eigenvectors. Moreover, since the eigenvalues\nHiare bounded by |2(1−2aii)|, the largest absolute eigenvalue of L−2Mis given by 2Pd\ni=1|1−2aii|.\n28"    },    {        "title": "2402.15287v1.Fractional_phase_jumps_in_stochastic_systems_with_tilted_periodic_double_well_potentials.pdf",        "content": "Fractional phase jumps in stochastic systems with tilted periodic double-well\npotentials\nMartin ˇZonda,1,∗Wolfgang Belzig,2Edward Goldobin,3and Tom´ aˇ s Novotn´ y1,†\n1Department of Condensed Matter Physics, Faculty of Mathematics and Physics,\nCharles University, Ke Karlovu 5, 121 16 Praha 2, Czech Republic\n2Fachbereich Physik, Universit¨ at Konstanz, D-78457 Konstanz, Germany\n3Physikalisches Institut, Center for Quantum Science (CQ) and in LISA+,\nUniversitat Tubingen, Auf der Morgenstelle 14, D-72076 Tubingen, Germany\n(Dated: February 26, 2024)\nWe present a theoretical investigation of the stochastic dynamics of a damped particle in a tilted\nperiodic potential with a double well per period. By applying the matrix continued fraction tech-\nnique to the Fokker-Planck equation in conjunction with the full counting statistics and master\nequation approaches, we determine the rates of specific processes contributing to the system’s over-\nall dynamics. At low temperatures, the system can exhibit one running state and two distinct\nlocked metastable states. We focus primarily on two aspects: the dynamics of phase jumps, which\nare rare thermally induced particle jumps over potential maxima, and their impact on the overall\nvelocity noise; and the retrapping process, involving the transition from the running to the locked\nmetastable states. We demonstrate the existence of fractional (in units of 2 π) phase slips that differ\nqualitatively from conventional 2 πjumps observed in single-well systems. Fractional phase slips sig-\nnificantly influence the system dynamics even in regimes dominated by dichotomous-like switching\nbetween running and locked states. Furthermore, we introduce a simple master equation approach\nthat proves effective in analyzing various stages of the retrapping process. Interestingly, our analysis\nshows that even for a system featuring a well-developed double-well periodic potential, there exists\na broad parameter range where the stochastic dynamics can be accurately described by an effective\nsingle-well periodic model. The techniques introduced here allow for valuable insights into the com-\nplex behavior of the system, offering avenues for understanding and controlling its steady-state and\ntransient dynamics, which go beyond or can be complementary to direct stochastic simulations.\nI. INTRODUCTION\nThe stochastic dynamic motion of a particle in tilted\nperiodic potentials plays an important role in the stud-\nies of various physical phenomena including Josephson\njunctions (JJs) [1–5], microparticles confined in shaped\nlaser beams [6–13], dynamics of charge density waves [14],\ncrystal surface melting [15, 16], ratchet and molecular\nmotors [17–19], cold atoms in optical lattices [20], differ-\nent bio-physical processes [21] as well as in investigations\nof phenomena such as anomalous diffusion and memory\neffects [22–26]. A well-paid effort has already been in-\nvested in the theoretical analysis of such systems [27–30],\nyet there are still plenty of open questions often inspired\nby the recent experimental realizations in different sys-\ntems, e.g., Refs. [4, 13, 31, 32]. Moreover, experimental\nprogress in different fields called for readdressing some\nold problems such as escape and retrapping of the Brow-\nnian particle from or to potential minima [24, 33–37], the\nstatistics of thermally activated jumps of the particle by\ninteger multiples of 2 π(i.e., phase jumps) [35, 38–40] and\nmultistability [24, 41, 42].\nIn this respect, of special interest are systems where the\nstochastic dynamics of the particle is affected by a bihar-\n∗martin.zonda@matfyz.cuni.cz\n†tomas.novotny@matfyz.cuni.czmonic potential containing two local minima per period\nU(φ) =−φ ib+U0(φ), (1)\nU0(φ) =−αcosφ+1\n2cos 2φ,\nwhere ibis a static external bias force, i.e., a potential\ntilt,U0(φ) is the untilted periodic double-well poten-\ntial, where the coefficient αtunes the ratio of the first\nto the second harmonic contribution and φ= 2πx/λ ,\nwhere xis the coordinate (position) and λthe period.\nConsistent with papers focusing, for example, on the\nJosephson junctions, we call the variable φphase. The\npotential (1) plays an important role in the theoret-\nical description of numerous physical systems, includ-\ning JJs with substantial second harmonic in the current\nphase relation (CPR) [4, 43–56] in particular in Joseph-\nson diodes [31, 57, 58] or various ratchet systems [59–61]\nand molecular motors [62, 63].\nA careful analysis of the motion of the particle in the\ntilted double-well potential has already led to some im-\nportant results. For example, it explained the existence\nof two critical escape currents from the superconduct-\ning to the resistive state observed experimentally for\nthe JJs with doubly degenerate ground state (i.e., φ-\nJunctions) [48], pointed to rather nontrivial retrapping\ndependencies of the phase which can lead to a butterfly\neffect [50] and showed the existence of chaotic phase tra-\njectories in various generalizations of the famous RCSJ\nmodel [64, 65]. The case α= 0, where only the secondarXiv:2402.15287v1  [cond-mat.stat-mech]  23 Feb 20242\nharmonic is present in the potential, plays an important\nrole in the studies of unconventional junctions which can\nundergo the so-called 0- πtransition [54, 66]. Moreover, a\nBrownian particle moving in the potential (1) with addi-\ntional harmonic terms, characterized by asymmetric mo-\nbility considering bias force, is an archetypal model of\nthe ratchet and diode systems [57–59, 61].\nA crucial component in all of the above phenomena is\nnoise. In combination with different dampings, a system\nwith potential (1) can show a wide variety of regimes,\neach important for different physical realizations. Here,\nwe provide a systematic analysis of a general case. We\nstart with the simple strong damping parameter regime\nand proceed to the more complicated intermediate and\nweak damping regimes. We use velocity-noise to iden-\ntify three main dynamic regimes [35]: the thermal noise\nregime; the phase-jumps (PJs) regime, and the switching\nregime. In each, we focus on the analysis of the dominant\ndynamics. In particular, we investigate the statistics of\nthe phase jumps and their contribution to the overall\nvelocity noise in the PJs regime and the escape and re-\ntrapping processes in the switching one. For this pur-\npose, we combine the matrix continued-fraction (MCF)\nmethod with other techniques. In particular, in Sec. II\nwe introduce a combination of the matrix continued-\nfraction (MCF) technique [27] with full counting statis-\ntics (FCS) [67–69] for the analysis of the multiple PJs. Its\nbiggest advantage over the stochastic simulations is that\nit is straightforward to access the steady state. As such,\nit is suitable for the calculation of rates for rare events.\nTherefore, in the relevant regime, this method allows de-\ncomposition of the phase dynamics into independent ele-\nmentary processes [70–72], constituted by single or mul-\ntiple phase jumps. Using this method, we demonstrate\nthe existence of the so-called fractional PJ in Sec. III B 1.\nIn Sec. III B 2 we investigate retrapping processes in the\nswitching regime. Here we combine the MCF with an\neffective master equation approach describing the transi-\ntion of the system between its three metastable regimes.\nOn top of the steady-state studies, we also discuss in\nSec. III B 3 a dynamical retrapping scenario.\nThere is a particular conclusion of our research that is\nworth foreshadowing here. Namely, for a broad range of\nparameters, even a system with a well-developed double-\nwell potential can be faithfully described by a single-well\nmodel.\nII. MODEL AND METHODS\nA. Model and dynamical regimes\nWe consider a stochastic motion of a particle in a po-\ntential U(φ) described by the dimensionless Langevine\nequations\n 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2\n 0  0.5  1  1.5  2α = 0.4\nΘ = 0.1a\nABvγ\nibγ = 5.0\nγ = 0.2\n2πdΓΑ\n−1\nΓΑ\n1ΓΒ\n−1 ΓΒ\n1ΓΒ\n2 cU(ϕ)\nϕAm Am+1 Am-1 Bm-1 Bm Bm+1b\nam am+1 am-1bm-1 bmU(ϕ)Figure 1. (a) Examples of v−ibcharacteristics represent-\ning the strong damping ( γ= 5 plotted with red lines) and\nthe weak damping regime ( γ= 0.2 plotted with black lines).\nThe thin dashed lines show the deterministic (noise-free) so-\nlutions. The solid thick lines represent stationary solutions\nfor temperature Θ = 0 .1. (b) -(c) Illustrations of the poten-\ntial Eq. (1) with ΓA\nnand ΓB\nnbeing the rates of the forward /\nbackward phase jumps over nlocal maxima. In general the\ndistance dis not equal to π.\n∂v(τ)\n∂τ=−γv(τ)−∂U(φ)\n∂φ+ζ(τ),\nv(τ) =∂φ/∂τ,(2)\nwhere vis the velocity of the particle, γis the fric-\ntion coefficient and ζrepresents a Gaussian white noise\nwith the zero mean ⟨ζ(τ)⟩= 0 and correlation function\n⟨ζ(τ1)ζ(τ2)⟩= 2γΘδ(τ1−τ2) where Θ is the dimension-\nless temperature. The associated Fokker-Planck equa-\ntion [27] of Eqs. (2) for the probability distribution func-\ntionW(φ, v, τ ) in the case of potential (1) reads\n∂\n∂τW(φ, v;τ) =∂\n∂v(γv+αsinφ−sin 2φ−ib)W\n−v∂\n∂φW+γΘ∂2\n∂v2W (3)\n≡LFPW(φ, v;τ).\nThe derivation of the average velocity\n⟨v⟩=2πˆ\n0dφ∞ˆ\n−∞dvvW stat(φ, v) (4)\nfollows closely the standard MCF method [27, Sec. 11.5]\nand the derivation of (zero-frequency) velocity noise\nS=∞ˆ\n−∞dτ\u0000\n⟨v(τ)v(0)⟩ − ⟨v(τ)⟩⟨v(0)⟩\u0001\n(5)\nin the stationary state Wstat(φ, v)≡limτ→∞W(φ, v;τ)\nfollows that of Ref. [35].\nFigs. 1(b)-(c) illustrate the system for zero and finite\nbias force. The potential (1) has for α < 2 and zero3\nbias two types of minima marked as AandBand two\ntypes of maxima ( aandb). Consequently, there are also\ntwo critical bias forces: icAwhere the minimum Aand\nmaximum adisappear and icBwhere the minimum B\nand maximum bdisappear. The system can be in three\ndistinct stationary regimes. Namely, the particle can be\nrunning, i.e., at high bias force, or it can be locked in\none of the potential minima types. Consequently, for\nthe noiseless case, the initial position of the particle can,\ndepending on the damping, play an important role even\nfor the steady state, as illustrated in Fig. 1(a).\nTo explain this, let us consider two limiting cases, one\nwith a strong damping and the other with a weak damp-\ning. In both cases, first slowly (compared to any other\nprocess) ramp the bias force from ib= 0 to icBwith the\naim of observing the escape of the particle from a po-\ntential well. Afterwards, we assume a backward ramping\nfrom ib≫icBtoib= 0 with the aim of observing the\nrecapture (retrapping) of the running particle in one of\nthe wells.\nFor strong damping [ γ= 5 in Fig. 1(a)] there is only\na single escape bias force and it is identical to the re-\ntrapping force. Due to the strong damping, the initial\nposition of the trapped particle does not play a role in\nthis regime. The particle will start running only after the\nhigher maximum bdisappears and, vice versa, will be re-\ntrapped at the minimal bias force for which the maximum\nbappears. Therefore, both the critical escape force and\nthe retrapping force are equal to icB(red dashed line).\nIn contrast, there are two possible escape tilts for weak\ndamping ( γ= 0.2). If the particle initially is trapped at\nthe minimum A, it will obtain enough inertia to overcome\nthe still existing maximum balready at the first critical\nforce icA. However, if it is initially trapped at the min-\nimum Bit will stay there until the second critical force\nicBis reached. The retrapping is also more complicated\nthan in the strong-damping case. If the particle is already\nrunning, then it has enough inertia to overcome the local\nmaxima existing below the critical force icBor even icA\nup to the actual retrapping force ir[73], black dashed\nlines in Fig. 1(a). The retrapping force is determined\nby the energy balance between the energy supply of the\nbias force and the dissipation [27, 50]. This means that\nthere is a region of coexistence of the running and the\nlocked state solutions. The question in which minimum\nwill the particle be trapped requires careful analysis be-\ncause it is a parameter-sensitive process which can lead\nto a deterministic butterfly effect [50, 74].\nThe noise makes the dynamics even more complicated.\nAdditional processes, such as switching between running\nand locked states or occasional jumps of the particle over\nthe neighboring maxima [see Fig. 1(c)], are possible for\nnonzero temperature. These processes affect the station-\nary probability distribution function and, therefore, also\nthe relevant mean values. The solid curves plotted in\nFig. 1(a) represent the mean stationary velocity of the\nparticle for temperature Θ = 0 .1. As a consequence of\nthe noise, the strong damping case shows a smooth andshallow crossover from the locked to the running state,\nwhile the weak damping case exhibits a much sharper\ntransition placed between the retrapping and the lower\nof the escape bias forces.\nThere are three main components that contribute to\nthe overall velocity noise of this model [33, 35]. The ther-\nmal noise component dominates close to the equilibrium\nib= 0. The second component is the switching noise\ncoming from the switching between running and locked\nstates, which is, for low enough temperatures, an effective\ndichotomous-like process. The third component is the\nshot noise (PJs regime) related to rare jumps of the par-\nticle over single or multiple maxima. The regimes where\nparticular components prevail can be identified from the\nFano factor F≡S/(2π⟨v⟩) [35], which is a normalized\nnoise-to-signal ratio.\nB. Full-counting statistics for the phase jump\ndynamics\nThe phase jumps are, for low enough temperature and\nweak bias force, well defined distinct (rare) events that\nsignificantly influence the overall dynamics. To calcu-\nlate the rates of these events, we have adapted the full-\ncounting statistics technique previously used to study\njump probabilities in single-harmonic systems [35, 67–\n69]. The double-well character of the potential (1) re-\nquires some generalizations of this method which we\npresent here.\nIn the first step we have approximated the solution\nof the Fokker-Planck equation (3) for sufficiently low\nbiases and temperatures by a weighted sum of quasi-\nequilibrated sharp (Θ ≪1) Gaussian distributions [69]\naround the two types od local minima\nW(φ, v;τ)≈X\nmPA\nm(τ)w(φ−φA\nm, v)\n+X\nmPB\nm(τ)w(φ−φB\nm, v), (6)\nwhere: w(φ, v) =exp(−φ2/2Θ)exp(−v2/2Θ)\n2πΘ.\nHere φA\nmandφB\nmare the positions of the m-th poten-\ntial minima and PA\nm(τ) and PB\nm(τ) are the corresponding\ntime-dependent weights. These are assumed to satisfy\nthe (Markovian) master equations (ME)\ndPA\nm\ndτ=X\nn\u0000\nΓA\n2nPA\nm−n−ΓA\nnPA\nm+ ΓB\n2n−1PB\nm−n\u0001\n,(7)\ndPB\nm\ndτ=X\nn\u0000\nΓB\n2nPB\nm−n−ΓB\nnPB\nm+ ΓA\n2n+1PA\nm−n\u0001\n,(8)\nwhere ΓA\nnis the rate of a phase jump from the potential\nwellAand ΓB\nnfrom the well Bover npotential local\nmaxima to another local minimum. Rates with even n\nbelong to phase jumps between minima of the same kind\n(A→A,B→B) and odd ones to phase jumps between4\nminima of different kinds ( A→B,B→A). The neg-\native n’s correspond to the jump rates in the direction\nopposite the slope of the bias (up the hill). Following the\nstandard FCS methodology [69], we can evaluate the k-\ndependent cumulant generating function (CGF) for long\ntimes from the ME and equate it with the CGF of the\nfull model.\nF(k;τ→ ∞ )≡ln∞ˆ\n−∞dφeikφ∞ˆ\n−∞dvW(φ, v;τ→ ∞ )\n(9)\ncalculated by MCF as explained in the Appendix of\nRef. [35]. The approximate probability density follow-\ning from the ME reads\nexp\u0002\nFPJ(k;τ)\u0003\n=eikφAX\nmPA\nmeik2πm\n+eikφBX\nmPB\nmeik2πm\n≡PA(k, τ) +PB(k, τ). (10)\nThe probability densities must satisfy the matrix equa-\ntion\nd\ndτ\u0012\nPA(k, τ)\nPB(k, τ)\u0013\n=\u0012\nHPJ\n11HPJ\n12\nHPJ\n21HPJ\n22\u0013\u0012\nPA(k, τ)\nPB(k, τ)\u0013\n,(11)\nwhere the matrix HPJ(k) reads\nHPJ(k) =\n−P\nnΓA\nn+ ΓA\n2neik2πnP\nnΓB\n2n−1eik(2πn−d)\nP\nnΓA\n2n+1eik(2πn+d)−P\nnΓB\nn+ ΓB\n2neik2πn\n.\n(12)\nThedfactor in the exponents of above equations is the\ndistance between the neighboring minima d=φB\nm−φA\nm\n[see Fig. 1(c)] which depends on the potential parameters\nαandib. In contrast to even phase jumps, where the\ndistance traveled is always an integer multiple of 2 π, odd\njumps overcome a distance of 2 πn+dwhere in general\nd̸=π, therefore, we call these jumps fractional .\nAnalogously to Appendix in Ref. [35] we use MCF to\ncalculate the two eigenvalues λ0(k), λ1(k) with the largest\nreal parts and related eigenvectors u0(k, φ, v ),u1(k, φ, v )\n[27, Sec. 9.3]. The above two eigenvalues are in the rele-\nvant regime well separated from all the others.\nWe construct two component vectors from the eigen-\nvectors by integration over the basins of attraction of the\nrespective nonequivalent local potential minima\nUn(k) =\n´∞\n−∞dv´φa\nm\nφb\nm−1dφun(k, φ, v )\n´∞\n−∞dv´φb\nm\nφamdφun(k, φ, v )\n. (13)\nThey can be used to reconstruct the matrix HPJ(k) =\nU(k)LPJ(k)U−1(k) where\nLPJ(k) =\u0012\nλ0(k) 0\n0λ1(k)\u0013\n(14)\n-9-6-3 0-3 0 3\n-0.5 0 0.5 1 1.5\n10-5ka\nλ0λ1\nRe λ(k)Im λ(k)-9-6-3 0\n-0.5  0  0.5  1  1.5α=0.06\nib=0.3\nγ=5\nΘ=0.0810-5\nbRe λ(k)\nkλ0λ1\n-3 0 3\n-0.5  0  0.5  1  1.510-5\ncIm λ(k)\nkλ0λ1Figure 2. The two eigenvalues with the biggest real part plot-\nted at the Riemann surface (a) and their projections to the\nreal (b) and the imaginary (c) plane.\nandU(k) is the square matrix of vectors U0(k) and U1(k).\nNote that due to the second harmonics in the potential\nand similarly to other non-Hermitian Hamiltonian sys-\ntems [75, 76], the eigenvalues (14) can have complicated\ntopological properties, as shown in Fig. 2. The two eigen-\nvalues can, for small enough α, connect at kc= 1/2 +m,\nwhere mis an integer, and smoothly continue each other\non the Riemann surface; see Fig. 2(a). Therefore, the\nreal parts of these two eigenvalues, plotted as functions\nofk, touch on kc[Fig. 2(b)], and there is a discontinu-\nity in the imaginary part of the eigenvalues at the same\npoints [Fig. 2(c)]. These discontinuities must be treated\nwith care in the numerical evaluation of the eigenvalues\nand eigenvectors.\nHaving the reconstructed matrix HPJone can evaluate\nthe rates of the even phase jumps (2 πn- PJ between the\nsame kind of minima) using the transformations\nΓA\n2n=1/2ˆ\n−1/2HPJ\n11e−ik2πndk,\nΓB\n2n=1/2ˆ\n−1/2HPJ\n22e−ik2πndk, (15)\nand the odd ones between the different minima types as\nΓA\n2n+1=1/2ˆ\n−1/2HPJ\n21e−ik(2πn+d)dk,\nΓB\n2n−1=1/2ˆ\n−1/2HPJ\n12e−ik(2πn−d)dk. (16)\nThis method also provides a simple tool to check its\nvalidity. The approximated mean velocity and velocity5\n 0 0.5 1 1.5 2\n 0 0.2  0.4  0.6  0.8  1 1.2  1.4γ = 5Θ=0.08aF\nibα=1.0\nα=0.1\nα=0.0\n10-610-510-410-3\n 0 0.05  0.1  0.15  0.2  0.25  0.3γ = 5Θ=0.08b\nib=0.3Γ\nαΓA\n1\nΓB\n1\nΓD\n10.50.60.70.80.91\n 0  0.2  0.4F\nαib=0.2\nib=0.3\nib=0.5\nFigure 3. (a) Examples of typical low-temperature Fano fac-\ntorF=S/2π⟨v⟩for strong damping plotted as a function\nof bias force for different parameters α. The inset illustrates\nhow the 0 .5 Fano factor plateau transits into Fano plateau\nwith value one as αincrease. (b) A comparison of the α-\ndependence of the single-phase-jumps rates evaluated numer-\nically (solid lines) with the analytical Kramers formula re-\nsult for the overdamped case [69, 77] (dashed lines). The\ngreen bullets represent rates obtained via the simplified model\nwhere only 2 πnPJ were taken into account.\nnoise can be calculated directly from the rates by evalu-\nating the formulas\n⟨v⟩= lim\nτ→∞−i\nτ∂FPJ(k, τ)\n∂k\f\f\f\f\nk=0, (17)\nS= lim\nτ→∞−1\nτ∂2FPJ(k, τ)\n∂k2\f\f\f\f\nk=0. (18)\nThese results can be compared with the mean velocity\nand overall noise obtained directly from the MCF method\ncalculations [35].\nIII. RESULTS\nA. Strong damping\nWe start our analysis with the strong damping case.\nThe main reason is that in this regime the phase dynam-\nics is much simpler than for the intermediate and weak\ndamping. Nevertheless, it is still far from trivial.\n1. Fano factor and phase jumps\nIn Fig. 3(a) we show the Fano factor in the strongly\ndamped regime represented by γ= 5 for three values of\nαat temperature Θ = 0 .08. The Fano factor exhibits a\ncharacteristic divergence close to the equilibrium ( ib= 0)\ndue to the finite thermal noise. For α= 0, it follows for-\nmula F=1\n2coth( πib/2Θ) [69, Eq. (28)] for small iband\nα= 0 with a plateau at the Poissonian value of F= 0.5\n(note that we still normalize Fto 2π). This is in agree-\nment with the dominant contribution to the noise fromthe single PJ [illustrated in Fig. 1(c)] over a distance\nd=πbetween the equivalent neighboring minima. The\nsituation for the finite αis more complicated. The range\nin which single PJs are the dominant source of the over-\nall noise is not marked by a clear plateau. Rather for\n0< α≪1 [α= 0.1 in Fig. 3(a)] we observe a slight\nslope in the Fano factor. This reflects the fact that for\nfinite αthe prevailing velocity noise contribution consists\nof a nontrivial combination of the single phase jumps for-\nward over the distance of φB\nm−φA\nm=dand backward\nbyφA\nm+1−φB\nm= 2π−d[see Fig. 1(c)] where ddepends\nboth on αandib. As already stated, because d̸=πwe\nrefer to these events as the fractional phase jumps .\nWith increasing αthe Fano factor increases smoothly\nin this region from 0 .5 signaling only PJs over the dis-\ntance πto 1 suggesting a single PJs over 2 πas shown in\nthe inset of Fig. 3(a). Interestingly, for Θ = 0 .08 the Fano\nfactor approaches one even for values of ibthat are still\nwell below icA, therefore, in a regime where both minima\nstill exist. For example, the critical current icAis approx-\nimately 0 .396 at α= 1, yet its Fano factor follows the\nanalytical result F= coth( πib/Θ), valid for systems with\nonly the first harmonic, even below this value. Neverthe-\nless, this can be understood as a consequence of the large\nγand can be explained using the FCS method together\nwith a simplified model of the elementary PJ processes.\nBecause only jumps over a single (uneven) maxima\nare realized in the locked state of the strong-damping\ncase the overall dynamics of this regime can be de-\nscribed using just four rates: ΓA\n1and ΓB\n1for the for-\nward single jumps and ΓA\n−1and ΓB\n−1for the backward\nsingle jumps. Moreover, the backward rates can be ne-\nglected if the bias force is strong enough. The typical\ndependencies of ΓA\n1and ΓB\n1onαin this regime rep-\nresented by the bias force ib= 0.3 and the damping\nγ= 5 are plotted in the Fig. 3(b) (solid lines). Both\nΓA\n1and ΓB\n1closely follow the Kramers formula for es-\ncape across the adjacent barrier for overdamped case\nΓX\n1=1\n2πp\n|U′′(φx)|U′′(φX)e−∆UX/Θ[69, 77] (where\n∆UX=U(φx)−U(φX), and x=a, b;X=A, B) plotted\nwith dashed lines of adequate colors. Note that because\nof the increasing difference ∆ UB−∆UAthe ratio\nΓA\n1/ΓB\n1∼exp [(∆ UB−∆UA)/Θ] (19)\nincreases exponentially with α. Consequently, for high\nenough αand low enough Θ the average waiting time for\nthe escape from minimum A τA→B= 1/ΓA\n1is negligible\ncompared to the waiting time for the escape from min-\nimum B(τB→A= 1/ΓB\n1). Therefore, in the long term,\nevery single phase jump Bm−1→Amis immediately fol-\nlowed by a single phase jump Am→Bm. The combina-\ntion of these two fractional PJs is effectively a complete\n2πphase jump. This is shown in Fig. 3(b) by the green\nbullets that were obtained via a simplified model where6\nonly the 2 πnphase jumps were considered [35]\nΓD\nn=1/2ˆ\n−1/2λ0(k)e−2πikndk. (20)\nTheir match with the ΓB\n1rate for high enough αis con-\nsistent with the Fano-factor value of one in the inset of\nFig. 3(a). This has interesting physical consequences. If\nthe temperature is low enough, then, because Θ is in the\ndenominator of the exponent of Eq. (19), any strongly\ndamped φ-junction or another equivalent system describ-\nable by a double harmonic potential with a finite bias ib\nwill in the steady state resemble a simple single harmonic\nsystem. As such, it can be described by the analytical\nformulas derived for the single-harmonic potential.\nB. Intermediate and weak damping\nThe stochastic dynamics of the particle in the tilted\ndouble-well periodic potential in the regime of interme-\ndiate and weak damping is significantly richer than in\nthe strong damping case. For example, for weak bias\nforce, there are phase jumps over multiple maxima, and\nfor stronger bias, a complicated switching between run-\nning and locked solutions is the prevailing source of the\nvelocity noise. Even the retrapping of a particle from the\nrunning to locked regime has complex dynamics, as it is\ndiscussed below.\nThe dependencies F−ibfor the underdamped case\n[plotted in Fig. 4(a)] differ qualitatively from the strongly\ndamped case [Fig. 3(a)]. The dominant feature of the\nFano factor is a huge peak [note the logarithmic scale\nin Fig. 4(a)] for finite ib. As was shown in our previous\nstudy of the RCSJ model with single harmonics CPR [35],\nthis peak is a consequence of the switching process be-\ntween coexisting, but well separated, running and locked\nstates in this range of ib. This interpretation is also valid\nfor the double-well potential, where, however, there exist\ntwo locked states as illustrated in Fig. 1.\nWe support this claim in Fig. 4(b) where an example\nof the stationary distribution function W(v, φ) is plotted\nfor the parameters (see figure description) close to the\nmaximum of the peak. Here, the two distinct peaks at\nv≈0 centered around potential minima signal the two\nlocked states. The continuous ridge that spreads above\nthem is the running phase. In this regime, the prevailing\ncontribution to the overall velocity noise comes from the\nswitching between these three well-separated metastable\nstates.\nWhen we lower ibto the regime in between the thermal\ndivergence at ib→0 and the switching maximum, multi-\nple fractional phase jumps (MPJ) become the prevailing\nsource of the velocity noise. We now analyze the regimes\nof phase jumps and switching separately and then show\nhow they transition smoothly into each other.\n100101102103\n 0  0.1  0.2  0.3  0.4  0.5 γ = 0.2Θ = 0.1aF\nibα=0.4\nα=0.2\nα=0.0\n-0.200.20.40.6\n-1 -0.5  0  0.5  1v\nϕ/π\n-0.200.20.40.6\n-1 -0.5  0  0.5  1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6\nΓrAΓrBΓeAΓeB\nΓAB\nΓBAbW(v,ϕ)Figure 4. (a) Three examples of a typical low-temperature\nFano factor F=S/2πvdependence on the bias force for weak\ndamping. The lines were calculated by the MCF method, and\nthe bullets show the ME results from Eq. (18). Vertical arrows\nmark the related noiseless retrapping forces. (b) An illustra-\ntion of the stationary distribution function W(v, φ) in the\nswitching regime calculated for γ= 0.2,α= 0.02,ib= 0.38,\nand Θ = 0 .08. The arrows illustrate the processes and their\nrates as used in Eq. (22). Namely, the red arrows represent\nescape from the locked states to the running one, the black\narrows represent retrapping of the particle, and the blue ar-\nrows are the phase jumps between the locked states AandB.\nThe gray dashed curves show the separatrices between the\nrunning solution and the locked solution in the well A(solid\nline) and/or well B(dashed line), respectively.\n1. Phase-jumps\nIn Fig. 5 we show the rates of multiple phase jumps\n(MPJs) for weak damping γ= 0.2, low temperature\nΘ = 0 .1, and different values of αandib, where nis\nthe number of maxima bridged by a single process. The\nblue columns represent the rates of jumps from the min-\nimaAand the red ones from the minima B. The n\ndependence of the PJ rates for α= 0 shown in panel (a)\nis the same as that for the simple single-harmonic po-\ntential [35] up to a redefinition of the jump length. We\nuse it as one of the tests of our extended FCS method.\nIn the inset of Fig. 5(a) we show the ratios Γ −n/Γnfor\nn= 1,2,3 obtained numerically with FCS (bullets and\ncrosses) and analytically from the detailed balance con-\ndition Γ −n/Γn= exp( −πnib/Θ) with the potential drop\nofπibnalong the phase jumps (lines). There is a perfect\nmatch over seven orders of magnitude for each n= 1,2,3\nproving the reliability of the FCS plus MCF method.\nThe profile of the rates for potentials with finite αis\nmore complicated. The ndependencies of the PJs rates\ndiffer qualitatively between jumps starting in different\nminima, as well as between odd and even n’s. We show\nthis in Figs. 5(b)-(c). The rates Γ A nand Γ B ndiffer\nby several orders of magnitude for odd n’s [see α= 0.2\ncase in Fig. 5(c)], but are comparable for even n’s. This\ncan be rationalized by analyzing the “trajectories” of the\nparticular PJs following the illustrations in Fig. 1(b)(c).\nThe jumps Am→Am+n/2andBm→Bm+n/2, where\nnis an even integer, are indeed comparable. Here, the\nparticle had to overcome the same number of maxima7\n10-710-610-510-4\n-2  0  2  4  6  8  10a\nα=0.0\nib=0.1\nγ =0.2\nΘ=0.1Γ\nn10-710-610-510-410-3\n-2  0  2  4  6  8  10b\nα=0.1\nib=0.14Γ\nnΓA\nn\nΓB\nn\n10-710-610-510-410-3\n-2  0  2  4  6  8  10c\nα=0.2\nib=0.14Γ\nnΓA\nn   \nΓB\nn   \nΓD\nn/2 10-610-510-410-310-210-1\n 0  0.1  0.2  0.3n=1\nn=2\nn=3Γ-n / Γn\nib\nFigure 5. The rates of MPJ (of order n) for different values of the bias force iband ratio α. The inset in (a) is a verification of\nthe detailed balance condition. The blue columns describe jumps starting at minimum A, red at minimum B(a small horizontal\nshift is introduced for visibility), and the green columns in panel (c) show the rates obtained via the simplified model where\nonly 2 πnPJ were considered (analogously to Fig. 3(b)).\naas well as b, namely n/2. In the idealized case, they\ntraveled the same distance πn. Therefore, also the re-\nlated rates are equivalent. However, this is not true for\nodd jumps representing fractional PJs between minima of\ndifferent kinds. In the case of an odd nthe particle over-\ncomes ( n+ 1)/2 of the lower maxima but only ( n−1)/2\nof the higher ones during a ΓA\nnprocess. The opposite is\ntrue for the ΓB\nnprocess. Furthermore, travel distances\ndiffer by 2 π−2d. Consequently, the “odd” rates differ\nsignificantly (ΓA\nn≫ΓB\nn).\nTaking into account this dynamics, the result that it is\nmore probable for a particle to travel over n+ 1 maxima\nthan just over n[for example, ΓA\n3>ΓA\n2and ΓB\n2>ΓB\n1\nin Fig. 5(b),(c)] seems rather paradoxical. However, this\nis a problem of “retrapping” of a particle. Basically, if\nthe particle has already overcome the higher maximum b,\nit will also obtain enough inertia in this regime to over-\ncome the next lower maximum a. This is a parameter-\ndependent process, and the retrapping scenario can be\nrather complicated [50, 74] as is also discussed below in\nSec. III B 3.\nIf the single-jump rate ΓA\n1(slip over single smaller max-\nima) is much larger than the rate of any other process,\nas is typical for high αand high bias force [Fig. 5(c)], it\nis again possible to capture the dynamics of the system\nusing the simplified model described by Eq. (20). This\nis shown in Fig. 5(c), where the green columns represent\nthe rates of the jumps 2 nπ, where the double-well char-\nacter of the potential is ignored. This model agrees well\nwith the even rates, which do not reflect the difference\nbetween the two kinds of minima. The underlying reason\nis that the waiting time for an escape from the minima\nof type Ais negligible compared to other time scales.\nBefore moving to the switching regime, it is worth\nstressing that the FCS method works well up to sur-\nprisingly high Fano factors ( F∼102). This is shown\nin Fig. 4(a) where the bullets represent the Fano fac-\ntor obtained directly from the rates by Eq. (18). They\nare aligned with the curves obtained by the full MCF\nmethod up to the values of ibthat are higher than theretrapping forces of the noiseless scenario [marked with\nthe arrows at the bottom of Fig. 4(a)]. The mathemat-\nical explanation of this agreement is that for ibbetween\nthe phase jump regime and the switching regime, the\nfirst two eigenvalues with the largest real parts are still\nsufficiently separated from the next ones. In addition,\nthe comparison in Fig. 4(a) also shows that the multiple\nphase jumps smoothly change into the running phase as\nwe enter the switching regime. Nevertheless, a different\napproach is needed to investigate the dynamics in this\nregime.\n2. Switching processes\nThe phase jumps within the double well play an impor-\ntant role even in the regime of larger bias forces, where\nthe switching between running and locked solutions takes\nplace. To show this and to evaluate the escape and re-\ntrapping rates between locked and running metastable\nstates, we again introduce a simplified model. We divide\nthe full probability distribution function into three well-\nseparated regions and calculate the occupation probabil-\nities for each of them. The sharp borders between the re-\ngions are defined by separatrices of the noiseless (Θ = 0)\ndissipative steady-state solution. In plain words, we de-\ntermine the momentary regime of the particle by identi-\nfying the deterministic steady state in which the particle\nwould end with its current position and velocity for the\nnoiseless case.\nA trivial example of such a division is plotted in\nFig. 4(b). There, the solid gray curve separates the A-\nvalley locked state with the associated time-dependent\noccupation probability PA(τ), the dashed curve separates\ntheB-valley locked state with the associated occupation\nprobability PB(τ) and, consequently, the rest of the area\nbelongs to the running state with occupation probabil-\nityPR(τ). The associated occupation probabilities are8\nassumed to satisfy the master equation\nd\ndτ\nPR\nPA\nPB\n=MS\nPR\nPA\nPB\n, (21)\nwith the rate matrix\nMS=\n−(ΓrA+ ΓrB) Γ eA ΓeB\nΓrA −(ΓeA+ ΓAB) Γ BA\nΓrB ΓAB −(ΓeB+ ΓBA)\n,\n(22)\nwhere Γ eA(ΓeB) is the escape rate from potential well\nA(B) to the metastable running state; Γ rAand Γ rBare\nthe retrapping rates from metastable running state to\nthe potential well AorB, respectively, and Γ AB(ΓBA)\nis the total rate of the phase jumps (of any length) from\npotential well AtoB(BtoA) as it is illustrated in\nFig. 4(b).\nThese rates can be obtained by reconstructing the\nswitching matrix MS=PLSP−1, with the diagonal ma-\ntrix\nLS=\nλ00 0\n0λ10\n0 0 λ2\n (23)\ncontaining the three eigenvalues with the largest real\nparts of the full Fokker-Planck operator calculated via\nMCF [27]. They represent the stationary state λ0= 0\nand the first two excited states λ1andλ2. The matrix\nPis a square matrix of the three related eigenvectors.\nThe three components of each eigenvector are obtained\nby integrating the full MCF eigenfunctions over the areas\nbounded by the separatrix curves of particular solutions.\nThe separatrices are 2 πperiodic and, as illustrated in\nFig. 6, can be quite complex when γ→0. As a conse-\nquence, dynamical processes, such as the retrapping of\nthe particle, can be very complicated [50].\nA typical example of the rate dependencies on the bias\nforce and related stationary occupations of particular\nstates for α= 0.2 and α= 0.02 are plotted in Fig. 7.\nThey represent two different regimes.\nIn panel (a), where α= 0.2 and Θ = 0 .1, the phase\njump from minimum Ato minimum Bhas a rate Γ AB\n(solid blue line) that is of the same order as the escape\nrate Γ eA(solid red line) and much higher than the escape\nrate Γ eB(dashed red line) at small bias. The rate of the\nopposite phase jump from BtoA(blue dashed line) is\nnegligible. In addition, the retrapping rate Γ rB(dashed\nblack line) is almost two orders of magnitude larger than\nretrapping rate Γ rA(solid black line) in the entire plotted\nrange. This means that the mean lifetime of a particle\ntrapped in the locked state Ais negligible compared to\nthe mean lifetime of the two other states. Consequently,\n-2-1 0 1 2\n-1 1α = 0.5,  γ = 0.2,  ib=0.27v\nϕ/π-2-1 0 1 2\n-1 1 -1 1α = 0.5,  γ = 0.1,  ib=0.14\nϕ/π -1 1 -1 1α = 0.5,  γ = 0.04,  ib=0.06\nϕ/π -1 1Figure 6. Separatrices (black lines) shown for different damp-\nings and bias forces. Panels can be used to illustrate compli-\ncated retrapping dynamics of the noiseless case. Any particle\nwithin the white region will keep running forever, particles\nwithin blue and red region will get trapped following paths\nbounded by the regions of the same color as their initial state\n(they can not cross separatrix). For example, a motionless\nparticle placed initially close to the top of the higher poten-\ntial maxima, but just right to the running-locked separatrix\nas marked by the green bullets ( v= 0, φ=φb+δ), will end\nin a minima marked by the green cross.\n10-610-510-410-310-2\n 0.2  0.25  0.3  0.35  0.4  0.45  0.5Γ\nibγ=0.2\nΘ=0.1\nα=0.02cΓrAΓrBΓeAΓeBΓABΓBA\n0.00.20.40.60.81.0\n 0.2  0.25  0.3  0.35  0.4  0.45P\nibγ=0.2\nΘ=0.1\nα=0.02dPRPAPB10-710-610-510-410-310-210-1\n 0.25  0.3 0.35  0.4 0.45  0.5 0.55  0.6Γ\nibγ=0.2\nΘ=0.1\nα=0.2aΓrAΓrBΓeAΓeBΓABΓBA\n0.00.20.40.60.81.0\n 0.25  0.3 0.35  0.4 0.45  0.5P\nibγ=0.2\nΘ=0.1\nα=0.2bPRPAPB\nFigure 7. (a),(c) Dependencies of the escape, retrapping, and\nphase jump rates [illustrated in Fig. 4(b)] on the bias force\nwithin the switching regime calculated using the simplified\nmodel (21). (b),(d) Related steady-state occupation proba-\nbilities. Two qualitatively different cases are presented: pan-\nels (a) and (b) show α= 0.2 where in the steady state the\nsystem behaves effectively as a single well, and panels (c) and\n(d) show α= 0.02 where the double-well character is evident.\nin the steady state, the system exhibits bistable behav-\nior reminiscent of a system governed by a simple single-\nharmonic potential. This is also evident in Fig. 7(b),\nwhere the stationary occupation probabilities are plot-\nted. For low bias forces, the particle is predominantly\ntrapped in the state B, while a sharp transition to the\nrunning state is observed near the Fano factor maximum\n(Fig. 4, blue curve). Throughout the range depicted in\nthe figure, the locked state Aexhibits minimal influence\non the system dynamics.\nThe situation is qualitatively different for α= 0.02. In\nthis case, Γ ABand Γ BAare comparable and considerably9\nsmaller than those of the escape and retrapping processes\nwithin the range where switching occurs. However, this\ndoes not mean that phase jumps between the two locked\nstates are irrelevant. As we discuss later in detail, it\nonly means that their influence is apparent only on much\nlonger timescales. At shorter ones, the phase dynamics is\ngoverned by the escape and retrapping processes. Due to\nthe small α, the rates of escape and retrapping processes\nare comparable (Γ rA∼ΓrBand Γ eA∼ΓeB). Together,\nthis leads to steady-state occupation probabilities where\nboth locked states are relevant, as illustrated in Fig. 7(d).\nInterestingly, even with the low value of α= 0.02, there\nremains a significant difference between the occupation\nprobabilities of states AandBin the steady state.\nIn addition to steady state, the simplified model (21)\nallows us to easily investigate the time evolution and\nto understand its particular time regimes. This can be\nuseful not only by itself but also as a supporting tool\nfor full-scale Langevin simulations. Let us illustrate this\nby investigating a retrapping process after a parameter\nquench.\n3. Retrapping\nIt is often difficult to reach the steady state in ex-\nperimental realizations or realistic Langevin simulations.\nThis is especially problematic for systems with low noise\nand weak damping. However, once we have the rates\nof the most relevant processes, we can investigate even\nlong-time processes via the deterministic Eq. (21).\nAs an illustration, we analyze the scenario of a sud-\nden quench of the parameters. We employ the master\n0.00.20.40.60.81.0\n10-1100101102103104ib = 0.2\nγ  = 0.2\nα  = 0.02\nΘ  = 0.1 Psteady\nB\nPsteady\nAP\nτPRPAPB \n \n  \n \n \nFigure 8. Example of the time dependence of the occupation\nprobabilities of the three metastable states for ib= 0.2 with\ninitial condition PR= 1, PA= 0 and PB= 0. This models a\nsudden quench of very large bias force to bias force just above\nthe retracing force of the noiseless case. Solid lines are results\nof the simplified model (21), dashed lines show full Langevin\nsimulations for two different initial conditions.equation (21) and also large-scale Langevin simulations.\nFor the latter, we used the modified Euler-Heun method\nLambaEulerHeun from the Julia package DifferentialE-\nquations.jl with adaptive integration step. In the simu-\nlations, we used ≈6.5×104particles and, as before, their\nimmediate regime was identified via the separatrices for\nthe model parameters after the quench.\nWe focus on the case with the steady state illustrated\nin Fig. 7(d), i.e., α= 0.02,γ= 0.2 and Θ = 0 .1. We\nprepare the system in a state where it is fully running,\nfor example, with a high bias force. For the simplified\nmodel, this means setting PR(τ= 0) = 1 and PA(τ=\n0) = PB(τ= 0) = 0. For the Langevin simulation, the\nsystem was first thermalized at ib= 0.5, which also gives\nPR(τ= 0)≈1, as evident from Fig. 1. In the next step,\nwe abruptly change the bias force to ib= 0.2, which is\njust below the actual retrapping force ir. For ib= 0.2 the\nsystem is already almost exclusively in the locked steady\nstate, yet the running state is still possible and can be\nidentified by its separatrix.\nThe calculated time dependence of the occupation\nprobabilities are shown in Fig. 8 by the solid (master\nequation) and dashed (Langevin dynamics) black ( PR),\nblue ( PA) and red ( PB) lines. The logarithmic time scale\nreveals several metastable regimes. At short times, there\nare the largest differences between the simplified model\nand the full simulation, as expected. In particular, the\nchange in initial occupations is faster for the simplified\nmodel than in the simulation. This is due to the de-\ntails of the initial state, that is, the initial distribution of\nthe particles. Full dynamics starts with a relatively high\nmean velocity of the particles, and therefore it takes a\nwhile to slow them down.\nTo illustrate that this is indeed the case, we also show\na simulation with different initial conditions. The dashed\nlines in dark gray ( PR), cyan ( PA) and orange ( PB)\nstarted in a steady state with ib= 0.3 and Θ = 0 .01\nand, therefore, significantly lower mean velocity. The\nshort-time dynamics of this case approaches that of the\nsimplified model. Nevertheless, the simplified model al-\nmost perfectly predicts, and, more importantly, explains,\nthe full dynamics of these simulations at longer times.\nFirst, particles are quickly caught in the minima\naround τ∼101, because the retrapping rates are Γ rA∼\nΓrB≈10−1. The system is close to be completely\ntrapped before τ∼102. However, although relatively\nstable, the occupation probabilities of the locked states\nare far from their steady-state values (horizontal dashed-\ndotted lines) obtained by the MCF method discussed\nabove. This metastable state exists due to the large sep-\naration of the retrapping rates from the escape and phase\njumps ones, see Fig. 7(c). Probabilities start to approach\nthe true steady state only for τ≳104where first the es-\ncape processes with rates Γ eA∼ΓeB∼10−4and then\nphase jumps with Γ AB∼0.5×10−4start to be relevant.\nThis difference of more than two orders of magnitude\nbetween the time of retrapping and reaching the steady\nstate, as well as the existence of the metastable region,10\nis an important observation. The long-lived metastable\nregime can be easily mistaken for the steady state in ex-\nperiments or simulations. Furthermore, although not ev-\nident from the images presented, the Langevin simula-\ntions are becoming unstable for τ∼105due to the ac-\ncumulation of numerical errors. The simple model does\nnot suffer from this problem and has the benefit of being\ndirectly interpretable.\nIV. CONCLUSION\nIn conclusion, we have presented a theoretical study\nof the stochastic dynamics of a particle in the periodic\ndouble-well potential. A combination of the matrix con-\ntinued fraction technique applied to solve the Fokker-\nPlanck equation combined with the full counting statis-\ntics and simple master equation models allowed us to\ndetermine the role of particular processes in the overall\ndynamics.\nFor the strong damping, the analysis of the velocivy\nFano factor revealed a region where single-phase jumps,\nincluding fractional ones, are the prevailing source of the\nvelocity noise. We have shown that with decreasing tem-\nperature, the steady-state properties of the overdamped\njunction approach an effective single well system for any\nfinite α.\nIn the intermediate and weak damping regime, the FCS\nanalyzes showed complex phase dynamics related to the\nsingle and multiple phase jumps. The revealed large dif-\nferences between the rates of odd (fractional) and even\nphase jumps can be explained by analyzing the parti-\ncle trajectories. Even in this regime we have identified\nparameter ranges, for which a single-harmonic analysis\nis sufficient for the description of the main phase slip\nstatistics.\nIn the analysis of the switching regime, we have pre-\nsented a simple master equation method to calculate the\nescape and retrapping rates. We have focused on both\nthe regime where the double well character plays an im-\nportant role in the steady state statistics and the regime\nwhere it does not. We have shown how this property is\nrelated to the retrapping, escape, and phase jump rates.\nWe have demonstrated how these rates evolve with the\nbias force and determined the probabilities of steady-\nstate occupations. The important observation is that\nretrapping, escape, and phase jumps can have rates that\ndiffer by orders of magnitude. This sets distinct time\nscales in realistic retrapping processes.To illustrate this, we have investigated a quench, where\nthe system in the fully running steady state is quenched\nto a bias force near the critical lower retrapping bias. We\nhave shown that the results of the simplified model are in\nagreement with full Langevin simulations at longer times\nand that the differences at the short times are due to the\ndetails of the initial state used. The advantage of the sim-\nplified model is its stability at long times and, more im-\nportantly, its straightforward interpretability. Each time\nregime can be related to particular rates. In this way,\nwe have been able to identify a metastable locked state,\nwhich can be easily mistaken for a steady state, due to\nthe low escape rates and low rates of the jumps between\nthe minima.\nTo wrap it up, besides providing a simple technique\nfor analyzing the statistical properties of stochastic sys-\ntems with double-well periodic potentials and analyzing\ntheir properties, we have shown two rather general fea-\ntures of such systems that can be crucial for the analysis\nof experimental setups. First, for a broad range of pa-\nrameters in a strong and intermediate damping regime,\nthe steady-state system can be approached with a simple\nsingle-harmonic model. Consequently, a two-well charac-\nter of real potential can be hidden in the averaged data\nwhen the escape rate from one of the minima significantly\nexceeds the other rates. This can be true even for wells\nof similar depths at low noise.\nOn the other hand, the occupations of the two min-\nima in the fully locked steady state can be significantly\ndifferent even for nearly equal minima if the damping is\nlow. In addition, these occupations are highly parameter\nsensitive. What is also important for analysis of exper-\niments and numerical simulations is the realization that\nthe retrapping time and the time when the steady-state\noccupation is finally reached can differ by several orders\nof magnitude.\nAcknowledgments.— M.ˇZ. and T.N. acknowledge sup-\nport by the Czech Science Foundation through Project\nNo. 23-05263K. This work was supported by the Min-\nistry of Education, Youth and Sports of the Czech Re-\npublic through the e-INFRA CZ (ID:90254). E.G. thanks\nDeutsche Forschungsgemeinschaft (DFG project GO-\n1106/6) for financial support of this collaboration. 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We find a steady order-by-order convergence and realistic descriptions of scattering\nobservables up to a laboratory scattering energy of approximately 100 MeV. We also compare per-\nturbative and non-perturbative calculations for phase shifts and cross sections and quantify how\nunitarity is gradually restored at higher orders. The perturbative approach offers an important\ndiagnostic tool for any power counting and our results suggest that the breakdown scale in chi-\nral effective field theory might be significantly lower than estimates obtained in non-perturbative\ncalculations.\nI. INTRODUCTION\nNuclear potentials used in ab initio [1] computations\nof atomic nuclei [2] are almost exclusively derived using\nchiral effective field theory ( χEFT) [3–5] based on Wein-\nberg power counting (WPC) [6, 7]. Such potentials [8–\n14], now derived up to the fifth chiral order [15–17], have\nfurnished a wide range of structure and reaction predic-\ntions across the nuclear chart [18, 19], but at the same\ntime they grapple with the renormalization challenge in-\nherent to chiral nuclear forces [20]. Indeed, numerical\nstudies [21] of the nucleon-nucleon scattering amplitude\nhave shown that the contact operators, accounting for\nunresolved short-range physics, already at leading order\n(LO) in WPC are not sufficient to renormalize the singu-\nlar nature [22] of the one pion-exchange potential. Con-\nsequently, LO predictions based on WPC exhibit an un-\nphysical dependence on the cutoff Λ that regularizes the\namount of high-momentum (or short-range) physics that\nis resolved.\nSeveral PCs leading to renormalization-group (RG) in-\nvariant nucleon-nucleon amplitudes have been proposed\nin the past two decades [23–35]. They can collectively be\nreferred to as modified Weinberg power countings (MW-\nPCs). However, we typically know very little about their\npredictive power for nuclei beyond the lightest-mass sys-\ntems [36]. The one exception is the recent study by\nYang et al. [37] that presented the first ab initio pre-\ndictions of binding energies in4He,6Li, and16O using\nχEFT potentials up to next-to-leading order (NLO) in\nseveral different MWPCs. The calculations in that work\nrevealed an α-decay instability in the ground states in6Li\nand16O. Subsequent analyses brought forward probable\ncauses for this instability as originating in ( i) overfitting\nof the low-energy constants (LECs) that parameterize the\nshort-range interactions [38] and ( ii) underestimating the\nimportance of few-nucleon forces [39] at LO in MWPC.\n∗oliver.thim@chalmers.seThe notable absence of MWPC-based predictions for\nheavier-mass nuclei is likely due to a variety of factors.\nFirstly, potentials based on WPC are easier to imple-\nment in present ab initio computer codes as one straight-\nforwardly sum leading and sub-leading corrections to\nthe potential before solving the Schr¨ odinger equation,\nwhereas in MWPC sub-leading corrections should be\nadded in perturbation theory [40]. Secondly, there ex-\nists several widely available computer codes for evaluat-\ning matrix elements of chiral nucleon-nucleon and three-\nnucleon potentials, as well as currents, to very high or-\nders in WPC. Finally, it is currently prohibitively costly\nto converge ab initio predictions of nuclear properties at\nthe large values of the cutoff required for analyzing RG-\ninvariance in MWPC.\nIn light of these facts we certainly see the utility of\nWPC, which might provide a consistent EFT framework\nprovided that renormalization is interpreted in a fashion\nwhere the cutoff never exceeds the order of the break-\ndown scale [41–44]. However, the existence of MW-\nPCs, where renormalization does allow for the cutoff\nto be taken far beyond the breakdown scale, calls for\na continued effort. Given the fundamental importance\nof RG-invariance it should be seriously explored whether\nMWPC approaches can furnish a realistic and predictive\nframework for ab initio nuclear physics.\nIn this paper, we contribute to the meager list of quan-\ntitative predictions grounded in RG-invariant formula-\ntions of χEFT. To the best of our knowledge, and some-\nwhat surprisingly, nucleon-nucleon scattering observables\nhave not been computed in MWPC beyond LO [41].\nHere, we present predictions for integrated and differ-\nential cross-sections, as well as polarization observables,\nfor elastic neutron-proton ( np) scattering up to next-to-\nnext-to-next-to-leading order (N3LO) in the MWPC of\nLong and Yang [30, 32, 45], where higher-order correc-\ntions to the potential are treated perturbatively [21, 40].\nThis work serves as an important step in the develop-\nment and uncertainty quantification of any model of the\nnuclear interaction [46–50].arXiv:2402.15325v1  [nucl-th]  23 Feb 20242\nIn Section II we review how to construct potentials\nin the PC of Long and Yang, describe how to numeri-\ncally compute the scattering amplitude in distorted-wave\nperturbation theory, and explain how we calibrated LEC\nvalues. In Section III we present results for scattering ob-\nservables up to N3LO, and we summarize and conclude\nin Section IV.\nII. FORMALISM\nInχEFT, scattering amplitudes are expanded in a di-\nmensionless ratio ( Q/Λb)ν. Here, νindicates the chi-\nral order, Λ bis the underlying high-momentum scale of\nχEFT, and Qdenotes the relevant low-energy scale. For\nnucleon-nucleon scattering, we assume Q≈max( p, m π),\nwhere pis the relative momentum in the center of mass\n(c.m.) frame of the interacting nucleons, and the pion\nmass mπis the relevant low-energy mass scale. In\nthis work we adopt a nomenclature where LO scales as\n(Q/Λb)0while sub-leading orders are denoted by their\nrelative scaling to LO. As such, NLO scales as ( Q/Λb)1,\nnext-to-next-to-leading order (N2LO) as ( Q/Λb)2and so\non. In what follows, we summarize relevant details re-\ngarding the MWPC that we use in this work, define the\npotential terms V(ν)entering at each chiral order, and\nexplain how we performed the perturbative calculations\nof scattering amplitudes.\nA. The nucleon-nucleon interaction potential in\nthe Long and Yang power counting\nWe employ the MWPC of Long and Yang [30, 32, 40,\n51], which adheres to the following overarching principles:\n•The chiral order of a pion-exchange diagram, along\nwith the necessary counterterms for renormaliz-\ning pion loops, is determined by the naive dimen-\nsional analysis (NDA) of its non-analytic part. This\nfollows the same principle as in Weinberg Power\nCounting (WPC).\n•Counterterms are promoted to lower chiral order\nonly when needed to fulfill the requirement of RG-\ninvariance.\n•All corrections to the potential beyond LO are in-\ncluded perturbatively to obtain RG-invariant am-\nplitudes.\nOne-pion exchange (OPE) enters at LO in χEFT and\nmust be treated non-perturbatively, at least in the low\npartial waves where it is sufficiently strong. The singu-\nlar nature of OPE is increasingly alleviated by the cen-\ntrifugal barrier. Thus, at some point in the partial-wave\nexpansion there is sufficient angular momentum ℓto fur-\nnish a perturbative treatment of OPE [29, 52, 53] and\nconsider it sub-leading.TABLE I. Potential contributions at each in channels where\nOPE is treated non-perturbatively (column three) and per-\nturbatively (column four). Detailed expressions for the po-\ntentials can be found in Appendix A.\nnon-perturbative (at LO) purely perturbative\norder potential channels channels\nLO V(0)V(0)\n1π+V(0)\nct 0\nNLO V(1)V(1)\nct V(0)\n1π\nN2LO V(2)V(2)\n2π+V(2)\nct 0\nN3LO V(3)V(3)\n2π+V(3)\nct V(2)\n2π\nAt LO in the MWPC by Long and Yang, the OPE\npotential V(0)\n1πis considered non-perturbative in the1S0,\n3P0,1P1,3P1,3S1−3D1and3P2−3F2channels. OPE\nis attractive in3P0and3P2. Renormalization requires\npromotion of counterterms to the corresponding chan-\nnels of the LO contact potential V(0)\nct[21], thereby ex-\ntending it beyond the canonical non-derivative1S0and\n3S1counterterms. At sub-leading orders ( ν > 0), two\npion-exchange, V(ν)\n2π, as well as higher-order contact po-\ntentials, V(ν)\nct, enter perturbatively according to the prin-\nciples presented in the beginning of this subsection. The\ncontributions to the potential up to N3LO in the1S0,\n3P0,1P1,3P1,3S1−3D1and3P2−3F2channels are listed\nin the third column of Table I labeled ”non-perturbative\n(at LO) channels”.\nSee Appendix A for detailed expressions of the poten-\ntials appearing in Table I. Following Long and Yang, we\ndo not consider any higher-order corrections to OPE and\nemploy potential expressions where pion loops are treated\nin dimensional regularization. For the sub-leading two-\npion exchange potential V(3)\n2πwe use pion-nucleon LECs\nc1, c3, c4with central values from the Roy-Steiner analy-\nsis in Ref. [54].\nLet us now turn to the channels with ℓ >1 (and with-\nout any coupling to ℓ≤1). For these channels we con-\nsider OPE to be perturbative and consequently set it to\nzero at LO. We follow Ref. [52] and suppress two-pion ex-\nchanges by the same chiral power as OPE. Up to N3LO,\nthere are no contact potentials in the perturbative chan-\nnels, and the contributions are listed in the last column\nof Table I. Other suggestions for the PC in perturbative\nchannels are discussed by, e.g., Pav´ on Valderrama et al.\n[27].\nB. A perturbative treatment of nucleon-nucleon\nscattering amplitudes\nThe perturbative computation of nucleon-nucleon scat-\ntering amplitudes proceeds in two steps. First, we solve\nthe Lippmann-Schwinger (LS) equation for the LO am-\nplitude in the1S0,3P0,1P1,3P1,3S1−3D1and3P2−3F2\nchannels. Note that the LO potential is identically zero3\nin all other channels. Second, we perturbatively include\nhigher-order potential corrections to the amplitude, ac-\ncounting for the distortion due to the non-perturbative\nLO solution where necessary. In the following, we explain\nthis procedure in detail, see also Refs. [30, 32, 53].\nThe neutron-proton Hamiltonian in the center-of-mass\n(c.m.) frame can be written\nH=p2\nmN+VI+VII, (1)\nwhere pdenotes the c.m. momentum and mN=\n2mnmp/(mn+mp) the nucleon mass. The projectile en-\nergy in the laboratory frame will be denoted Tlab. Fur-\nthermore, VIdenotes the LO potential, and VIIdenotes\nthe sum of all sub-leading potentials, which formally can\nbe infinitely many. The PC helps us identify important\nand less important contributions to the scattering ampli-\ntude Tand therefore facilitates a meaningful truncation\nofVII. With the notation for the chiral potentials V(ν)\nintroduced in Section II A, VIandVIIread\nVI=V(0), (2)\nVII=∞X\nν=1V(ν). (3)\nThe LO amplitude, T(0), is obtained (non-\nperturbatively) by solving the LS-equation\nT(0)=V(0)+V(0)G+\n0T(0), (4)\nwhere the free resolvent is given by\nG+\n0= (E−H0+iϵ)−1, (5)\nandH0=p2/mN. We use a notation where we suppress\nthe explicit dependence on the c.m. scattering energy, E,\nfor the resolvents and amplitudes.\nIn WPC, higher-order corrections are accounted for\nnon-perturbatively by solving the LS-equation for the\nsumVI+VII. In MWPC, however, potentials beyond LO,\ni.e., the corrections ( VII), enter in perturbation theory to\nobtain RG invariant results [40]. Indeed, higher-order\ncorrections should be amenable to a perturbative treat-\nment. If not, they are non-perturbative in nature and\nbelongs at LO.\nDistorted-wave perturbation theory has been applied\nto compute scattering amplitudes in several previous\nstudies, see, e.g., Refs. [28, 30, 32, 51, 53, 55]. The pertur-\nbation series for the scattering amplitude can be derived\nand expressed in various ways. The one that we find\nmost instructive follows Refs. [56, 57]. First, using the\ntwo-potential trick, the T-operator for the Hamiltonian\nin Eq. (1) is written in the form\nT=T(0)+ Ω†\n−VII∞X\nn=0\u0000\nG+\n1VII\u0001nΩ+, (6)where the Møller wave operators are defined as\nΩ+=1+G+\n0T(0), (7)\nΩ†\n−=1+T(0)G+\n0, (8)\nand the full LO resolvent reads\nG+\n1= Ω +G+\n0. (9)\nInserting Eq. (3) in Eq. (6) gives for the full T-operator\nT=T(0)+ Ω†\n−\"∞X\nν=1V(ν)#∞X\nn=0\"\nG+\n1 ∞X\nν′=1V(ν′)!#n\nΩ+.\n(10)\nExpanding both sums and organizing terms according to\ntheir chiral orders νyields the expressions for the first-,\nsecond-, and third-order corrections to the LO amplitude\nas\nT(1)= Ω†\n−V(1)Ω+ (11)\nT(2)= Ω†\n−\u0010\nV(2)+V(1)G+\n1V(1)\u0011\nΩ+ (12)\nT(3)= Ω†\n−\u0010\nV(3)+V(2)G+\n1V(1)+V(1)G+\n1V(2)+\n+V(1)G+\n1V(1)G+\n1V(1)\u0011\nΩ+. (13)\nA diagrammatic representation of amplitudes up to NLO\nis presented in Fig. 1. Note that the full amplitude at,\ne.g., third order (N3LO) is given by the sum T(0)+T(1)+\nT(2)+T(3). Clearly, the distorted-wave corrections in\nEqs. (11) to (13) simplify dramatically when applied to\nthe channels where OPE is perturbative such that T(0)=\n0, Ω +=1, and Ω†\n−=1. In these channels we therefore\nrecover ordinary perturbation theory.\nThe distorted-wave corrections to the amplitudes\nT(ν>0)can alternatively be obtained as solutions to a set\nof modified LS-type equations, discussed in more detail\nin Refs. [58, 59], which read\nT(ν)=V(ν)+νX\ni=1V(i)G+\n0T(ν−i)+V(0)G+\n0T(ν).(14)\nWe use this formulation to verify our numerical imple-\nmentation of Eqs. (11) to (13). We note that the alter-\nnative approach of modified LS-equations requires a ma-\ntrix inversion at each order, whereas the distorted-wave\napproach requires matrix multiplications only. However,\nthe number of matrix multiplications increases rapidly\nas the chiral order is increased. For example, at ν= 10,\nEqs. (11) to (13) require an order of magnitude more\nmatrix multiplications than the modified LS equations\nin Eq. (14). In this study we only go to ν= 3 for which\nthe number of matrix multiplications of the two formu-\nlations are similar.\nC. Numerical implementation\nWe project potentials and amplitudes to a partial-wave\nbasis of states |p, ℓ, s, j ⟩following the prescription in Ref.4\nT(0)= = +\nT(1)= + + +\nFIG. 1. Diagrammatic representation of the LO neutron-proton amplitude T(0)(hatched oval), obtained by solving the LS-\nequation, as well as the first correction T(1)given in Eq. (11). The grey (black) solid blobs represent the potentials V(0)(V(1)).\n[60]1. Here, p=|p|, while s, ℓ, j denote the quantum\nnumbers of the two-nucleon spin, orbital angular momen-\ntum, and total angular momentum, respectively. Partial-\nwave matrix elements are denoted by\nVjs\nℓ′ℓ(p′, p) =⟨p′, ℓ′, s, j|V|p, ℓ, s, j ⟩, (15)\nwhere the conserved quantum numbers sandjare given\nas superscripts.\nIn the LS-equation, as well as in Eqs. (11) to (13),\ninfinite momentum integrals appear and all potentials\nare regulated according to\nVjs\nℓ′ℓ(p′, p)→fΛ(p′)Vjs\nℓ′ℓ(p′, p)fΛ(p), (16)\nwhere we choose a regulator function\nfΛ(p) = exp\u0014\n−p6\nΛ6\u0015\n(17)\nat all orders up to N3LO. In the calibration of the\nLECs, we use the cutoff values Λ = 500 MeV and\nΛ = 2500 MeV.\nUsing Eqs. (7) to (9), the terms in Eqs. (11) to (13) can\nbe expanded to sums of products of the form A1G+\n0A2,\nof varying length. The Ai’s are either T(0)orV(ν)with\nν= 1,2,3. For example, the NLO correction in Eq. (11)\nreads\nT(1)=V(1)+T(0)G+\n0V(1)+V(1)G+\n0T(0)\n+T(0)G+\n0V(1)G+\n0T(0). (18)\nClearly, the fundamental matrix elements that need to be\nevaluated at sub-leading orders are always of the form\n⟨p′, ℓ′|A1G+\n0A2|p, ℓ⟩, (19)\nwhere we omit the sandjquantum numbers that are\nidentical for the ket and the bra. In Appendix B we show\nhow to evaluate Eq. (19) using ordinary matrix products\nand Gauss-Legendre quadrature. Longer products, e.g.,\n1Note the mistake in Eq. (4.22) pointed out in Ref. [4].of the form A1G+\n0A2G+\n0A3, are straightforwardly reduced\nto the form in Eq. (19) by the associativity of matrix\nproducts. Knowing this, and the distributive property\nwith respect to addition, we can also reduce the compu-\ntational complexity of evaluating the perturbation series\nforTby computing and storing the composite operators\nΩ†\n−, Ω+, and G+\n1.\nFor separable potentials of Yamaguchi type [61], both\nthe distorted-wave series and the LS equation can be\nsolved analytically. We exploit this to verify our numer-\nical implementation and to inspect the stability of the\nperturbative expansion. Numerical and analytical results\nfor semi-realistic and separable Yamaguchi potentials in\nthe1S0and3S1−3D1channels agree to at least single\nprecision.\nD. Calibrating the low-energy constants\nOur focus in this work is to predict and analyze the\ndescription of npscattering observables in MWPC and\nspecifically the PC of Long and Yang. To enable quan-\ntitative calculations, we calibrate the values of the un-\nknown LECs using the same approach as Long and Yang,\ni.e., by tuning the contact LECs to achieve a good repro-\nduction of the low-energy Nijmegen phase shifts [62] at\nselected scattering energies.\nBefore discussing the details of the calibration, it is im-\nportant to remember that the order-by-order amplitudes\nT=T(0)+T(1)+T(2)+. . . (20)\nare computed perturbatively and their sum is unitary\nonly up to perturbative corrections. To obtain real-\nvalued phase shifts in the calibration of the LECs we must\ncompute phase shifts perturbatively by expanding the np\nS-matrix and matching to chiral orders, see Appendix C\nfor details. If one instead solves for the partial-wave S-\nmatrix non-perturbatively from the order-by-order sum\nofT(ν)amplitudes, the corresponding phase shifts will\nhave a non-zero imaginary part that increases with scat-\ntering energy. Indeed, Figure 2 shows phase shifts com-\nputed perturbatively and non-perturbatively in the two\nchannels1D2and3D2. There are no LECs that need to\nbe calibrated in these channels at the orders considered5\nTABLE II. Laboratory scattering energies Tlab(in MeV) of\nthe Nijmegen phase shifts [62] used to calibrate the values\nof the LECs at each chiral order. In total, we employed 33\nsingle-energy phase shifts—the same as the total number of\ncontact LECs in the chiral expansion of Long and Yang up to\nN3LO.\nChannel LO NLO N2LO N3LO\n1S0 5 5, 25 5, 25, 50 5, 25, 50, 75\n3P0 25 - 25, 50 75, 100\n1P1 - - 50 50\n3P1 - - 50 50\n3S1−3D13S1: 30 -3S1: 30,50.3S1: 30,50.\nϵ1: 50 ϵ1: 50\n3P2−3F23P2: 30 -3P2: 30,50.3P2: 30,50.\nϵ2: 50 ϵ2: 50\nin this work. The imaginary part of the non-perturbative\nphase shift increases with scattering energy. As that hap-\npens, the real part of the phase shift and the (real-valued)\nperturbative phase shift differ progressively. This is con-\nsistent with observations in Ref. [63].\nIn the calibration of LECs, we do not account for un-\ncertainties stemming from the Nijmegen phase shifts or\nthe truncation of the χEFT expansion. While we are\naware of the potential risk of overfitting in doing so, we\nopted for a simple approach to establish a first quanti-\ntative potential and a baseline understanding. The ap-\nplication of Bayesian inference methods [47–49] to quan-\ntify the posterior probability distributions for the values\nof the LECs in MWPC [38], though more robust, re-\nquires considerably more efforts. In this work, we focus\non studying the effectiveness of MWPC for realistic de-\nscription of elastic npscattering.\nTheTlabvalues of the Nijmegen phase shifts used as\ncalibration data are listed in Table II for each channel and\norder. The calibrated LECs up to N3LO are compiled in\nTable III in Appendix A. We use a naming-convention\nwhere capital letters C, D, E, . . . denote LECs with di-\nmension MeV−2, MeV−4, MeV−6, . . ., respectively.\nEach LEC receives perturbative corrections at subse-\nquent orders from where it was first introduced. As an\nexample, the LO LEC C1S0is expanded into contribu-\ntions\nC1S0=C(0)\n1S0+C(1)\n1S0+C(2)\n1S0+. . . , (21)\nwhere the superscript enumerates the perturbative cor-\nrection and not the chiral order. In the following we will\nexemplify the calibration procedure by discussing in de-\ntail how we calibrated the LECs in the1S0channel.\nAt LO we calibrate the LEC C(0)\n1S0such that the LO\n1S0phase shift, δ(0), reproduces the Nijmegen phase shift\natTlab= 5 MeV. Two LECs are present in the1S0chan-\nnel of the NLO potential: D(0)\n1S0andC(1)\n1S0. The latter\nis a perturbative correction to the LO LEC. These two\nLECs are calibrated such that the LO phase shift plus\nthe perturbative NLO correction, i.e., δ(0)+δ(1), repro-duce the Nijmegen phase shifts at Tlab= 5 and 25 MeV.\nThe role of C(1)\n1S0is to ensure that the NLO correction\nvanishes for Tlab= 5 MeV. At N2LO we have the LECs\nC(2)\n1S0, D(1)\n1S0, E(0)\n1S0calibrated to phase shifts at energies\nTlab= 5,25 and 50 MeV. Finally, at N3LO the LECs\nC(3)\n1S0, D(2)\n1S0, E(1)\n1S0, F(0)\n1S0are calibrated to reproduce the\nphase shifts at Tlab= 5,25,50 and 75 MeV. An analo-\ngous scheme is employed for the remaining partial waves\nand LECs. We calibrate all LECs for two different mo-\nmentum cutoffs: Λ = 500 and 2500 MeV.\nFor the channels where OPE is perturbative there\nare no LECs present that need to be calibrated. As a\nconsistency check we compute and reproduce the scat-\ntering phase shifts of Ref. [52]. Figure 3 shows our\nfit of the phase shifts in the channels where OPE is\nnon-perturbative. The bands indicate the variation due\nto the two different cutoff values. There is an overall\norder-by-order convergence in all channels up to around\nTlab= 100 MeV and we can reproduce the known results\nof [30, 32, 51]. The degree of cutoff sensitivity varies no-\ntably among different channels. For instance, channels\nlike1P1and3F2show minimal sensitivity to the cutoff\nvalue, while3P2andϵ1demonstrate a more pronounced\ndependency. The calibration in the3P0channel was par-\nticularly challenging at the higher chiral orders and the\ncalibration energies needed to be shifted to relatively high\nvalues at N3LO, as seen in Table II.\nIII. NEUTRON-PROTON SCATTERING\nOBSERVABLES\nHere we predict selected npscattering observables up\ntoTlab≈100 MeV using the potentials that were defined\nand calibrated in Section II. We compute scattering ob-\nservables from the partial-wave amplitudes by first con-\nstructing the spin-scattering matrix, M, by [56, 64, 65]\nMs\nm′sms(p0, θcm, ϕ) =√\n4π\n2ip0X\nj,ℓ,ℓ′iℓ−ℓ′(2j+ 1)√\n2ℓ+ 1\n×\u0012\nℓ′s j\nms−m′\nsm′\ns−ms\u0013\u0012\nℓ s j\n0ms−ms\u0013\n(22)\n×Yℓ′\nms−m′s(θcm, ϕ)\u0010\nS(ν)js\nℓ′ℓ(p0, p0)−δℓ′ℓ\u0011\n.\nThe angles θcm∈[0, π] and ϕ∈[0,2π] are the polar and\nazimuthal scattering angles, respectively where the lat-\nter is set to zero by cylindrical symmetry. The on-shell\nscattering momentum, p0, is given from the laboratory\nscattering energy Tlabusing Eq. (B2) in Appendix B. We\ncompute S(ν)js\nℓ′ℓ(p0, p0), i.e., the S−matrix for a potential\nup to some chiral order ν, by summing the perturba-\ntively computed T-matrix amplitudes to order ν. Using\nthe conventions applied in this work, the partial-wave re-\nlation between the on-shell S- and T-matrix elements is6\n0510δ1D2(deg)NLONijm pert. non-pert. real non-pert. imag.\n0510N2LO\n0510N3LO\n0 200\nTlab(MeV)02550δ3D2(deg)NLO\n0 200\nTlab(MeV)02550N2LO\n0 200\nTlab(MeV)02550N3LO\nFIG. 2. npscattering phase shifts in the1D2(top row) and3D2(bottom row) channels at NLO, N2LO, and N3LO using a\nmomentum cutoff Λ = 2500 MeV. Phase shifts computed using the perturbative method are shown with black solid lines. The\nred dashed and dot-dashed lines show the real and imaginary parts, respectively, of the phase shift computed by summing the\nT-matrix contribution and using the non-perturbative relation between phase shifts and the S-matrix. The black dashed lines\nshow phase shifts from the Nijmegen analysis [62].\nthus given by\nS(ν)js\nℓ′ℓ(p0, p0) =δℓ′ℓ−iπm Np0\n×h\nT(0)js\nℓ′ℓ(p0, p0) +···+T(ν)js\nℓ′ℓ(p0, p0)i\n. (23)\nWe focus our discussion on the differential npscattering\ncross section and two selected polarizations, and calculate\nthese from the spin-scattering matrix as\ndσ\ndΩ=1\n4TrMM†(24)\ndσ\ndΩ×Pb=1\n4TrMσ1nM†(25)\ndσ\ndΩ×Ayy=1\n4TrMσ1nσ2nM†(26)\nwhere σin≡σi·ˆnfor nucleon i,σiis the Pauli spin\nmatrices, and ˆnis normal to the scattering plane.\nFigure 4 shows our prediction for these scattering ob-\nservables in the energy range Tlab= 10 to 100 MeV for\nthe two cutoffs Λ = 500 MeV and Λ = 2500 MeV. For\nthe lower scattering energies ( Tlab≲60 MeV) we observe\nan order-by-order improvement for all considered observ-\nables. Interestingly, the N3LO predictions do not always\nperform better, but in general performs at least as well\nas N2LO. Indeed, for Tlab≈100 MeV (rightmost panels\nof Fig. 4), it appears that the order-by-order improve-\nment in the predictions of the differential cross section\nandPbpolarization deteriorates and N2LO can perform\nbetter than N3LO. This effect is visible also at the levelof phase shifts shown in Fig. 3. It is not clear at the\nmoment if this is due to overfitting and (or) an underly-\ning issue with the MWPC that we employ. Our N3LO\npredictions are certainly influenced by the adopted val-\nues of sub-leading πNLECs [54]. Calculations of other\nscattering cross observables show that the order-by-order\nconvergence demonstrated in Fig. 4 is representative for\nall elastic npscattering observables in the PC by Long\nand Yang. Two-pion exchange is clearly important for\nachieving a realistic description of scattering observables\nwith Tlab≲100 MeV.\nThe total cross section can be straightforwardly com-\nputed from the differential cross section as\nσtot(p0) = 2 πZ1\n−1d(cosθcm)dσ\ndΩ(p0, θcm), (27)\nand predictions for scattering energies up to Tlab=\n150 MeV are shown in Fig. 5. Also for this obvservable,\nthe agreement with experimental data typically improves\norder-by-order, at least up to N2LO. The improvement\nof N3LO over N2LO is not obvious. At very low ener-\ngies, the higher-order predictions for the total cross sec-\ntion are much better than the lower-order predictions.\nThis result is somewhat peculiar for a low-energy EFT\nand likely due to overfitting at the phase shift level. For\nTlab≳100 MeV, roughly corresponding to 220 MeV rel-\native momentum, the agreement with data even deterio-\nrates at N3LO. This is analogous to what was found for\nthe angular-differential observables shown in Fig. 4 and7\n050δ(deg)\n1S0\n−10010δ(deg)\n3P0−2001P1\n−2003P1\n0100δ(deg)3S1\n−2003D1\n0510/epsilon11\n0 100 200\nTlab(MeV)050δ(deg)3P2\n0 100 200\nTlab(MeV)0123F2\n0 100 200\nTlab(MeV)05/epsilon12LO\nNLON2LO\nN3LONijm\nFIG. 3. Phase shifts in the channels where OPE is non-perturbative and the amplitudes are computed using full distorted-wave\nperturbation theory. The bands indicate the envelope of the variation due to the two different cutoff values; 500 MeV (dashed\nline) and 2500 MeV (solid line). Note that LO and NLO results coincide for all channels except1S0, which is why the blue\nNLO band appears to be missing in several panels. The black solid lines show phase shifts from the Nijmegen analysis [62] and\nthe diamond markers indicate the calibration data at Tlabvalues from Table II.\nconsistent with the observation in Fig. 3 that the phase\nshifts at N3LO might suffer from overfitting at the higher\nenergies. Alternatively, the observed decline in predictive\npower might indicate the presence of an additional mass\nscale at 200-300 MeV. Thus, it will be very interesting to\nstudy the effects of accounting for the ∆(1232)-isobar in\ntwo-pion exchange in this MWPC.\nNext, we analyze how the perturbative breaking of uni-\ntarity in χEFT affects the predictions of total cross sec-\ntions. Indeed, the computation of S-matrix elements us-\ning Eq. (23), where the order-by-order contributions of\nthe scattering amplitudes are summed directly to the S-\nmatrix, leads to a perturbative breaking of unitarity. In\ncontrast, amplitudes computed non-perturbatively, i.e.,\nwhen the potential terms are summed before solving for\nthe scattering amplitude (as is done in WPC), are uni-\ntary by construction. In this case, the probability flux inthe scattering process is also conserved exactly and the\noptical theorem can be safely used to compute the total\ncross section as, e.g.,\nσtot(p0) =2π\np0Im [a(θcm= 0) + b(θcm= 0)] , (28)\nwhere a(θcm) and b(θcm) are Saclay-amplitudes computed\nfrom the M-matrix [68].\nWe use the difference between total cross sections cal-\nculated using Eq. (27) and Eq. (28) to measure the effects\nof unitarity breaking. In Fig. 6 we show the relative dif-\nference between the cross sections computed using exact\nintegration and the optical theorem as a function of scat-\ntering energy. The figure demonstrates how unitarity is\nrestored perturbatively as we go to higher chiral orders.\nIndeed, the relative difference between the two cross sec-\ntion calculations is limited to 10% for scattering energies8\n72.575.077.580.0dσ/d Ω, 10.0 MeV (mb/sr)\n101520dσ/d Ω, 47.5 MeV (mb/sr)\n51015dσ/d Ω, 96.0 MeV (mb/sr)\n0.000.020.04Pb, 14.1 MeV\n0.00.2Pb, 60.0 MeV\n0 50 100 150\nθc.m.(deg)0.00.20.4Pb, 100.0 MeV\n0 50 100 150\nθc.m.(deg)0.00.10.2Ayy, 23.1 MeV\n0 50 100 150\nθc.m.(deg)−0.250.000.250.50Ayy, 50.0 MeV\nLO\nNLO\nN2LON3LO\nexp.\nFIG. 4. Selection of npscattering observables in the energy interval Tlab= 10 to 100 MeV. Experimental data from Refs.\n[66, 67]. The bands indicate cutoff variation in the same way as in Fig. 3.\nup to 40 MeV at NLO, 70 MeV at N2LO, and 120 MeV\nat N3LO, respectively. The bands in the figure reflect\ndifferences coming from using two cutoff values 500 MeV\nand 2500 MeV. The bands for NLO and N2LO increase\nsmoothly with the scattering energy. The band at N3LO\nshows an artifact from the two different calculations for\nΛ = 2500 MeV intersecting at some energies leading to\nvery small relative errors. We also note that the cutoff\ndependencies for the N2LO and N3LO calculations do\nnot vanish as the scattering energy approaches zero.\nWe can also discuss this result in terms of the EFT\ntruncation error. For a given chiral order, we argue that\nthe results from the two different cross section calcula-\ntions should not become significantly different until we\nreach an energy where the next (omitted) order in the chi-\nral low-energy expansion becomes relevant. This should\ncorrespond to the scattering energy for which the trunca-\ntion error is significant. Breaking unitarity implies that\nthe norm of the partial-wave S-matrix in Eq. (23) devi-\nates from unity as\u0000\nS(ν)\u0001†S(ν)= 1−C(Q/Λb)ν+1, where\nwe also expect Cto be of natural size. This scaling of\nunitarity breaking should be revisited when probabilitydistributions of the LEC values and the hyperparameters\nof the EFT truncation error have been inferred using a\nBayesian approach.\nIV. SUMMARY AND OUTLOOK\nThis work presents a comprehensive analysis of np\nscattering observables (cross sections and polarizations)\nutilizing an RG-invariant formulation of χEFT by Long\nand Yang. We calibrated the LECs by reproducing Ni-\njmegen phase shifts at specific scattering energies, and\ncarried out calculation up to N3LO for two values of the\nmomentum-space cutoffs, 500 MeV and 2500 MeV. The\nPC that we employed is fairly representative of a broad\nclass of MWPCs in which corrections beyond LO, based\non one-pion exchange, are included perturbatively and\nthe short-range contact potential incorporates countert-\nerms promoted to renormalize the long-range pion con-\ntributions to the scattering amplitudes. A key result of\nthis paper was a quantitative demonstration that RG-\ninvariant χEFT exhibits a steady order-by-order conver-9\n0.0 0.2 0.4 0.6\nTlab(MeV)5101520σ(b)\n50 100 150\nTlab(MeV)0.050.100.15σ(b)\n0 25 50 75 100 125 150\nTlab(MeV)0.000.250.500.751.001.251.501.75σ(b) hello(a)\n(b)\n(c)LO\nNLO\nN2LON3LO\nexp.\nFIG. 5. Total npcross sections computed by integrating the differential cross sections (27). Panel ( a) shows cross sections for\na large interval of scattering energies, Tlab= 5–150 MeV. Panels ( b) and ( c) expand results at low- and high-energy intervals,\nrespectively. The bands indicate cutoff variation as in Fig. 3. Experimental data from Refs. [66, 67].\n0 25 50 75 100 125 150\nTlab(MeV)10−510−410−310−210−1100rel. diff\nFIG. 6. The relative difference between total npcross sec-\ntions ( σ) computed by integrating of the differential cross sec-\ntion (27) and the optical theorem (28). The bands indicate\ncutoff variation as in Fig. 3. The color coding for the orders is\nthe same as Fig. 3. The horizontal dashed line marks a 10%\ndifference.\ngence in the description of scattering observables, start-\ning already at LO. A second key result was the realistic\nreproduction of experimental scattering data in an en-\nergy range up to Tlab= 100 MeV at N2LO. We also\nfound that N3LO predictions do not always improve overN2LO.\nA perturbative approach exposes the deficiencies of any\nPC, not only the possible lack of RG-independence. In\nfact, using a perturbative approach we found that the\naccuracy of our N3LO predictions for the total npcross\nsection declines as one approaches Tlab≳100 MeV. This\ncorresponds to a relative scattering momentum of 220\nMeV and might suggest the presence of an additional\nmass scale at 200–300 MeV. This finding is in accordance\nwith the known mass splitting between the nucleon and\nthe ∆(1232) resonance, but is markedly lower than con-\nventional estimates of the breakdown scale of χEFT re-\nsiding in the vicinity of the ρ-meson mass. The latter\nestimate has also been corroborated in a Bayesian study\nof non-perturbative WPC predictions of nucleon-nucleon\nscattering observables [69].\nBased on our comparison of perturbative and non-\nperturbative calculations of phase shifts, we speculated\nthat the magnitudes of the imaginary component of the\nnon-perturbative phase shift and the χEFT truncation\nerror are linked. We also investigated the breaking of\nunitarity at the level of total npcross sections. The con-\nnection between perturbative unitarity breaking and the\ntruncation error deserves further attention.\nFuture work will focus on quantifying posterior prob-\nability distributions for the LECs and the EFT trunca-\ntion error, making predictions beyond the two-nucleon\nsystem, and the effects of including the ∆(1232) res-10\nonance in the two-pion exchange potential. Fast and\naccurate emulators [70], adapted to perturbative com-\nputations, will likely be essential for rigorous testing of\nRG-invariant χEFT against nuclear data and to address\ncritical questions regarding, e.g., the construction of LO,\nthe importance of promoting higher-order pion exchanges\nand many-nucleon forces as one increases the mass num-\nber, and the level of fine-tuning in χEFT.ACKNOWLEDGMENTS\nO.T. thanks C.-J. Yang, B. Long, and R. Peng for help-\nful discussions and for providing detailed benchmarks.\nThe authors also thank Daniel Phillips for feedback on a\ndraft version of the manuscript. 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Meissner, Nuclear\nforces from chiral Lagrangians using the method of uni-\ntary transformation. 2. The two nucleon system, Nucl.\nPhys. A 671, 295 (2000), arXiv:nucl-th/9910064.\n[73] M. I. Haftel and F. Tabakin, Nuclear saturation and the\nsmoothness of nucleon-nucleon potentials, Nucl. Phys. A\n158, 1 (1970).\n[74] R. H. Landau, Quantum mechanics. Vol. 2: A second\ncourse in quantum theory (1990).\n[75] J. Hoppe, C. Drischler, R. J. Furnstahl, K. Hebeler, and\nA. Schwenk, Weinberg eigenvalues for chiral nucleon-\nnucleon interactions, Phys. Rev. C 96, 054002 (2017),\narXiv:1707.06438 [nucl-th].\n[76] H. P. Stapp, T. J. Ypsilantis, and N. Metropolis, Phase\nshift analysis of 310-MeV proton proton scattering ex-\nperiments, Phys. Rev. 105, 302 (1957).13\nAppendix A: Nuclear potentials in the Long and Yang power counting\nThe orders at which potentials appear in the Long and Yang PC in channels where OPE is treated non-perturbatively\nare shown in Table I. Similarly, for the channels where OPE is treated perturbatively, we follow the PC of Ref. [52]\nalso shown in Table I. In this appendix, we list the expressions for the potentials appearing in Table I. The potential\ncontributions will be listed using the following decomposition convention [4]\nV(p′,p) =VC+τ1·τ2WC\n+ [VS+τ1·τ2WS]σ1·σ2+\n+ [VLS+τ1·τ2WLS] (−iS·(q×k))\n+ [VT+τ1·τ2WT]σ1·qσ2·q\n+ [VσL+τ1·τ2WσL]σ1·(q×k)σ2·(q×k),(A1)\nwhere\nq=p−p′,k=1\n2(p+p′),S=1\n2(σ1+σ2) (A2)\nandσidenotes the Pauli spin matrix for the respective nucleon.\nThe one-pion exchange potential takes the form\nV(0)\n1π= (τ1·τ2) (σ1·qσ2·q)WT, (A3)\nWT=−\u0012gA\n4fπ\u001321\nq2+m2π, (A4)\nwhere gA= 1.29 is the axial coupling, fπ= 92.1 MeV the pion decay constant, mπ= 138 .039 MeV is the average\npion mass and q=|q|. For the two-pion exchange potentials, we employ expressions computed with dimensional\nregularization (DR). The leading two-pion exchange potential takes the form [4, 71, 72]\nV(2)\n2π=τ1·τ2WC+σ1·σ2VS+σ1·qσ2·qVT, (A5)\nWC=−L(q)\n384π2f4π\"\n4m2\nπ\u0000\n5g4\nA−4g2\nA−1\u0001\n+q2\u0000\n23g4\nA−10g2\nA−1\u0001\n+48g4\nAm4\nπ\nw2#\n, (A6)\nVS=3g4\nAL(q)q2\n64π2f4π, (A7)\nVT=−1\nq2VS=−3g4\nAL(q)\n64π2f4π, (A8)\nwith\nL(q) =w\nqlnw+q\n2mπ, w =p\n4m2π+q2. (A9)\nThe sub-leading two-pion exchange potential takes the form of Eqs. (4.13) - (4.20) in [4]. We apply the power\ncounting ( Q/m N) = ( Q/Λb)2for (1 /mN) corrections, which means that all terms proportional to 1 /mNvanish at\norder ( Q/Λb)3(N3LO). The non-zero contributions read\nV(3)\n2π=VC+ (τ1·τ2) (σ1·qσ2·q)WT, (A10)\nVC=−3g2\nA\n16πf4πh\n2m2\nπ(2c1−c3)−q2c3i\n˜w2A(q), (A11)\nWT=−1\nq2WS=−g2\nAA(q)\n32πf4πc4w2, (A12)\n(A13)\nwith\nA(q) =1\n2qarctanq\n2mπ,˜w=p\n2m2π+q2. (A14)14\nFor the πNLECs c1, c3, c4, appearing in V(3)\n2π, we employ numerical values determined in a Roy-Steiner analysis at\nNLO: c1=−0.74 GeV−1,c3=−3.61 GeV−1andc4= 2.44 GeV−1[54].\nThe potential contributions at each order in the channels where OPE is treated non-perturbatively are listed in\nTable III. We denote counterterms in coupled channels by a 2 ×2 matrix representing ℓ′=j∓1 (rows) and ℓ=j∓1\n(columns). Table III expands upon Table I in Ref. [30] to also explicitly show the perturbative corrections to LECs\npresent at each order. Table IV summarizes the number of LECs present at each order, excluding the three πNLECs\nat N3LO from the total number.\nTABLE III. Potential contributions at each chiral order in the channels where OPE is treated non-perturbatively. This table\ncomplements the information in Table I.\nOrder Pion contribution Contact terms\nLO V(0)\n1π V(0)\nct:\nC(0)\n1S0,\nC(0)\n3S10\n0 0\n,D(0)\n3P0p′p,\nD(0)\n3P2p′p0\n0 0\n\nNLO - V(1)\nct:\nD(0)\n1S0(p′2+p2),C(1)\n1S0\nN2LO V(2)\n2π V(2)\nct:\nE(0)\n1S0p′2p2,D(1)\n1S0(p′2+p2),C(2)\n1S0,\n\nD(0)\n3S1(p′2+p2)D(0)\nSDp2\nD(0)\nSDp′20\n,\nC(1)\n3S10\n0 0\n,\nE(0)\n3P0p′p(p′2+p2) ,D(1)\n3P0p′p,\np′p\nE(0)\n3P2(p′2+p2)E(0)\nPFp2\nE(0)\nPFp′20\n,\nD(1)\n3P2p′p0\n0 0\n,\nD(0)\n1P1p′p,D(0)\n3P1p′p\nN3LO V(3)\n2π, (includes V(3)\nct:\nπNLECs: c1, c3, c4)F(0)\n1S0p′2p2(p′2+p2),E(1)\n1S0p′2p2,D(2)\n1S0(p′2+p2),C(3)\n1S0,\n\nD(1)\n3S1(p′2+p2)D(1)\nSDp2\nD(1)\nSDp′20\n,\nC(2)\n3S10\n0 0\n,\nE(1)\n3P0p′p(p′2+p2),D(2)\n3P0p′p,\np′p\nE(1)\n3P2(p′2+p2)E(1)\nPFp2\nE(1)\nPFp′20\n,\nD(2)\n3P2p′p0\n0 0\n,\nD(1)\n1P1p′p,D(1)\n3P1p′p\nAppendix B: Numerical implementation of distorted-wave perturbation theory\nThis appendix gives some more details regarding the implementation of the equations for higher-order corrections\nto the scattering amplitude in Eqs. (11) to (13). Since all operator products reduce to the form in Eq. (19), the\nimplementation can be done in complete analogy with the solution of the partial-wave Lippmann-Schwinger equation\nusing Gauss-Legendre quadrature [73, 74].15\nTABLE IV. The number of LECs at each order in the Long and Yang PC.\nChiral order New LECs Pert. correction Total up to order\nLO 4 – 4\nNLO 1 1 6\nN2LO 8 5 19\nN3LO 1 (+3a) 13 33\naSub-leading πNLECs: c1, c3, c4excluded from the total in the last column.\nIn this appendix we suppress the conserved quantum numbers sandj, and write the resolution of identity in the\npartial wave basis as\n1=X\nℓZ∞\n0dk k2|k, ℓ⟩⟨k, ℓ|. (B1)\nFurthermore, for a stationary proton (mass mp) and an incoming neutron (mass mn) with kinetic energy Tlabin the\nlaboratory frame of reference, the modulus of the c.m.momentum, p0, is given by\np2\n0=m2\npTlab(2mn+Tlab)\n(mn+mp)2+ 2mpTlab. (B2)\nBy inserting the resolution of identity in Eq. (19) and discretizing the integral using Gauss-Legendre quadrature with\nmomentum points and weights, {ki, wi}N\ni=1, we obtain\n⟨p′, ℓ′|A1G+\n0A2|p, ℓ⟩=X\nℓ′′,ℓ′′′Z∞\n0dk1k2\n1Z∞\n0dk2k2\n2⟨p′, ℓ′|A1|k1, ℓ′′⟩⟨k1, ℓ′′|G+\n0|k2, ℓ′′′⟩⟨k2, ℓ′′′|A2|p, ℓ⟩=\n=X\nℓ′′Z∞\n0dk1k2\n1⟨p, ℓ′|A1|k1, ℓ′′⟩mN\np2\n0−k2\n1+iϵ⟨k1, ℓ′′|A2|p, ℓ⟩= (B3)\n=X\nℓ′′NX\ni=1k2\niwi⟨p, ℓ′|A1|ki, ℓ′′⟩mN\np2\n0−k2\ni+iϵ⟨ki, ℓ′′|A2|p, ℓ⟩. (B4)\nHere, p0denotes the on-shell momentum for a given scattering energy Tlabgiven by Eq. (B2). Doing some manipu-\nlations and converting the + iϵprescription to a principal value we obtain [65, 74]\n⟨p′, ℓ′|A1G+\n0A2|p, ℓ⟩=X\nl′′NX\ni=1k2\niwi⟨p′, ℓ′|A1|ki, ℓ′′⟩mN\np2\n0−k2\ni⟨ki, ℓ′′|A2|p, ℓ⟩\n− ⟨p′, ℓ′|A1|p0, ℓ′′⟩⟨p0, ℓ′′|A2|p, ℓ⟩\"\nmNp2\n0NX\ni=1wi\np2\n0−k2\ni+iπm Np0\n2−mNp0arctanh\u0012p0\n˜Λ\u0013#\n.(B5)\nAll potentials are regulated using Eq. (16) and at sufficiently high momentum, ˜Λ, all potential matrix elements are\nessentially zero. This means that the integral in Eq. (B4) is well represented by the discretized sum where the\nmomentum points and weights {ki, wi}N\ni=1are chosen using Gauss-Legendre quadrature in the interval [0 ,˜Λ]. The\nlast term in the bracket in Eq. (B5) implements the principal-value integral on the interval [ ˜Λ,∞] analytically since\nthe grid is just doing the integration on [0 ,˜Λ] [75]. It is possible to have a grid that extends to numerical infinity, but\nthis generally leads to slower convergence with N. For the calculations in this study, we employ ˜Λ = Λ + 1500 MeV,\nfor both Λ = 500 MeV and Λ = 2500 MeV, which we find sufficient for numerical convergence.\nEquation (B5) can be expressed in a simpler form using matrix products, which speeds up the computations. We\ndefine the propagator matrix as\n[G+\n0]ij=δijFi, F i=(mN\np2\n0−k2\ni, i= 1, ..., N\n−f(p0), i=N+ 1,(B6)\nwhere\nf(p0) =mNp2\n0NX\ni=1wi\np2\n0−k2\ni+iπm Np0\n2−mNp0arctanh\u0012p0\n˜Λ\u0013\n. (B7)16\nSimilarly, we make the following definitions of matrices for Aµ,µ= 1,2,\n[Aℓ′ℓ\nµ]i,j =ki√wi⟨ki, ℓ′|Aµ|kj, ℓ⟩kj√wj, i, j = 1, . . . , N (B8)\n[Aℓ′ℓ\nµ]i,j=N+1 =ki√wi⟨ki, ℓ′|Aµ|p0, ℓ⟩, i= 1, . . . , N (B9)\n[Aℓ′ℓ\nµ]i=N+1,j =⟨p0, ℓ′|Aµ|kj, ℓ⟩kj√wj, j = 1, . . . , N (B10)\n[Aℓ′ℓ\nµ]i=N+1,j=N+1=⟨p0, ℓ′|Aµ|p0, ℓ⟩, (B11)\neffectively including an extra momentum-grid point kN+1≡p0with weight√wN+1=p−1\n0. Using these definitions\nand defining D=A1G+\n0A2, Eq. (B5) can be written using ( N+ 1)×(N+ 1) matrix products\n[Dℓ′ℓ]ij=X\nℓ′′N+1X\nn,m=1[Aℓ′ℓ′′\n1]in[G+\n0]nm[Aℓ′′ℓ\n2]mj, i, j = 1, . . . , N + 1. (B12)\nFor coupled channels, we further eliminate the sum over ℓ′′in Eq. (B12) by defining (2 N+2)×(2N+2) block-matrices,\nwhich for A1reads\n[A1] =\u0012\n[A−−\n1] [A−+\n1]\n[A+−\n1] [A++\n1]\u0013\n. (B13)\nThe±notation represents ℓ=j±1. The propagator is diagonal in ℓand can be written as\n[G+\n0] =\u0012\n[G+\n0] 0\n0 [G+\n0]\u0013\n. (B14)\nWe can finally write Eq. (B12) as\n[D] = [A1][G+\n0][A2]. (B15)\nNote that the simplification of Eq. (B5) to an ordinary matrix product in Eq. (B15) is only possible due to the\nspecific structure of having G+\n0in between A1andA2. This structure gives rise to the last “on-shell” term in (B5)\nthat can be incorporated by adding the grid point kN+1=p0, which then extends the sum in Eq. (B5) to N+ 1.\nEquation (B12) can now be used recursively to compute longer products such as ⟨p′, ℓ′|A1G+\n0A2G+\n0A3|p, ℓ⟩.\nAs an example, the first-order correction to the T-matrix in Eq. (11) can be expressed as the matrix equation\n[T(1)] =\u0010\n1+ [T(0)][G+\n0]\u0011\n[V(1)]\u0010\n1+ [G+\n0][T(0)]\u0011\n. (B16)\nAppendix C: Perturbative phase shifts\nIn this appendix we discuss how to obtain phase shifts given perturbative corrections to the T-matrix computed\nfrom Eqs. (11) to (13). We will follow the method outlined in Ref. [32] and add some additional details. For uncoupled\nscattering channels, the 1 ×1S-matrix can be parameterized by\nS= exp (2 iδ), (C1)\nwhere δis the phase shift. We expand both the phase shifts and the on-shell S-matrix with the contributions at each\nchiral order obtaining\nS(0)+S(1)+S(2)+S(3)+O(Q3) = (C2)\nexp\u0010\n2ih\nδ(0)+δ(1)+δ(2)+δ(3)+O(Q3)i\u0011\n. (C3)\nPerforming a Taylor expansion of both sides, and matching chiral orders, gives\nS(0)= exp\u0010\n2iδ(0)\u0011\n(C4)\nS(1)= 2iδ(1)exp\u0010\n2iδ(0)\u0011\n(C5)\nS(2)=\u0014\n2iδ(2)−2\u0010\nδ(1)\u00112\u0015\nexp\u0010\n2iδ(0)\u0011\n(C6)\nS(3)=\u0014\n2iδ(3)−4δ(1)δ(2)−4i\n3\u0010\nδ(1)\u00113\u0015\nexp\u0010\n2iδ(0)\u0011\n(C7)17\nFrom these equations, we straightforwardly obtain explicit expressions for the LO phase shift δ(0)(trivial), and all\ncorrections {δ(ν)}ν>0. We note that all corrections are real valued. To obtain the total phase shift at, e.g., N2LO,\none has to sum δ(0)+δ(1)+δ(2). The S-matrix corrections are obtained from the T-matrix corrections as\nS(ν)\nℓ′ℓ=−iπm Np0T(ν)\nℓ′ℓ, ν > 0, (C8)\nfor a given on-shell momentum, p0.\nFor coupled channels we use the Stapp-parametrization [76] for the on-shell 2 ×2S-matrix\nS=\u0012\ncos(2 ϵ)e2iδ1isin(2ϵ)ei(δ1+δ2)\nisin(2ϵ)ei(δ1+δ2)cos(2 ϵ)e2iδ2\u0013\n, (C9)\nwhere the three phase shifts δ1,δ2andϵparameterize the amplitude for a given channel. We now proceed completely\nanalogous to the uncoupled case, dividing the S-matrix and phase shifts into chiral orders as\nS=∞X\nν=0=S(ν), δ 1=∞X\nν=0δ(ν)\n1, δ 2=∞X\nν=0δ(ν)\n2, ϵ=∞X\nν=0ϵ(ν). (C10)\nFor convenience, we define the functions\nf11(ϵ, δ1) = cos(2 ϵ)e2iδ1, (C11)\nf12(ϵ, δ1, δ2) =isin(2ϵ)ei(δ1+δ2), (C12)\nf22(ϵ, δ2) = cos(2 ϵ)e2iδ2, (C13)\nwhich are the constituents of the matrix in Eq. (C9). Inserting the expansions in Eq. (C10) into Eq. (C9), Taylor\nexpanding and matching chiral orders, gives the perturbative corrections to the phase shifts. Expanding the upper\nleft matrix element of Sgives\nS(0)\n11=f11 (C14)\nS(1)\n11=∂ϵf11×ϵ(1)+∂δf11×δ(1)(C15)\nS(2)\n11=∂ϵf11×ϵ(2)+∂δf11×δ(2)\n+g(2)\n11(ϵ(1), δ(1)) (C16)\nS(3)\n11=∂ϵf11×ϵ(3)+∂δf11×δ(3)\n+g(3)\n11(ϵ(1), δ(1), ϵ(2), δ(2)) (C17)\nwhere the functions g(ν)\n11are introduced to capture all non-linear terms in the expansion\ng(2)\n11(ϵ(1), δ(1)) =1\n2∂2\nϵf11×\u0010\nϵ(1)\u00112\n+1\n2∂2\nδf11×\u0010\nδ(1)\u00112\n+∂ϵ∂δf11×δ(1)ϵ(1)(C18)\ng(3)\n11(ϵ(1), δ(1), ϵ(2), δ(2)) =∂2\nϵf11\u0010\nϵ(1)ϵ(2)\u0011\n+∂ϵ∂δf11\u0010\nϵ(1)δ(2)+ϵ(2)δ(1)\u0011\n+∂2\nδf11\u0010\nδ(1)δ(2)\u0011\n+1\n6∂3\nϵf11\u0010\nϵ(1)\u00113\n+1\n2∂δ∂2\nϵf11\u0010\nϵ(1)\u00112\nδ(1)\n+1\n2∂2\nδ∂ϵf11ϵ(1)\u0010\nδ(1)\u00112\n+1\n6∂3\nδf11\u0010\nδ(1)\u00113\n. (C19)\nSince f11depends on ϵandδ1the index one is suppressed. The function f11and all its derivatives are evaluated at\n(ϵ(0), δ(0)\n1).\nFor the lower right matrix element described by f22the expressions are completely analogous to Eqs. (C18)\nand (C19), but with δ2instead of δ1. For the off-diagonal elements we get\nS(0)\n12=f12 (C20)\nS(1)\n12=∂ϵf12×ϵ(1)+∂δ1f12×δ(1)\n1+∂δ2f12×δ(1)\n2 (C21)\nS(2)\n12=∂ϵf12×ϵ(2)+∂δ1f12×δ(2)\n1+∂δ2f12×δ(2)\n2+g(2)\n12(ϵ(1), δ(1)\n1, δ(1)\n2) (C22)\nS(3)\n12=∂ϵf12×ϵ(3)+∂δ1f12×δ(3)\n1+∂δ2f12×δ(3)\n2+g(3)\n12(ϵ(1), δ(1)\n1, δ(1)\n2, ϵ(2), δ(2)\n1, δ(2)\n2), (C23)18\nwhere the functions g(ν)\n12capture the non-linear terms\ng(2)\n12(ϵ(1), δ(1)\n1, δ(1)\n2) =1\n2∂2\nϵf12×\u0010\nϵ(1)\u00112\n+1\n2∂2\nδ1f12×\u0010\nδ(1)\n1\u00112\n+1\n2∂2\nδ2f12×\u0010\nδ(1)\n2\u00112\n+∂ϵ∂δ1f12ϵ(1)δ(1)\n1+∂ϵ∂δ2f12ϵ(1)δ(1)\n2+∂δ1∂δ2f12δ(1)\n1δ(1)\n2 (C24)\ng(3)\n12(ϵ(1), δ(1)\n1, δ(1)\n2, ϵ(2), δ(2)\n1, δ(2)\n2) =∂2\nϵf12ϵ(1)ϵ(2)+∂2\nδ1f12δ(1)\n1δ(2)\n1+∂2\nδ2f12δ(1)\n2δ(2)\n2\n+∂ϵ∂δ1f12\u0010\nϵ(1)δ(2)\n1+ϵ(2)δ(1)\n1\u0011\n+∂ϵ∂δ2f12\u0010\nϵ(1)δ(2)\n2+ϵ(2)δ(1)\n2\u0011\n+∂δ1∂δ2f12\u0010\nδ(1)\n1δ(2)\n2+δ(2)\n1δ(1)\n2\u0011\n+1\n2∂2\nϵ∂δ1f12\u0010\nϵ(1)\u00112\nδ(1)\n1+1\n2∂2\nϵ∂δ2f12\u0010\nϵ(1)\u00112\nδ(1)\n2\n+1\n2∂ϵ∂2\nδ1f12ϵ(1)\u0010\nδ(1)\n1\u00112\n+1\n2∂δ2∂2\nδ1f12δ(1)\n2\u0010\nδ(1)\n1\u00112\n+1\n2∂ϵ∂2\nδ2f12ϵ(1)\u0010\nδ(1)\n2\u00112\n+1\n2∂δ1∂2\nδ2f12δ(1)\n1\u0010\nδ(1)\n2\u00112\n+\n+∂ϵ∂δ1∂δ2f12ϵ(1)δ(1)\n1δ(1)\n2\n+1\n6∂3\nϵf12\u0010\nϵ(1)\u00113\n+1\n6∂3\nδ1f12\u0010\nδ(1)\n1\u00113\n+1\n6∂3\nδ2f12\u0010\nδ(1)\n2\u00113\n. (C25)\nThe function f12and all its derivatives are evaluated at ( ϵ(0), δ(0)\n1, δ(0)\n2). Note that all the functions g(ν)\n∗∗vanish if\nthe NLO corrections ( δ(1)\n1, δ(1)\n2, ϵ(1)) are zero. This is the case for all coupled channels where OPE is treated non-\nperturbatively as seen in Table III. Furthermore, in all channels where OPE is treated perturbatively the LO phase\nshifts are all zero, which makes many of the terms in the expressions for g(ν)\n∗∗vanish due to vanishing derivatives.\nThus, in both the perturbative and non-perturbative cases, Eqs. (C18), (C19), (C24) and (C25) can be simplified\nsubstantially. The phase shift corrections ( ϵ(ν), δ(ν)\n1, δ(ν)\n2) for ν= 1,2,3.are finally obtained by solving a system of\nlinear equations\nNLO :\nS(1)\n11\nS(1)\n12\nS(1)\n22\n =\n∂ϵf11∂δ1f11 0\n∂ϵf12∂δ1f12∂δ2f12\n∂ϵf22 0∂δ2f22\n\nϵ(1)\nδ(1)\n1\nδ(1)\n2\n (C26)\nN2LO:\nS(2)\n11−g(2)\n11\nS(2)\n12−g(2)\n12\nS(2)\n22−g(2)\n22\n=\n∂ϵf11∂δ1f11 0\n∂ϵf12∂δ1f12∂δ2f12\n∂ϵf22 0∂δ2f22\n\nϵ(2)\nδ(2)\n1\nδ(2)\n2\n (C27)\nN3LO:\nS(3)\n11−g(3)\n11\nS(3)\n12−g(3)\n12\nS(2)\n22−g(3)\n22\n=\n∂ϵf11∂δ1f11 0\n∂ϵf12∂δ1f12∂δ2f12\n∂ϵf22 0∂δ2f22\n\nϵ(3)\nδ(3)\n1\nδ(3)\n2\n. (C28)"    },    {        "title": "2402.15338v1.Effect_of_temperature_and_copper_doping_on_the_heterogeneous_Fenton_like_activity_of_Cu__x_Fe___3_x__O__4__nanoparticles.pdf",        "content": "Effect of  temperature and copper dop ing on the heterogeneous Fenton -like \nactivity of 𝑪𝒖 𝒙𝑭𝒆 𝟑−𝒙𝑶𝟒 nanoparticles   \nNahuel Nuñez1,2,3*, Enio Lima Jr.1,2, Marcelo Vásquez  Mansilla1,2, Gerardo F. Goya4,5, Álvaro Gallo -\nCordova6, María del Puerto Morales6, Elin L. Winkler1,2,3* \n1 Laboratorio de Resonancias Magnéticas, Gerencia de Física, Centro Atómico Bariloche, Av. \nBustillo 9500, (8400) S. C. de Bariloche (RN), Argentina.  \n2 Instituto d e Nanociencia y Nanotecnología (CNEA -CONICET), Nodo Bariloche, Av. Bustillo 9500, \n(8400) S. C. de Bariloche (RN), Argentina.  \n3 Instituto Balseiro, CNEA -UNCuyo, Av. Bustillo 9500, (8400) S. C. de Bariloche (RN), Argentina  \n4 Dept. Física de la Materia Conden sada, Universidad de Zaragoza, C/ Pedro Cerbuna 12, 50009, \nZaragoza, Spain  \n5 Instituto de Nanociencia y Materiales de Aragón, CSIC -Universidad de Zaragoza, C/ Mariano \nEsquillos S/N , 500 18, Zaragoza, Spain  \n6 Instituto de Ciencia de Materiales de Madrid, ICM M/CSIC, C/  Sor Juana Inés de la Cruz 3, 28049, \nMadrid, Spain  \n *Corresponding authors  : nahuel.nunez@ib.edu.ar   \nKeywords: copper doped iron oxide nanoparticles; Heterogeneous Fenton reaction; free radicals ; \nmagnetic catalyst ; organic dyes  \n  Abstract  \nFerrite nanoparticles serve as potent heterogeneous Fenton -like catalysts, producing reactive \noxygen species (ROS) for decomposing organic pollutants. We investigated the impact of \ntemperature and copper content on the catalytic activity of nanoparticles w ith different oxidation \nstates of iron. Via solvothermal synthesis, we fabricate d copper -doped magnetite (Cu xFe3-xO4) with \na Fe2+/Fe ratio ~0.33 for the undoped system. Using a microwave -assisted method, we produced \ncopper -doped oxidized ferrites, yielding  a Fe2+/Fe ratio of ~0.11 for the undoped nanoparticles. \nThe ROS generated by the catalyst were identified and quantified by electron paramagnetic \nresonance, while optical spectroscopy allowed us to evaluate its effectiveness for the degradation \nof a model  organic dye. At room temperature, the magnetite nanoparticles exhibited the most •OH \nradical production and achieved almost 90% dye discoloration in 2 hours. This efficiency \ndecreased with increasing Cu concentration, concurrently  with a decrease in •OH g eneration. \nConversely, above room temperature, Cu -doped nanoparticles significantly enhance the  dye \ndegradation, reaching 100% discoloration at 90°C. This enhancement is accompanied by a \nsystematic increase in the kinetic constants, obtained  from reaction equations, with Cu doping. \nThis study highlights the superior stability and high -temperature catalytic advantages of copper \nferrite holding promise for enhancing the performance of nanocatalysts for decomposing organic \ncontaminants.  \n \n \n \n \n \n 1. Introduction  \n \nEnvironmental remediation process es based on the catalytic activity of m agnetic nanoparticles \nhave gained considerable attention in recent years.1–3 One promising example on this is the use \nof iron oxide nanoparticles as heterogeneous catalysts in the Fenton  reaction s that generates \nreactive oxygen species (ROS) .4,5 These species allow s the degradation of various organic \npollutants  through an advanced oxidation process .6 In the homogeneous  Fenton reaction, •OH and \n•OOH radicals are generated by soluble Fe2+ and Fe3+ ions through  the following reactions : Fe2++\nH2O2→Fe3++•OH+OH− (kcat=63 M−1 s−1) and  \n Fe3++H2O2→Fe2++•OOH +H+ (kcat=0.001  M−1 s−1).7 In the literature this second equation \nis known as Fenton -like reaction, bei ng the limiting reaction due to  the lower rate constant .8,9 These \nfree radicals oxidize  organic molecules and pollutants, reducing their toxicity. Compared to the \nhomogeneous Fenton reaction, the use of heterogeneous catalysts offers several advantages, \nsuch as increased stability, recyclability, and reduced secondary pollution .10 It is expected that in  \nhomogeneous catalysis , the solubility of catalyst ions is the process limiting factor , while in the \nheterogeneous one, the adsorption steps as well as the diffusion of the reactive species in the \nmedium are dominant .11 Therefore, heterogeneous  catalysis provides  greater degrees of freedom \nto tune and gain control over  the subsequent advanced oxidation reaction .  \nLike any thermally activated process, the kinetics of th e Fenton reaction is accelerated by \ntemperature .12 For this reason, numerous studies aimed to design multifunctional nanoparticles \ncapable of inducing local heating to enhance its catalytic activity.13,14 This provi des an additional \nparameter to control and optimize the efficiency of catalysts in Fenton processes. In fact, studies \nfocused on decontaminating landfill leachate using copper catalysts concluded that the optimum \ntemperature for the reaction was 70 °C.15   \nWhile it is true that increasing the temperature of the system can elevate the cost of wastewater \ntreatment, in some instances, the waste is already above room temperature, condition that can be \nleveraged to enhance the efficiency of the nanocata lyst. For example, it was reported that the temperature of effluents from textile wet processing is in the range of 30°C to 60°C .16 Similarly, \nvinasse, a byproduct of bioethanol distillation, is generated at high temperatures, approximately \naround 90°C.17,18 Another example is the paper industry, whe re the chemical treatment of  wood \ngenerates a black liquor residue at about  60-70°C .19 On the other hand, w hile increasing the \ntemperature accelerates the kinetics of th e reaction, improving the catalyst's efficiency, it may also \nproduce undesirable effects The thermally accelerated decomposition of hydrogen peroxide can \ndecrease the degradation efficiency by Fenton  and Fenton -like processes .20 Moreover , it has been \ndemonstrated that Fe3O4 nanoparticles undergo a transformat ion into γ-Fe2O3 after continuous \nFenton reaction due to the different kcat of Fe2+ and Fe3+ ions,  resulting in a subsequent loss of \ntheir catalytic activity .21 This oxidation of iron catalysts  also could be promoted by increasing the \ntemperature. Therefore, various effects of the operating temperature must be taken into account \nwhen the optimal working temperature for wastewater treatment is defined.  \nTo optimize the  catalytic efficiency of iron oxide nanoparticles  in advanced oxidation reactions , \nvarious ions have been incorporated in to the ferrite lattice in order to modify their  surface \nreactivity .7,22,23 Among them, copper has received special attention due to its strikingly similar redox \nproperties like iron.24–32 Both the monovalen t Cu+ and divalent Cu2+ oxidation states easily react \nwith H2O2 analogous to the Fe2+/H2O2 and Fe3+/H2O2 systems, the reaction constants of these \nFenton -like process es are kcat=104 M−1 s−1 and kcat=4.6 ∗102 M−1 s−1, respectively .7 Due to \nits enhanced activity, copper nanocatalysts have been previously  employed to efficient ly degrade \ncontaminants like phenols ,33 insecticides ,34 pharmaceuticals35–37  and also as antibacterial agent.38 \nMoreover, a synergistic effect was observed when copper was introduced into a mesoporous iron \noxide catalyst for the Fenton -like process .15,26,39,40 In this case, t he proposed mechanism for the \ngood performance of copper -iron catalysts is the regeneration of Fe2+ ions through the reaction \nFe3++Cu1+→Fe2++Cu2+. An other interesting capability of copper catalysts is  its good \nperformance after recycling . Hussain et al. found that zirconia -supported copper catalysts \nmaintained their activity when recycled, unlike iron catalysts wh ose activity diminished .40 For these reasons  copper catalysts are interesting materials to be tested in Fenton -like processes at high \ntemperature reactions.  \nIn this context, in the present work we investigate the heterogeneous Fenton -like catalytic activity \nof copper -doped  iron oxide nanoparticles with the aim of studying  the role played by the surface \nactive ions in the generation of ROS, and determine the proper doping condition  and surface \noxidation state  for different working temperature in wastewater treatment .  For this , copper -doped \nmagnetite and maghemite nanoparticles  were synthesized by two different polyol method s: \nsolvothermal and microwave -assisted, respectively . The as -synthesized nanoparticles were \ncharacterized by various techniques, including X -ray diffraction  (XRD) , transmission electron \nmicroscopy  (TEM)  and X -ray photoelectron spectroscopy  (XPS)  in order to determine their \ncrystal line structur e, morpholog y and composition. Subsequently,  the production of free radicals \ncatalyzed by each sample was identified and quantified by means of electron paramagnetic \nresonance (EPR) spectroscopy assisted with a spin trap molecule . Finally , the catalytic \nperformance of the nanoparticles was evaluated by measuring their degradation efficiency of \nmethylene blue (M B) at different reaction temperatures using optical spectroscopy . These results \ndemonstrate th at, while the magnetite is most efficient nanocatalyst at room temperature, it \noxidizes and los es its activity at higher temperature s; copper doping  being  essential  to maintain, \nand even surpass, its performance at higher temperature s. \n \n2. Materials and methods  \n2.1. Materials  \nThe reagents used in this work are: iron (III) nitrate nonahydrate ( Fe(NO 3)3.9H2O, 98% Sigma -\nAldrich ), copper (II) sulfate pentahydrate ( CuSO4.5H2O, 99% Merk ), triethylene glycol  (99% Sigma -\nAldrich) , diethylene glycol  (99% Sigma -Aldrich) , ethan ol (96%) , methylene blue  (>82% Sigma -\nAldrich) , 5,5 -dimethyl -1-pyrroline N -oxide (DMPO , >97%  Cayman ), dimethyl sulfoxide (DMSO, \n>99%) and H 2O2 aqueous solution (30% Sigma -Aldrich).  2.2. Synthesis of the catalysts  \nFor this study we synthesized two series of copper doped iron oxide nanoparticles. The first one \nis co mposed of almost stoichiometric copper ferrite nanoparticles ( Cu𝑥2+Fe1−𝑥2+Fe23+O4). It was \nobtained by the solvothermal method using a polyol as solvent. The second batch was obtained \nby the microwave assisted method, also using a polyol as solvent. In this case the nanoparticles \nwere overoxidized by adding water into the synthesis rea ctor.  \nSolvothermal synthesis : In this method  4 mmol of metals were dissolved in 40 mL of triethylene \nglycol. The  proportions of the precursors Fe(NO 3)3.9H2O and CuSO4.5H2O were adjusted \naccording to the required stoichiometry  by the relations: 𝑚𝐹𝑒(𝑁𝑂 3)3=(3−𝑥)∗241 .86 mg and \n𝑚𝐶𝑢𝑆𝑂 4=𝑥∗249 .68 mg, where x determines the Cu content of atoms in a molecule (Cu xFe3-xO4). \nThe solution was heated at 100 °C for 1 h to evaporate the water from the system and then \ntransferred to a 100 mL teflon autoclave and k ept at 260 °C for 4 h. Once the synthesis is finished, \nthe obtained material was repeatedly washed by magnetic separation with ethanol and then dried \nat 70 °C. The nanoparticle samples were named STX, w here X= 0, 1, 2 an d 3 represents  the \nnominal conce ntration x= 0, 0.1, 0.2 and 0.3 , respectively.  \nMicrowave -assisted synthesis : In this synthesis method , the microwave oven Monowave 300 \n(Anton Paar GmbH, Graz, Austria) with a built -in magnetic stirrer  was used to synthesize  the \ncatalysts,  working at 2.45 GH z and following a protocol similar to that presented by Gallo -Cordova \net al41. Briefly , 1.75 mmol of metals were dissolved in a solution of diethyle ne glycol (18.3 mL) and \nwater (0.7 mL), adjusting the proportions of precursors Fe(NO 3)3.9H2O and CuSO4.5H2O2 \naccording to the required stoichiometry  following  the relationship previously mentioned in the \nsolvothermal synthesis method. The solution was tra nsferred to a 30 mL vial and the temperature \nwas raised to 230 °C with a ramp of 5.25 °C/m in, where it was maintained for 2 h and then cooled \nabruptly. It is important to mention that the  pressure increases due to  the presence of water that \nlowers the boiling point of the solvent mixture (170 ºC) , reaching up to 30 Bar in some cases. After \nthe synthesis, the nanoparticles were repeatedly washed  by magnetic separation  with alcohol and redispersed in water. In this case, the nanoparticle samples were name d MWX, where X= 0, 1, 2, \n3 and 4 represents  the nominal concentration x= 0, 0.1, 0.2, 0.3 and  0.4, respectively.  \n2.3. Characterization of the samples and evaluation of catalytic activity  \nThe crystal structure of the synthesized samples was characterized by X -ray diffraction (XRD) \nusing the Bruker Advance D8 diffractometer (Cu -Kα radiation, λ=0.15406 nm). The incorporation \nof copper ions into the structure of the nanoparticles was verified by inductively coupled plasma \noptical emission spectroscopy  (ICP-OES) using  a Perkin Elmer apparatus (OPTIME  2100 DV ) and \nthe organic content quantified by thermogravimetric analysis (TGA) in a ATD/DSC/TG, Q600 from \nTA Instruments . X-ray photoelectron spectroscopy (XPS) was used to analyze the oxidation state \nof iron ions in the n anoparticles, employing the Kratos AXIS Supra . The values of Fe2+ and Fe3+ for \neach sample w ere obtained by fitting each XPS spectrum with the peaks of the Fe2+ (lower binding \nenergy) and Fe3+ (higher binding energy) multiple ts using the software CASA XPS. The size and \nmorphology of the NPs were studied by transmission electron microscopy (TEM)  in a Philips CM -\n200 microscope operating at 200 kV. The DC magnetization of the samples was studied with a \nLakeShore 7300 vibrating sample magnetometer (VSM). Magneti zation versus applied field  (M(H))  \ncycles w ere acquired at room temperature with the VSM up to  an applied field of  ±10 kOe. \nThe generation of free radicals by the catalysts was studied by electron paramagnetic resonance \n(EPR) working in the X -band (9.5 GHz ) at room temperature with a BRUKER ELEXSYS II -E500 \nspectrometer  using the nitrone -based DMPO as a spin trap . Measurements were performed with \na modulation signal of 100 kHz and 3 G of amplitude , and using the resonance of Mn2+ impurities \nin a MgO crystal as a pattern signal to normalize the free radical production of each sample .21 In \nthese experiments, 0.1 mg/mL of catalyst  was dispersed in 100 mM acetate buffer  at pH=5 , then \n50 μL of a 1 mg:6 mL s olution of DMPO:DMSO were added . Then 5 μL of 30% H2O2 aqueous \nsolution were added and EPR spectra were taken at intervals of no more than 10 minutes . About \n90 μL of the reaction solution was contained in a quartz tube  and the height of the measured region \nwas 30 mm . To evaluate the efficiency of the synthesiz ed nanoparticles for degrading organic compounds, \ncolorimetric experiments were performed with methylene blue dye . The methylene blue \nconcentration was selected from previous works42, and the catalyst and hydrogen peroxide dosage \nwere fixed in common values reported for Fenton and Fenton -like processes43,44. A pH=5 was \nchosen, acid ic enough to favor the Fenton reaction45, but no t so low to avoid leaching, which is \ndrastically enhanced for pH<5 .46,47 \nIn these experiments, 1 mg/mL of nanoparticles and 100 ppm of dye were dispersed in a pH= 5 \n100 mM acetate buffer , and the system was kept under agitation for two hours to ensure adsorption \nof the dye onto the nanoparticles' surface. Then, 10 μL/mL of 30% H2O2 aqueous mixture  were \nadded to the solution, and the degradation efficiency was determined by measuring the \nabsorbance at 663 nm at different time intervals (5 , 15, 30, 60 and 120 min) and considering a \ncalibration curve absorbance vs  MB concentration . These experiments were carried out in 2 mL \nreactors, using different reactors for each time (six in total for a complete measurement).  \nMeasurements were taken at room temperature , at 60 °C and at 90°C using the NUMAK 721 UV -\nVis spectrophotometer.   \n3. Results and discussion  \n3.1. Nanocatalysts characterization  \nXRD  patterns of the copper doped iron oxide  nanoparticles synthe sized by the solvothermal and \nmicrowave -assisted method are shown in Fig.1a and Fig.1b , respectively. The main XRD \ndiffraction peaks observed can be  assigned to the Fd3m spinel structure, characteristic of copper \nferrite. Noticeably, the XRD pattern of the  x=0 microwave -assisted sample shows  also additional \npeaks at 32° and 34°, besides the ones indexed with the spinel structure mentioned above , which \nare characteristic of  the ordering of cation vacancies in the maghemite structure .48 This result \nconfirms  a higher degree of oxidation of  the samples fabricated by microwave -assisted method  in \ncomparison to those prepared by  the solvothermal one , attributed  to the presence of the water \nadded in the synthesis. The intensity of the mentioned peaks decreases as copper is introduced into the structure and almost disappears for x=0.2, which could be  explained by the copper \noccupancy of cation vacancies into the maghemite structure.  Minority m etallic copper segreg ation \nwas observed for the largest copper substitution , this occurs for x>0.3 and x>0.4 for nanoparticles \nfabricated by solvothermal and microwave -assisted method s, respectively. Due to the metallic \nphase segregation, this study was restricted to x ≦0.3 and x ≦0.4 for nanoparticles synthesized by \nsolvothermal and microwave route, respectively. The XRD pattern s of the solvothermal sample s \nshow a low intensity peak at 24°, attributed to the presence of iron hydroxide  traces  (JCPDS 00 -\n046-1436)  49. Additionally,  narrow peaks at 17° and 23° were observed in the x=0 sampl e which \nare ascribed to polymerized (poly -) ethylene glycol.50,51  \n \nFig.1: X-ray diffractogram of the copper -incorporated iron oxide nanoparticles obtained by : (a) the \nsolvothermal method and (b) the microwave –assisted method . \n \nThe incorporation of copper in the structure was measured by ICP -OES , and the results are \npresented in Table1 , where a systematic increase of Cu -concentration was determined, although \nslightly smaller than the nominal doping  for both methods.  The oxidation state of iron for all the \nnanoparticles system s was evaluated by XPS measurements.  Notice that although the XPS is a \nsurface sensitive technique, in this case it provides  information of almost the whole nanoparticle \nas the measuring range technique is about 4 nm.  From the XPS spectra (shown in Figs. S1 and \nS2 of the Supplementary Informat ion) the Fe2+/Fe ratio was determined for all the samples and the \nresults are presented in Table1 . In the solvothermal samples, the Fe2+/Fe ratio obtained by XPS \nis similar to the one expected for the stoichiometric  CuxFe3−xO4 nanoparticles, where x \ncorresponds  to the value determined by ICP -OES.  On the other hand, microwave -assisted \nsamples present much lower Fe2+/Fe ratio than the theoretical one, confirming that these samples \nare overoxidized. This result is consist ent with the maghemite phase detected by the XRD patterns.  \n \nMethod  Solvothermal  Microwave -assisted  \nSample  ST0 ST1 ST2 ST3 MW0  MW1  MW2  MW3  MW4  \n𝐱𝐧𝐨𝐦𝐢𝐧𝐚𝐥  0 0.1 0.2 0.3 0 0.1 0.2 0.3 0.4 \n𝐱𝐈𝐂𝐏 0 0.08 0.15 0.23 0 0.09 0.17 0.24 0.32 \nFe2+/Fe \nnominal  0.333  0.315  - 0.278  0.333  0.313  0.293  0.286  0.253  \nFe2+/Fe  \nby XPS  0.33 0.32 - 0.26 0.11 0.16 0.18 0.12 0.16 \n𝐝𝐓𝐄𝐌 (nm)  13(3)  12(3)  10(2)  9(2) 11(2)  12(2)  11(2)  12(4)  11(3)  \n(𝐦𝐍𝐏𝐬\n𝐦𝐬𝐚𝐦𝐩𝐥𝐞)𝐓𝐆𝐀 0.81 0.84 0.83 0.86 0.96 0.97 0.96 0.96 0.95 𝐌𝐬 by VSM \n(emu/g)  66(1)  67(1)  64(1)  62(1)  76(1)  74(1)  68(1)  51(1)  50(1)  \nTable1:  Results of the 𝐶𝑢𝑥𝐹𝑒3−𝑥𝑂4 nanoparticles characterization: Cu incorporation degree \ndetermined by ICP -OES ( 𝐱𝐈𝐂𝐏), Fe2+/Fe ratio determined by XPS measurements and theoretically \nby taking in account the copper incorporation obtained by ICP -OES, size obtained by TEM ( 𝐝𝐓𝐄𝐌), \nmass percentage of nanoparticles in the samples as obtained by TGA (𝐦𝐍𝐏𝐬\n𝐦𝐬𝐚𝐦𝐩𝐥𝐞), and the  saturation \nmagnetization (M s) obtained from the M(H) curves at room temperature.  \n \nTo study the size and morphology of the nanoparticles we performed TEM measurements of all \nsamples, which are presented in Fig.2  along with the corresponding size distributi on histogram \n(measured as diameter due to the spherical -like shape of the particles)  and fitted to a lognormal \nfunction . Solvothermal samples exhibit a slight dependence of size with copper content, ranging \nfrom 13(3) nm for x=0 to 9(2) nm for x=0.3  and irregular shape  (see Table1 ). In contrast, no size \ndependence with copper content was observed in microwave -assisted samples , all with rather \nspherical shape and  sizes between 11 - 12 nm , probably due to the higher pressure synthesis \ncondition.  \n  \nFig.2: Representative TEM images of the MWX and STX CuxFe3−xO4 nanoparticles and their \ncorresponding size distribution  histogram fitted with lognormal distribution.  \n \nThe DC field dependence of the magnetization, measured at room temperature, shows a reversible \nbehavior for all the samples, in agreement with a superparamagnetic regime of the nanoparticles \nat these conditions. Figures  3a and 3b show  the MvsH curves for the solvothermal and \nmicrowave -assisted samples, respectively . The sample ´s magnetization was corrected by \nconsidering the proportion of the organic compound in the as -made nanoparticles, determined \nfrom TGA measurement, as reported in Table1. \n \n \nFig.3: (a)  M vs H cycle of the solvothermal nanoparticles. The inset shows the  fit using the \nLangevin model  for the ST1 sample . (b) M vs H cycle of the Microwave -assisted nanoparticles. \nThe inset shows the fit using the Langevin model for the MW1 s ample.  \n \nThe M(H) curve can be fitted with a Langevin model for all the samples, in agreement with a \nsuperparamagnetic behavior . From the fitting, the corresponding saturation magnetization ( Ms) \ncan be determined. Table1  presents the values  of Ms obtained  for bo th the solvothermal and \nmicrowave -assisted samples. In both systems it is observed a decreases of Ms when the copper \nconcentration  increases  in the ferrite , as expected when the Fe2+ (~4B) is replaced by an ion \nwith smaller magnetic moment such as Cu2+ (~1B) and in agreement with previous results .52,53 \nFrom this results  it is also notice d that the microwave -assisted nanoparticles  (MW)  have larger \nmagne tization than those prepared by solvothermal method  (ST). This is striking considering that \nthe crystalline structure of MW nanoparticles corresponds to the maghemite oxidized phase with \nlower Ms than magnetite, i.e.   Ms(γ-Fe2O3)=74 emu/g   and Ms(Fe3O4)=84 emu/g  for bulk materials .54 \nThe origin of this result may be due to the higher pressure attained in the microwave synthesis \nthat improve the crystallinity of the nanoparticles,  reducing  magnetic disorder , and also due t o the \noverestimation of the magnetite  mass  because of the presence of iron hydroxide  impurities  in \nsolvothermal samples . At this point we would like to highlight  that these  system s show promising \nproperties to test their potential as magnetic catalysts. On one hand, the nanoparticles ' \nsuperparamagnetic behavior at room temperature reduces the  agglomeration and facilitates the \nnanoparticles dispersion in the solution; and on the other hand, their relative large magnetization, \ni.e. Ms=50-70 emu/g, would enable the nanoparticles magnetic ally-assisted harvesting from \nsolution after catalytic reactions, both properties particularly relevant for the design of magnetic \nnanocatalyst s for environmental remediation applications.  \n \n3.2. Catalytic evaluation  In order to  evalu ate the Fenton -like activity  of these nano catalysts, we performed EPR \nmeasurements of the nanoparticles dispersed in acetate buffer solution at pH=5, containing H 2O2 \nand using DMPO as spin trap. These measurements allow the  identif ication  and quantification of  \nthe free radicals produced in the reaction. Figure 4a  shows a representative EPR spectrum along \nwith its corresponding fitting curve.  The fitting curve results to the superposition of the resonance \nlines of the different free radicals generated in the rea ction. According to the NIEHS Spin -Trap \ndatabase,55  from the spectrum features and the fitting parameters, the resonance of four different \nfree radicals can be clearly recognized: •OH, •OOH, •CH 3, and •N. According to the Fenton -like \nreaction, the •OH and •OOH radicals were produc ed by  Fe2+/ Cu+ and  Fe3+/ Cu2+, respectively; \nwhile the •CH 3 signal comes from a secondary reaction between •OH and the DMSO  used to \ndissolve DMPO .23 Also, a small nitrone radical  signal is observed. However, this last one  is also \nobserved in the solution withou t the nanoparticles, being not resulting from oxidation process \nduring the studied reactions, so it was not considered in the analysis.23,56 All the spectra also shows \nthe resonance  corresponding to the MgO : Mn2+ sample used as a pattern to normalize the EPR \nintensity .   \nFig.4:  (a) Representative measurement of the free radicals generated by the catalysts, along with  \nthe spectrum fitting and  the corresponding deconvoluted spectra . Kinetic curves of the free radicals  \ncatalyzed  by the  Cu-doped oxidi zed nanoparticles fabricated by microwave -assisted method  (b-\nc); and, (d-f) by the cu -doped magnetite nanoparticles fabricated by  solvothermal  method .  \nThe kinetic reaction was followed by acquiring the EPR spectra for each set o f samples  as a \nfunction of the time . The quantity of radicals is directly proportional to the area of the EPR \nabsorption curve , which can be determined by the double integral of the measured spectrum.56 \nThe kinetic  of radical generation by  the Cu-doped magnetite  samples , ST0, ST1, and ST3 is shown \nin Figs.4d -f. From these figures it is notice that the •OH and •CH 3 are the main species  produced , \nwith a systematic decrease  of their concentration for increas ing copper content. Assuming that \nCu2+ replaces Fe2+ within the spinel structure, this result suggests  that Fe2+ ions have the highest \ncatalytic activity  at room temperature . On the other hand, the Cu-doped  oxidized samples , i.e. the \nbatch  fabricated by mi crowave assisted method;  produce at least ten times fewer  radicals than the  \nCu-doped magnetite , as depicted  in Figs.4b -c. This result is in agreement with the lower Fe2+/Fe \nratio in this set of samples measured by XPS, as compared with the Cu -doped magneti te. \nConsistently, a lower ROS concentration is expected due to the lower rate constant of Fe3+ than \nthat of Fe2+ in the Fenton reaction. Notice that, besides the  •OH and •CH 3, microwave -assisted \nsamples also produce an appreciable amount of •OOH radicals, as shown in Figs.4b -c. Actually, \nin sample MW4 the •OOH intensity exceeds that of the hydroxyl radical after 1 h of reaction time. \nThis response is attributed to the predomin ance of Fe3+ oxidation state of iron .13,23 Overall, the \nEPR measurements provide valu able insights into the mechanisms of free radicals generation by \nthe copper ferrite nanoparticles, highlighting the importance of Fe2+ ions and the impact of copper \nincorporation on the Fenton catalytic activity of the nanoparticles at room temperature.  \nIn order t o evaluate the ability of the synthesized nanoparticles to degrade organic compounds , \nwe conducted colorimetric  experiments using a  MB cationic dye . In these experiments the \nnano catalyst s and the MB were dis persed in a pH=5 acetate buffer and,  to ensure adsorption of \nthe dye onto the nanoparticles surface , the system was kept under agitation for two hours before \nadding  H2O2. Furthermore, control experiments, or blanks tests, were conducted for each condition \nto quantify MB degradation by hydrogen p eroxide alone, in absence of cat alyst. Figures 5 a and \n5b show  the MB discoloration  curves measured at room temperature using the nanocatalyst \nfabricated by  solvothermal and microwave -assisted method s, respectively. Nanoparticles \nfabricated by solvothermal route showed the  great performance for degrading MB, since the \nmagnetite ST0 sample exhibited a percentage of discoloration up to 80% in the first 15 minutes \nand up to 90% in 2 h. For this set of samples  the activity decreases with the copper incorporation,  \nfor example the ST3 nanoparticles  only exhibited an efficiency of 24% in 2 h. The activity observed \nat room temperature for this family of samples  can be attributed to the Fe2+ active ions at the \nnanoparti cle surface and, accordingly , the efficiency decreased with the substitution of iron by \ncopper. The EPR measurement s support this result as the Fe2+ was identified as the principal \nresponsible of  the free radical production at room temperature . On the othe r hand, none of the Cu -doped oxidized nanoparticles, fabricated b y microwave -assisted method,  were efficient for \ndegrading MB at room temperature  (Fig. 5b ). Resul t in agreement with the low Fe2+/Fe ratio \nobtained by XPS for these  samples  and the low free radical production measured by  EPR \nexperiments .  \n \n \nFig.5 : Methylene blue degradation experiments by solvothermal samples  measured  (a) at room \ntemperature with an adsorption time of t=2h;  (c) at T=60°C and adsorption time of t=2h ; and (e) at \nT=60°C with an  adsorption time of t=24h. Methylene blue degradation experiments by m icrowave -\nassisted samples measured (b) at room temperatur e with an adsorption time of t=2h; (d) at T=60°C \nwith an adsorption time of t=2h;  and (f) at T=90°C with an adsorption time of t=2h.  Control \nexperiments, indicated by dashed lines in each graph, were performed to quantify MB degradation \nby hydrogen peroxide alone, without the catalysts.  \n \nIt is well known that the catalytic reaction rate increases with the increasin g temperature due to \nthe higher kinetic energy of the molecules. Besides, a s mentioned at the introduction, different \nresidual effluents are produced above room temperatures, as it is the case of textile and paper \nindustries  16–19 Therefore, it is interesting to take advantage of this additional kinetic energy pr esent \nin some waste water  to increase the performance of the nanocatalysts to decompose organic \ncontaminants.  To anal yze the temperature dependence o n the reactions  and the stability of the \nmaterials , we performed colorimetric experiments up to 9 0°C with a similar protocol to the one \nmentioned above.  \n \nFigure 5 c shows t he MB oxidation experiments at 60°C using solvothermal samples . Surprisingly, \nthe MB  degradation  is lower at 60°C than the obtained at room temperature for the ST0 sample.  \nThis result can be explained by considering the accelerated oxidation from Fe2+ to Fe3+ as a \nconsequence of the  temperature , which  decreases the kinetic rate for the ROS production . For the \nST3 sample a slight increase of activity was observed at higher temperature , attributed to the \ncombined effect of iron oxidation and kinetic promotion of Cu catalyzed degradation pathways, \nexpected to rapidly increase with increasing temperature . This assumption  was confirmed  by \nrepeating the discoloration curve after  incubati ng the nanoparticles for  24 h at 60°C, to comple te \nthe surface oxidation  (Fig.5e). In this case the efficiency of ST0 sample was even lower than the \npresented in Figs. 5a and 5c, consistent ly with the lower kinetic of Fe3+ to catalyze the H 2O2 \ndecomposition in ROS . On the other hand, the ST3 activity remained almost the same  for both of \nthe adsorption times tested , suggesting a better response  and stability of copper  doped \nnanoparticles  to high temperature condition s.  \nThis observation is clearly confirmed by the discoloration experiments using the Cu -doped \nmaghemite  nanoparticles  (samples fabricated by microwave route) above room temperature. Figure 5d  shows the experiments carried out at 60°C, where it is observed that almost all the \nsamp les showed catalytic activity to degrade MB and its efficiency increased with the copper \ncontent.  When the MB oxidation experiment  is running at 90°C  the response of the nanocatalysts \nimproves significantly , reaching up to 90% percentage of MB discoloratio n in the first 30 minutes \nand almost 100% in 2 h. This result shows that the copper ferrite exhibits superior thermal stability \nand catalytic performance under elevated temperature conditions, maintaining its catalytic activity \nwithout significant degradat ion or loss of performance. It is noteworthy that increasing the \ntemperature in reactions mediated by iron -based catalyst is not always beneficial, and it is crucial \nto know the chemical kinetic of the active ions involved in the ROS generation, for the de sign of \nproper nanocatalys for specific work conditions.   The key role of co pper ions in enhancing the \ncatalytic activity of oxidized nanoparticles can be elucidated through an examination of the \nelectronic properties of Cu -doped maghemite. Employing Densi ty Functional Theory (DFT) \nalongside  with the functional Perdew, Burke and Ernzerhof (PBE) 57, Pires et al. 58 calculated th e \nelectronic density map of -Fe2O3 and Cu/-Fe2O3  at the (311) surface plane. These computations \nrevealed a lower electronic density at Cu sites, these more positive region exhibit heightened \nsusceptibility to interact with hydrogen peroxide molecules, thereby facilitating the generation of \n•OH radicals. Despite the advancements achieved through computational methods, a \ncomprehensive investigation of the electron transfer process at interfaces is still lacking to fully \nunderstand the peroxidase decomposi tion mechanism at the nanocatalyst's surface.59  \n \nNanocatalyst stability is also an important property desired in materials for environmental \napplications. With the aim to evaluate the stability of the samples with time, the free radical \nproduction  was followed by EPR  after different storage conditions .  Figure S3 of the \nSupplementa ry Information shows the •OH radical s produced by Cu 0.1Fe2.9O4 (MW1) and \nCu0.4Fe2.6O4 (MW4) nanoparticles upon synthesis and after 4 mon ths of storage in air and water. \nThe •OH production in the as -made samples decreases with the copper content due to copper \nsubstituting the most active  Fe2+ ion. For powder samples stored in air condition, Fe2+ oxidized to Fe3+, decreasing MW1's activity. Conversely, MW4, already highly oxidized and copper -activity \nreliant, retains its •OH production after 4 months in air. This effect is amplified in water storage, \nwhere after four months, the trend reverses, and •OH production increases with copper content . \nThese results signal that the Fe2+-dependent catalysts would become less efficient upon reuse, \nwhile copper -dependent catalysts are expected to maintain their efficiency  for longer times .  \nOn the other hand, the stability of the catalysts before and after the MB degradation experiments, \nwere conducted by evaluating the changes in the structure and morphology of the nanoparticles. \nThe degradation was carried out under the most efficient conditions for each catalyst, i.e. room \ntemperature for magnetite (ST0) and high temperature (60°C) for Cu 0.4Fe2.6O4 (MW4 ). Figure S4 -\na and c, included in the Supplementa ry Information , illustrate the diffractograms comparison  before \nand after catalysis for samples ST0 and MW4, respectively. For ST0, the spinel phase \ncharacteristic peaks remain  unchanged. Also it is observed that the  peaks at 17° and 23° ascribed \nto polymerized ethylene glycol are no longer observed after reaction, likely due to the high solubility \nof the ethylene glycol in water. Additionally, the pea k at 24° , attributed to traces of iron hydroxide , \nvanishes  as expected , given its lower stability compared to the ferrite structure of the nanoparticles. \nNo structural changes are observed in the MW4 sample after the reaction , according to the X -ray \ndiffra ctograms. TEM micrographs of samples ST0 and MW4 after catalysis are presented in Fig. \nS4-b and d, respectively , along with their corresponding size distribution pre - and post -catalysis. \nThe size distributions for ST0 and MW4 before and after catalysis show no significant differences, \nsuggesting minimal or no leaching during the process.  \n \nThe Fenton -like reaction assumes that the free radicals produced in the nanoparticles' surface \nreact with the organic molecules adsorbed on t he surface of the nanocatal yst. Th is oxidation \nreaction can be described by the nth -order equation:  \n𝑑𝐶\n𝑑𝑡=−𝑘𝑛𝐶𝑛 \nwhere C represent s the organic molecu les concentration, kn is the reaction constant and n is the \norder of the reaction.  Usually , the Fenton -like oxidation follows a pseudo -second order dependence  𝐶0/𝐶=𝑘2𝐶0𝑡+1  , i.e. the kinetic depends on the pollutant or dye concentrations .60,61 \nOn the contrary, if the reaction is independent of the  substrates concentration,  the kinetics follow s \na pseudo -first order dynamics 𝑙𝑛(𝐶0\n𝐶)=𝑘1𝑡.8,62 Figures S5 and S6 in the Supplementary \nInformation include the analysis of the resul ts, adjusted with the pseudo -first order and pseudo -\nsecond order models, respectively; and the corresponding fitting parameters are reported in Table \n2.  At room temperature, the system with higher pseudo -first order kinetic rate is the magnetite  \n(ST0) , reaching k1=0.07(2) min-1 value, while in the doped samples the activity is negligible. \nInstead, the constant rate of the MB degradation by over oxidized MWX samples at 90 °C, \nsystematically increases with the Cu concentration up to k1= 0.10(2) min-1 for the sample MW4 \n(Cu0.4Fe2.6O4). The obtained k1 values confirm the good response of the materials to degrade MB \ncompared to the previous reports of heterogeneous Fenton and Fenton -like catalysts which are \nreported in the k1 =0.0053 min-1- 0.1455 min-1 range. 63–68 In the case of the adjustment with a \nsecond order equation, similar trends are obtained, with k2=0.0020(3) min-1mg-1L for the magnetite \nat room temperature, and k2=0.0039(4) min-1mg-1L for the sample Cu0.4Fe2.6O4 (MW4) m easured \nat 90  °C.  \nThe regression coefficients  obtained from the fitting, R2, are in the range R2=0.82-0.991 for the \npseudo -first order reaction, and R2=0.95 -0.99997 for the  pseudo -second order reaction model,  \nwhich signal t hat the kinetic is better described with the pseudo -second order reaction equation . \nThis result suggests that the amount of MB molecules adsorbed on the catalyst depends on its \ninitial  concentration and determines the kinetic of the reaction.9 \n \nSample  ST0 ST3 MW1 MW2 MW4 \nTemp.  RT 60°C RT 60°C 60°C 90°C 60°C 90°C 60°C 90°C \nPFO k1 \n(min-1) 0.07(2)  0.010(1)  0.009(1)  0.0032(2)  0.0028(7)  0.032(4)  0.0044(5)  0.065(9)  0.009(1)  0.10(2)  \nR2 0.85 0.97 0.94 0.991 0.82 0.96 0.97 0.94 0.96 0.91 PSO k2 \n(min-1 \nmg-1L) 0.0020(3)  0.00011(1)  0.00017(3)  0.000034(2)  0.000029(8)  0.000047(4)  0.00010(1)  0.00049(2)  0.00159(7)  0.0039(4)  \nR2 0.95 0.9994  0.9990  0.99997  0.9995  0.998 0.9998  0.996 0.9993  0.97 \n \nTable 2. Reaction constants for methylene blue degradation: Data fitted using Pseudo -First-Order \n(PFO) and Pseudo -Second -Order (PSO) models. The table also includes the corresponding \nregression coefficients for each fit.  \n \n4. Conclusions  \nWe have evaluated the catalytic efficiency of Cu doped  magnetite and maghemit e nanoparticles \nwith average size of  11 nm, obtained from solvothermal and microwave -assisted methods, \nrespectively. We found t hat the magnetite nanoparticles present an excellent performance for \noxidizing the organic dye methylene blue at room temperature, producing up to a 90% of \ndiscoloration  in 2 h. The catalytic activity was attributed to Fe2+ centers  that generate •OH in the  \nheterogeneous Fenton reaction , which are the main specie s responsible for the MB oxidation.  This \ndegradation efficiency decreased notably with increasing Cu content. Consistently, th e •OH  \nspecie s is systematic ally reduced when the  Cu concentration  increases . However, a bove room \ntemperature, the magnetite loss es its  efficiency as nanocatalyst due to the surface oxidation and \nbecause of the low rate constant of the Fe3+/Fe2+ redox cycle  in the Fenton reaction. In this \ncondition the role of copper bec ame relevant, with the samples with the highest percentage of \ncopper being those that presented a better efficiency as catalysts. In fact, at 90°C the CuₓFe ₃₋ₓO₄ \nwith x=0.4 produce a 90% of MB discoloration in the first 30 minutes and a complete discolorat ion \nin less than 2 h.  \nThese results show that the nanoparticle composition and the oxidation state of the surface active \nions, determine the nanoparticle reactivity and the nature of the free radicals produced, providing a tool to engineering more efficien t nanocatalyst s. In particular,  we have demonstrate d that while \nmagnetite nanocatalyst is effective to degrade dyes at room temperature, it is also highly unstable \ncompound, easily prone to oxidation under normal conditions, and particularly at high \ntemperatures. Instead, copper ferrite has an advantage over magnetite in terms of stability being \na superior alternative for catalytic applications  at higher temper ature. Given that many industrial \nresidual effluents are produced above  room temperatures, as observed in textile and paper \nindustries, leveraging this additional kinetic energy becomes interesting for enhancing the \nperformance of nanocatalysts in decomposing organic contaminants. In summary, our research \ncontributes valuable  insights that could pave the way for the development of more efficient \nnanocatalysts, with practical applications in addressing environmental challenges associated with \nindustrial effluents.  \n \n5. Declaration  of interest  \nThe authors declare that they have no known competing financial interests or personal \nrelationships that could have appeared to influence the work reported in this paper.  \n6. CRediT authorship contribution statement  \nNahuel Nuñez: Writing – Original Draft , Conceptualization, Investigation, Visualization. Enio  Lima \nJr.: Investigation. Marcelo Vásquez Mansilla:  Investigation . Gerardo F. Goya:  Funding \nacquisition , Project administration.  Álvaro Gallo -Cordova:  Investigation. Maria del Puerto \nMorales:  Supervisi on, Resources. Elin L. Winkler:  Writing – Review & Editing,  Conceptualization, \nSupervision, Resources. The manuscript was written through contributions from all authors. All \nauthors have given approval to the final version of the manuscript.  \n7. Acknowledgemen t The authors thank  to the Argentine government agency ANPCyT for providing financial support for \nthis work through Grants No. PICT -2019 -02059, as well as to UNCuyo for their support through \nGrant No. 06/C029 -T1. Additionally, this research received partia l funding from Project PDC2021 -\n12109 -I00 (MICRODIAL) MCIN/AEI/10.13039/501100011033 through the European Union \n“NextGenerationEU/PRTR” . Furthermore, the authors acknowledge the support of Project H2020 -\nMSCA -RISE -2020 (NESTOR) PROJECT Nº 101007629, funded by the EU -commission.  \n \n8. 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Degradation of Methylene Blue in the \nPhoto -Fenton -Like Process with WO3 -Loaded Porous Carbon N itride Nanosheet Catalyst. \nWater  2022 , 14 (16), 2569. https://doi.org/10.3390/w14162569.  \n    \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n Supplementary Information  \nEffect of the temperature and copper doping on the heterogeneous Fenton \nactivity of 𝑪𝒖 𝒙𝑭𝒆 𝟑−𝒙𝑶𝟒 nanoparticles  \nNahuel Nuñez1,2,3*, Enio Lima Jr.1,2, Marcelo Vásquez Mansilla1,2, Gerardo F. Goya4,5,  \nÁlvaro Gallo -Cordova6, María del Puerto Morales6, Elin L. Winkler1,2,3*  \n \n1 Resonancias Magnéticas, Gerencia de Física, Centro Atómico Bariloche, Av. Bustillo 9500, (8400) S. C. \nde Bariloche (RN), Argentina.  \n2 Instituto de Nanociencia y Nanotecnología (CNEA -CONICET), Nodo Barilo che, Av. Bustillo 9500, (8400) \nS. C. de Bariloche (RN), Argentina.  \n3 Instituto Balseiro, CNEA -UNCuyo, Av. Bustillo 9500, (8400) S. C. de Bariloche (RN), Argentina  \n4 Dept. Física de la Materia Condensada, Universidad de Zaragoza, C/ Pedro Cerbuna 12, 50009,  Zaragoza, \nSpain  \n5 Instituto de Nanociencia y Materiales de Aragón, CSIC -Universidad de Zaragoza, C/ Mariano Esquillos \nS/N, 50018, Zaragoza, Spain  \n6 Instituto de Ciencia de Materiales de Madrid, ICMM/CSIC, C/ Sor Juana Inés de la Cruz 3, 28049, Madrid, \nSpain  \n *Corresponding authors : nahuel.nunez@ib.edu.ar  \n \n \n \n \n XPS characterization.  We p erformed XPS measurements of the Fe 2 P3/2 absorption of  \nsolvothermal and microwave -assisted samples , which are presented in Fig.S1  and Fig.S2  \nrespectively. The spectra obtained were fitted with two multiplets related to Fe2+ and Fe3+ \nrespectively, as done by Grosvenor et al.45, and the fitting parameters are presented in TableS1 .  \n \nFig.S1: XPS spectra and fit of the Fe 2 P3/2 absorption in the solvothermal samples. Peaks p1, p2 \nand p3 are from Fe2+ ion; peaks p4, p5, p6 and p7 are from Fe3+ ion. The peaks position for each \nof the 3 peaks related to the 𝐅𝐞𝟐+ and 4 peaks related to the 𝐅𝐞𝟑+ were extr acted from reference \n45 \n \n \nFig.S2: XPS spectra and fit of the Fe 2 P3/2 absorption in the microwave -assisted samples. Peaks \np1, p2 and p3 are from Fe2+ ion; pea ks p4, p5, p6 and p7 are from Fe3+ ion. The peaks position \nfor each of the 3 peaks related to the 𝐅𝐞𝟐+ and 4 peaks related to the 𝐅𝐞𝟑+ were extracted from \nreference . 45 \n \nIdentificat\nion 𝐅𝐞𝟐+ 𝐅𝐞𝟑+ Satellite  \n peak  p1 p2 p3 p4 p5 p6 p7 LE HE \nBE (eV)  708.50\n(2) 709.50\n(3) 710.40\n(1) 710.30\n(2) 711.30\n(2) 712.60\n(3) 714.00\n(4) 707.1\n(2) 718.9\n(5) \nSample  Area  𝐅𝐞𝟐+/\nFe \nratio  \nST0 583 600 209 984 895 636 305 662 3579  0.33 \nST1 125 129 45 218 199 141 67 0 2066  0.32 \nST3 406 419 146 954 868 617 295 0 1997  0.26 \nMW0  106 110 38 689 627 446 213 0 2555  0.11 \nMW1  334 344 120 1435  1306  928 444 0 5040  0.16 \nMW2  270 278 97 996 906 644 308 595 3754  0.18 \nMW3  85 87 30 540 491 349 167 0 2359  0.12 \nMW4  78 81 28 351 319 227 108 0 1350  0.16 Table S1:  Parameters used for the fitting of the XPS spectra. The peaks position for each of the 3 \npeaks related to the 𝐅𝐞𝟐+ and 4 peaks related to the 𝐅𝐞𝟑+ were extracted from reference 45. The \nrelationship between the area of the peaks of each multiplet  was fixed. Peaks LE and HE \ncorresponds to low and high energy satellites.  \n \n \nStability analysis.  We investigated the aging stability of selected catalysts under various \nconditions, as well as the structural changes in the materials after the catalytic reac tion. \nSpecifically, using EPR, we quantified the •OH free radicals production in the as made MW1 and \nMW4 samples, and again after four months of storage in air and water. These results are detailed \nin Fig.S3 . Additionally, we examined the structural altera tions of ST0 and MW4 samples post -\ncatalysis. TEM images of samples ST0 and MW4 after 6 hours of continuous catalysis are \npresented in Figs.S4 -b and -d, respectively. Correspondingly, Figs.S4 -a and -c show  a \ncomparative analysis of the X -ray diffraction pat terns before and after the catalytic reaction.  \n \nFig. S3:  Normalized •OH production, determined by EPR, of samples MW1 and MW4 in different \nconditions: as made (black) and after 4 months stored in air (red) and water (blue).  \n \nFig. S4:  (a) XRD patterns of ST0 before and after 6 hours of continuous catalysis. (b) TEM of ST0 \npost-catalysis with comparative size distributions before and after catalysis. (c)  XRD patterns of \nMW4  before and after 6 hours of continuous catalysis. (d) TEM of MW4  post-catalysis with \ncomparative size distributions before and after catalysis.  \nKinetic modelling. From the methylene blue degradation curves depicted in Fig.5 , different \nreaction kinetics models were explored. Specifically, the experimental data were fitted  to pseudo -\nfirst-order ( ln (𝐶0\n𝐶)=𝑘1𝑡) and pseudo -second -order (𝐶0\n𝐶=𝑘2𝐶0𝑡+1) reaction models. Fig.S5  \nillustrates the linearized pseudo -first-order fits for the various catalysts used under different \nexperimental conditions. The linearized pseudo -second -order fits are shown in Fig.S6. Table  2 \npresents the 𝑘1 and 𝑘2 constants derived from each model, along with their corresponding \nregression coefficients  𝑅2. \n \nFig.S5: Pseudo -First-Order kinetics for methylene blue degradation at different experimental \nconditions: (a) ST samples at room temperature (RT), (b) MW samples at 60°C, (c) ST samples \nat 60°C, and (d) MW samples at 90°C.  \n \nFig.S6: Pseudo -Second -Order kinetics for  methylene blue degradation at different experimental \nconditions: (a) ST samples at room temperature (RT), (b) MW samples at 60°C, (c) ST samples \nat 60°C, and (d) MW samples at 90°C.  \n"    },    {        "title": "2402.15344v1.Iteration_and_Stochastic_First_order_Oracle_Complexities_of_Stochastic_Gradient_Descent_using_Constant_and_Decaying_Learning_Rates.pdf",        "content": "Iteration and Stochastic First-order Oracle Complexities of\nStochastic Gradient Descent using Constant and Decaying Learning\nRates\nKento Imaizumi*1and Hideaki Iiduka*1*Equal contribution;1Department of Computer\nScience, Meiji University, Japan\nARTICLE HISTORY\nCompiled February 26, 2024\nABSTRACT\nThe performance of stochastic gradient descent (SGD), which is the simplest first-\norder optimizer for training deep neural networks, depends on not only the learning\nrate but also the batch size. They both affect the number of iterations and the\nstochastic first-order oracle (SFO) complexity needed for training. In particular, the\nprevious numerical results indicated that, for SGD using a constant learning rate,\nthe number of iterations needed for training decreases when the batch size increases,\nand the SFO complexity needed for training is minimized at a critical batch size and\nthat it increases once the batch size exceeds that size. Here, we study the relationship\nbetween batch size and the iteration and SFO complexities needed for nonconvex\noptimization in deep learning with SGD using constant or decaying learning rates\nand show that SGD using the critical batch size minimizes the SFO complexity. We\nalso provide numerical comparisons of SGD with the existing first-order optimizers\nand show the usefulness of SGD using a critical batch size. Moreover, we show that\nmeasured critical batch sizes are close to the sizes estimated from our theoretical\nresults.\nKEYWORDS\nSGD, batch size, iteration complexity, SFO complexity, nonconvex optimization\n1. Introduction\n1.1. Background\nFirst-order optimizers can train deep neural networks by minimizing loss functions\ncalled the expected and empirical risks. They use stochastic first-order derivatives\n(stochastic gradients), which are estimated from the full gradient of the loss function.\nThe simplest first-order optimizer is stochastic gradient descent (SGD) [1–5], which\nhas a number of variants, including momentum variants[6,7] and numerous adaptive\nvariants, such as adaptive gradient (AdaGrad) [8], root mean square propagation (RM-\nSProp) [9], adaptive moment estimation (Adam) [10], adaptive mean square gradient\n(AMSGrad) [11], and Adam with decoupled weight decay (AdamW) [12].\nSGD can be applied to nonconvex optimization [13–22], where its performance\nstrongly depends on the learning rate αk. For example, under the bounded variance as-\nsumption, SGD using a constant learning rate αk=αsatisfies1\nKPK−1\nk=0∥∇f(θk)∥2=\nCONTACT Kento Imaizumi. Email: ee207005@meiji.ac.jp, Hideaki Iiduka. Emal: iiduka@cs.meiji.ac.jparXiv:2402.15344v1  [stat.ML]  23 Feb 2024O(1\nK) +σ2[18, Theorem 12] and SGD using a decaying learning rate (i.e., αk→0)\nsatisfies that1\nKPK−1\nk=0E[∥∇f(θk)∥2] =O(1√\nK) [18, Theorem 11], where ( θk)k∈Nis\nthe sequence generated by SGD to find a local minimizer of f,Kis the number of\niterations, and σ2is the upper bound of the variance.\nThe performance of SGD also depends on the batch size b. The convergence analyses\nreported in [14,17,21,23,24] indicated that SGD with a decaying learning rate and large\nbatch size converges to a local minimizer of the loss function. In [25], it was numerically\nshown that using an enormous batch reduces both the number of parameter updates\nand model training time. Moreover, setting appropriate batch sizes for optimizers used\nin training generative adversarial networks were investigated in [26].\n1.2. Motivation\nThe previous numerical results in [27] indicated that, for SGD using constant or lin-\nearly decaying learning rates, the number of iterations Kneeded to train a deep neural\nnetwork decreases as the batch size bincreases. Motivated by the numerical results in\n[27], we decided to clarify the theoretical iteration complexity of SGD with a constant\nor decaying learning rate in training a deep neural network. We used the performance\nmeasure of previous theoretical analyses of SGD, i.e., min k∈[0:K−1]E[∥∇f(θk)∥]≤ϵ,\nwhere ϵ(>0) is the precision and [0 : K−1] :={0,1, . . . , K −1}. We found that,\nif SGD is an ϵ–approximation, i.e., min k∈[0:K−1]E[∥∇f(θk)∥]≤ϵ, then it can train a\ndeep neural network in Kiterations.\nIn addition, the numerical results in [27] indicated an interesting fact wherein dimin-\nishing returns exist beyond a critical batch size; i.e., the number of iterations needed\nto train a deep neural network does not strictly decrease beyond the critical batch\nsize. Here, we define the stochastic first-order oracle (SFO) complexity asN:=Kb,\nwhere Kis the number of iterations needed to train a deep neural network and bis\nthe batch size, as stated above. The deep neural network model uses bgradients of the\nloss functions per iteration. The model has a stochastic gradient computation cost of\nN=Kb. From the numerical results in [27, Figures 4 and 5], we can conclude that the\ncritical batch size b⋆(if it exists) is useful for SGD, since the SFO complexity N(b)\nis minimized at b=b⋆and the SFO complexity increases once the batch size exceeds\nb⋆. Hence, on the basis of the first motivation stated above, we decided to clarify the\nSFO complexities needed for SGD using a constant or decaying learning rate to be an\nϵ–approximation.\n1.3. Contribution\n1.3.1. Upper bound of theoretical performance measure\nTo clarify the iteration and SFO complexities needed for SGD to be an ϵ–\napproximation, we first give upper bounds of min k∈[0:K−1]E[∥∇f(θk)∥2] for SGD\nto generate a sequence ( θk)k∈Nwith constant or decaying learning rates (see The-\norem 3.1 for the definitions of CiandDi). As our aim is to show that SGD is an ϵ–\napproximation min k∈[0:K−1]E[∥∇f(θk)∥2]≤ϵ2, it is desirable that the upper bounds\nof min k∈[0:K−1]E[∥∇f(θk)∥2] be small. Table 1 indicates that the upper bounds be-\ncome small when the number of iterations and batch size are large. The table also\nindicates that the convergence of SGD strongly depends on the batch size, since the\nvariance terms (including σ2andb; see Theorem 3.1 for the definitions of C2andD2)\n2in the upper bounds of min k∈[0:K−1]E[∥∇f(θk)∥2] decrease as the batch size becomes\nlarger.\nTable 1. Upper bounds of min k∈[0:K−1]E[∥∇f(θk)∥2] for SGD using a constant or decaying learning rate\nand the critical batch size to minimize the SFO complexities and achieve min k∈[0:K−1]E[∥∇f(θk)∥]≤ϵ(Ci\nandDiare positive constants, Kis the number of iterations, bis the batch size, T≥1,ϵ >0, and Lis the\nLipschitz constant of ∇f)\nLearning Rate Upper Bound Critical Batch Size\nConstant α∈(0,2\nL)C1\nK+C2\nb2C2\nϵ2\na∈(0,1\n2)D1\nKa+D2\n(1−2a)Kab(1−a)D2\na(1−2a)D1\nDecay a=1\n2D1√\nK+\u00121√\nK+ 1\u0013D2\nb≈D2\nϵ2\nαk=1\n(⌊k\nT⌋+1)aa∈(1\n2,1)D1\nK1−a+2aD2\n(2a−1)K1−ab2a2D2\n(1−a)(2a−1)D1\n1.3.2. Critical batch size to reduce SFO complexity\nSection 1.3.1 showed that using large batches is appropriate for SGD in the sense of\nminimizing the upper bound of the performance measure. Here, we are interested in\nfinding appropriate batch sizes from the viewpoint of the computation cost. This is\nbecause the SFO complexity increases with the batch size. As indicated in Section\n1.2, the critical batch size b⋆minimizes the SFO complexity, N=Kb. Hence, we\nwill investigate the properties of the SFO complexity N=Kbneeded to achieve an\nϵ–approximation. Here, let us consider SGD using a constant learning rate. From the\n“Upper Bound” row in Table 1, we have\nmin\nk∈[0:K−1]E[∥∇f(θk)∥2]≤C1\nK+C2\nb≤ϵ2\n|{z }\n⇔K≥K(b):=C1b\nϵ2b−C2(b>C2\nϵ2).\nWe can check that the number of iterations, K(b) :=C1b\nϵ2b−C2, needed to achieve an\nϵ–approximation is monotone decreasing and convex with respect to the batch size\n(Theorem 3.2). Accordingly, we have that K(b)≥inf{K: min k∈[0:K−1]E[∥∇f(θk)∥]≤\nϵ}, where SGD using the batch size bgenerates ( θk)K−1\nk=0. Moreover, we find that the\nSFO complexity is N(b) =K(b)b=C1b2\nϵ2b−C2. The convexity of N(b) =C1b2\nϵ2b−C2(Theorem\n3.3) ensures that a critical batch size b⋆=2C2\nϵ2whereby N′(b⋆) = 0 exists such that\nN(b) is minimized at b⋆(see the “Critical Batch Size” row in Table 1). A similar\ndiscussion guarantees the existence of a critical batch size for SGD using a decaying\nlearning rate αk=1\n(⌊k\nT⌋+1)a, where T≥1,a∈(0,1\n2)∪(1\n2,1), and ⌊·⌋is the floor\nfunction (see the “Critical Batch Size” row in Table 1).\nMeanwhile, for a decaying learning rate αk=1q\n⌊k\nT⌋+1, although N(b) is convex\nwith respect to b, we have that N′(b)>0 for all b >D2\nϵ2(Theorem 3.3(iii)). Hence, for\n3this case, a critical batch size b⋆defined by N′(b⋆) = 0 does not exist. However, since\nthe critical batch size minimizes the SFO complexity N, we can define one as follows:\nb⋆≈D2\nϵ2. Accordingly, we have that N(b⋆)≥inf{Kb: min k∈[0:K−1]E[∥∇f(θk)∥]≤ϵ},\nwhere SGD using b⋆generates ( θk)K−1\nk=0.\n1.3.3. Iteration and SFO complexities\nLetF(n,∆0, L) be an L–smooth function class with f:=1\nnPn\ni=1fiandf(θ0)−f⋆≤\n∆0(see (C1)) and let O(b, σ2) be a stochastic first-order oracle class (see (C2) and\n(C3)). The iteration complexity Kϵ[21, (7)] and SFO complexity Nϵneeded for SGD\nto be an ϵ–approximation are defined as\nKϵ(n, b, α k,∆0, L, σ2) := sup\nO∈O(b,σ2)sup\nf∈F(n,∆0,L)inf\u001a\nK: min\nk∈[0:K−1]E[∥∇f(θk)∥]≤ϵ\u001b\n,\n(1)\nNϵ(n, b, α k,∆0, L, σ2) :=Kϵ(n, b, α k,∆0, L, σ2)b. (2)\nTable 2 summarizes the iteration and SFO complexities (see also Theorem 3.4). Corol-\nlaries 6 and 7 in [14] are the same as our results for SGD with a constant learning\nrate in Theorems 3.1 and 3.3, since the randomized stochastic projected gradient free\nalgorithm in [14] which is a stochastic zeroth-order (SZO) method that coincides with\nSGD and it can be applied to the situations where only noisy function values are avail-\nable. In particular, Corollary 6 in [14] gave the convergence rate of the SZO methods\nusing a fixed batch size, and Corollary 7 indicated the SZO complexity of the SZO\nmethod is the same as the SFO complexity. Hence, Corollaries 6 and 7 in [14] lead\nto the finding that the iteration complexity of SGD using a constant learning rate is\nO(1/ϵ2) and the SFO complexity of SGD using a constant learning rate is O(1/ϵ4).\nSince the positive constants CiandDidepend on the learning rate, we need to\ncompare numerically the performance of SGD with a constant learning rate with that\nof SGD with a decaying learning rate. Moreover, we also need to compare SGD with\nthe existing first-order optimizers in order to verify its usefulness. Section 4 presents\nnumerical comparisons showing that SGD using the critical batch size outperforms\nthe existing first-order optimizers. We also show that the measured critical batch sizes\nare close to the theoretical sizes.\n2. Nonconvex Optimization and SGD\n2.1. Nonconvex optimization in deep learning\nLetRdbe ad-dimensional Euclidean space with inner product ⟨x,y⟩:=x⊤yinducing\nthe norm ∥x∥andNbe the set of nonnegative integers. Define [0 : n] :={0,1, . . . , n }\nforn≥1. Let ( xk)k∈Nand ( yk)k∈Nbe positive real sequences and let x(ϵ), y(ϵ)>0,\nwhere ϵ >0.Odenotes Landau’s symbol; i.e., yk=O(xk) if there exist c >0 and\nk0∈Nsuch that yk≤cxkfor all k≥k0, and y(ϵ) =O(x(ϵ)) if there exists c >0 such\nthat y(ϵ)≤cx(ϵ). Given a parameter θ∈Rdand a data point zin a data domain\nZ, a machine-learning model provides a prediction whose quality can be measured\nin terms of a differentiable nonconvex loss function ℓ(θ;z). We aim to minimize the\nempirical loss defined for all θ∈Rdbyf(θ) =1\nnPn\ni=1ℓ(θ;zi) =1\nnPn\ni=1fi(θ), where\nS= (z1, z2, . . . , z n) denotes the training set (We assume that the number of training\n4Table 2. Iteration and SFO complexities needed for SGD using a constant or decaying learning rate to be an\nϵ–approximation (The critical batch sizes are used to compute KϵandNϵ)\nLearning Rate Iteration Complexity KϵSFO Complexity Nϵ(n, b, α k,∆0, L, σ2)\nConstant α∈(0,2\nL) O\u00121\nϵ2\u0013\n= sup\nf,OK(b⋆) O\u00121\nϵ4\u0013\n= sup\nf,O4C1C2\nϵ4\na∈(0,1\n2)O\u00121\nϵ2\na\u0013\n= sup\nf,OK(b⋆) O\u00121\nϵ2\na\u0013\n= sup\nf,O(1−a)1−1\naD2\na(1−2a)D1−1\na\n1ϵ2\na\nDecay a=1\n2O\u00121\nϵ4\u0013\n= sup\nf,OK(b⋆) O\u00121\nϵ6\u0013\n= sup\nf,O\u0012D1(D2+ 1)\nϵ2+D2\u00132D1+ 1\nϵ2\nαk=1\n(⌊k\nT⌋+1)aa∈(1\n2,1)O\u00121\nϵ2\n1−a\u0013\n= sup\nf,OK(b⋆)O\u00121\nϵ2\n1−a\u0013\n= sup\nf,O2a2−1\n1−a(1−a)−1D2\n(2a−1)D1−1\n1−a\n1 ϵ2\n1−a\ndata nis large) and fi(·) :=ℓ(·;zi) denotes the loss function corresponding to the i-th\ntraining data zi.\n2.2. SGD\n2.2.1. Conditions and algorithm\nWe assume that a stochastic first-order oracle (SFO) exists such that, for a given\nθ∈Rd, it returns a stochastic gradient Gξ(θ) of the function f, where a random\nvariable ξis independent of θ. LetEξ[·] be the expectation taken with respect to ξ.\nThe following are standard conditions.\n(C1) f:=1\nnPn\ni=1fi:Rd→RisL–smooth, i.e., ∇f:Rd→RdisL–Lipschitz contin-\nuous (i.e., ∥∇f(x)−∇f(y)∥ ≤L∥x−y∥).fis bounded below from f⋆∈R. Let\n∆0>0 satisfy f(θ0)−f⋆≤∆0, where θ0is an initial point.\n(C2) Let ( θk)k∈N⊂Rdbe the sequence generated by SGD. For each iteration k,\nEξk[Gξk(θk)] =∇f(θk), where ξ0, ξ1, . . .are independent samples and the random\nvariable ξkis independent of ( θl)k\nl=0. There exists a nonnegative constant σ2such\nthatEξk[∥Gξk(θk)− ∇f(θk)∥2]≤σ2.\n(C3) For each iteration k, SGD samples a batch Bkof size bindependently of kand\nestimates the full gradient ∇fas∇fBk(θk) :=1\nbP\ni∈[b]Gξk,i(θk), where b≤n\nandξk,iis a random variable generated by the i-th sampling in the k-th iteration.\nAlgorithm 1 is the SGD optimizer under (C1)–(C3).\n2.2.2. Learning rates\nWe use the following learning rates:\n(Constant rate) αkdoes not depend on k∈N, i.e., αk=α <2\nL(k∈N), where the\nupper bound2\nLofαis needed to analyze SGD (see Appendix A.2).\n(Decaying rate) (αk)k∈N⊂(0,+∞) is monotone decreasing for k(i.e., αk≥\nαk+1) and converges to 0. In particular, we will use αk=1\n(⌊k\nT⌋+1)a, where\n5Algorithm 1 SGD\nRequire: αk∈(0,+∞) (learning rate), b≥1 (batch size), K≥1 (iteration)\nEnsure: θK\n1:k←0,θ0∈Rd\n2:loop\n3:∇fBk(θk) :=1\nbP\ni∈[b]Gξk,i(θk)\n4:θk+1:=θk−αk∇fBk(θk)\n5:k←k+ 1\n6:end loop\n(Decay 1) a∈(0,1\n2)∨(Decay 2) a=1\n2∨(Decay 3) a∈(1\n2,1). It is guar-\nanteed that there exists k0∈Nsuch that, for all k≥k0,αk<2\nL. Furthermore,\nwe assume that k0= 0, since we can replace αkwithα\n(⌊k\nT⌋+1)a≤α <2\nL(k∈N),\nwhere α∈(0,2\nL) is defined as in (Constant) .\n3. Our Results\n3.1. Upper bound of the squared norm of the full gradient\nHere, we give an upper bound of min k∈[0:K−1]E[∥∇f(θk)∥2], where E[·] stands for the\ntotal expectation, for the sequence generated by SGD using each of the learning rates\ndefined in Section 2.2.2.\nTheorem 3.1 (Upper bound of the squared norm of the full gradient) .The sequence\n(θk)k∈Ngenerated by Algorithm 1 under (C1)–(C3) satisfies that, for all K≥1,\nmin\nk∈[0:K−1]E\u0002\n∥∇f(θk)∥2\u0003\n≤\n\nC1\nK+C2\nb(Constant)\nD1\nKa+D2\n(1−2a)Kab(Decay 1)\nD1√\nK+\u00121√\nK+ 1\u0013D2\nb(Decay 2)\nD1\nK1−a+2aD2\n(2a−1)K1−ab(Decay 3)\nwhere\nC1:=2(f(θ0)−f⋆)\n(2−Lα)α, C2:=Lσ2α\n2−Lα,\nD1:=2 (f(θ0)−f⋆)\nα(2−Lα), D2:=Tα2Lσ2\n2−Lα.\nTheorem 3.1 indicates that the upper bound of min k∈[0:K−1]E[∥∇f(θk)∥2] consists\nof a bias term including f(θ0)−f⋆and a variance term including σ2and that these\nterms become small when the number of iterations and the batch size are large. In\nparticular, the bias term using (Constant) isO(1\nK), which is a better rate than using\n(Decay 1) –(Decay 3) .\n63.2. Number of iterations needed for SGD to be an ϵ–approximation\nLet us suppose that SGD is an ϵ–approximation defined as follows:\nE\u0002\n∥∇f(θK∗)∥2\u0003\n:= min\nk∈[0:K−1]E\u0002\n∥∇f(θk)∥2\u0003\n≤ϵ2, (3)\nwhere ϵ > 0 is the precision and K∗∈[0 : K−1]. Condition (3) implies that\nE[∥∇f(θK∗)∥]≤ϵ. Theorem 3.1 below gives the number of iterations needed to be an\nϵ–approximation (3).\nTheorem 3.2 (Numbers of iterations needed for nonconvex optimization using SGD) .\nLet(θk)k∈Nbe the sequence generated by Algorithm 1 under (C1)–(C3) and let K:R→\nRbe\nK(b) =\n\nC1b\nϵ2b−C2(Constant)\n\u001a1\nϵ2\u0012D2\n(1−2a)b+D1\u0013\u001b 1\na\n(Decay 1)\n\u0012D1b+D2\nϵ2b−D2\u00132\n(Decay 2)\n\u001a1\nϵ2\u00122aD2\n(2a−1)b+D1\u0013\u001b 1\n1−a\n(Decay 3)\nwhere C1,C2,D1(> ϵ2), and D2are defined as in Theorem 3.1, the domain of Kin\n(Constant) isb >C2\nϵ2, and the domain of Kin(Decay 2) isb >D2\nϵ2. Then, we have\nthe following:\n(i)The above Kachieves an ϵ–approximation (3).\n(ii)The above Kis a monotone decreasing and convex function with respect to the\nbatch size b.\nTheorem 3.2 indicates that the number of iterations needed for SGD using constant\nor decay learning rates to be an ϵ–approximation is small when the batch size is large.\nHence, it is appropriate to set a large batch size in order to minimize the iterations\nneeded for an ϵ–approximation (3). However, the SFO complexity, which is the cost\nof the stochastic gradient computation, grows larger with b. Hence, the appropriate\nbatch size should also minimize the SFO complexity.\n3.3. SFO complexity needed for SGD to be an ϵ–approximation\nTheorem 3.2 leads to the following theorem on the properties of the SFO complexity\nNneeded for SGD to be an ϵ–approximation (3).\nTheorem 3.3 (SFO complexity needed for nonconvex optimization of SGD) .Let\n(θk)k∈Nbe the sequence generated by Algorithm 1 under (C1)–(C3) and define N:R→\n7Rby\nN(b) =K(b)b=\n\nC1b2\nϵ2b−C2(Constant)\n\u001a1\nϵ2\u0012D2\n(1−2a)b+D1\u0013\u001b 1\na\nb (Decay 1)\n\u0012D1b+D2\nϵ2b−D2\u00132\nb (Decay 2)\n\u001a1\nϵ2\u00122aD2\n(2a−1)b+D1\u0013\u001b 1\n1−a\nb(Decay 3)\nwhere C1,C2,D1andD2are as in Theorem 3.1, the domain of Nin(Constant) is\nb >C2\nϵ2, and the domain of Nin(Decay 2) isb >D2\nϵ2. Then, we have the following:\n(i)The above Nis convex with respect to the batch size b.\n(ii)There exists a critical batch size\nb⋆=\n\n2C2\nϵ2(Constant)\n(1−a)D2\na(1−2a)D1(Decay 1)\n2a2D2\n(1−a)(2a−1)D1(Decay 3)(4)\nsatisfying N′(b⋆) = 0 such that b⋆minimizes the SFO complexity N.\n(iii) For(Decay 2) ,N′(b)>0holds for all b >D2\nϵ2.\nTheorem 3.3(ii) indicates that, if we can set a critical batch size (4) for each of\n(Constant) ,(Decay 1) , and (Decay 3) , then the SFO complexity will be minimized.\nHowever, it would be difficult to set b⋆in (4) before implementing SGD, since b⋆in (4)\ninvolves unknown parameters, such as Landσ2(computing Lis NP-hard [28]). Hence,\nwe would like to estimate the critical batch sizes by using Theorem 3.3(ii) and (iii)\n(see Section 4.3). Theorem 3.3(ii) indicates that the smaller ϵis, the larger the critical\nbatch size b⋆in(Constant) becomes. Theorem 3.3(iii) indicates that the critical batch\nsize is close toD2\nϵ2when using (Decay 2) to minimize the SFO complexity N.\n3.4. Iteration and SFO complexities of SGD\nTheorems 3.2 and 3.3 lead to the following theorem indicating the iteration and SFO\ncomplexities needed for SGD to be an ϵ–approximation (see also Table 2).\nTheorem 3.4 (Iteration and SFO complexities of SGD) .The iteration and SFO\ncomplexities such that Algorithm 1 under (C1)–(C3) is an ϵ–approximation (3) are as\n8follows:\n(Kϵ(n, b⋆, αk,∆0, L, σ2),Nϵ(n, b⋆, αk,∆0, L, σ2)) =\n\n\u0012\nO\u00121\nϵ2\u0013\n, O\u00121\nϵ4\u0013\u0013\n(Constant)\n\u0012\nO\u00121\nϵ2\na\u0013\n, O\u00121\nϵ2\na\u0013\u0013\n(Decay 1)\n\u0012\nO\u00121\nϵ4\u0013\n, O\u00121\nϵ6\u0013\u0013\n(Decay 2)\n\u0012\nO\u00121\nϵ2\n1−a\u0013\n, O\u00121\nϵ2\n1−a\u0013\u0013\n(Decay 3)\nwhere Kϵ(n, b, α k,∆0, L, σ2)andNϵ(n, b, α k,∆0, L, σ2)are defined as in (1), the crit-\nical batch sizes in Theorem 3.3 are used to compute Kϵ(n, b⋆, αk,∆0, L, σ2)and\nNϵ(n, b⋆, αk,∆0, L, σ2). In(Decay 2) , we assume that b⋆=D2+1\nϵ2. (see also (4)).\nTheorem 3.4 indicates that the iteration and SFO complexities for (Constant) are\nsmaller than those for (Decay 1) –(Decay 3) .\n4. Numerical Results\nWe numerically verified the number of iterations and SFO complexities needed to\nachieve high test accuracy for different batch sizes in training ResNet [29] and Wide-\nResNet [30]. The parameter αused in (Constant) was determined by conduct-\ning a grid search of {0.001,0.005,0.01,0.05,0.1,0.5}. The parameters αandTused\nin the decaying learning rates (Decay 1) –(Decay 3) defined by αk=α\n(⌊k\nT⌋+1)a\nwere determined by a grid search of α∈ {0.001,0.1,0.125,0.25,0.5,1.0}andT∈\n{5,10,20,30,40,50}. The parameter awas set to a=1\n4in(Decay 1) anda=3\n4\nin(Decay 3) . We compared SGD with SGD with momentum (momentum), Adam,\nAdamW, and RMSProp. The learning rates and hyperparameters of these four opti-\nmizers were determined on the basis of the previous results [9,10,12] (The weight decay\nused in the momentum was 5 ×10−4). The experimental environment consisted of an\nNVIDIA DGX A100 ×8GPU and Dual AMD Rome7742 2.25-GHz, 128 Cores ×2CPU.\nThe software environment was Python 3.10.6, PyTorch 1.13.1, and CUDA 11.6. The\ncode is available at https://github.com/imakn0907/SGD_using_decaying .\n4.1. Training ResNet-18 on the CIFAR-10 and CIFAR-100 datasets\nFirst, we trained ResNet-18 on the CIFAR-10 dataset. The stopping condition of\nthe optimizers was 200 epochs. Figure 1 indicates that the number of iterations is\nmonotone decreasing and convex with respect to batch size for SGDs using a constant\nlearning rate or a decaying learning rate. Figure 2 indicates that, in each case of SGD\nwith(Constant) –(Decay 3) , a critical batch size b⋆= 24exists at which the SFO\ncomplexity is minimized.\nFigures 3 and 4 indicate that the number of iterations and the SFO complexity for\nfour different learning rates in achieving a test accuracy of 0.6 when training ResNet-\n18 on the CIFAR-100 dataset. The figures indicate that critical batch sizes existed\nwhen using (Constant) –(Decay 3) .\nFigures 5 and 6 compare SGD with (Decay 1) with the other optimizers in training\n9212223242526272829210211\nBatch Size104105106Iteration\nConstant\nDecay 1\nDecay 2\nDecay 3Figure 1. Number of iterations needed for SGD with\n(Constant), (Decay 1), (Decay 2), and (Decay 3) to\nachieve a test accuracy of 0 .9 versus batch size (ResNet-\n18 on CIFAR-10)\n212223242526272829210211\nBatch Size106107SFO\nConstant\nDecay 1\nDecay 2\nDecay 3Figure 2. SFO complexity needed for SGD with\n(Constant), (Decay 1), (Decay 2), and (Decay 3) to\nachieve a test accuracy of 0 .9 versus batch size (ResNet-\n18 on CIFAR-10)\n212223242526272829210211212\nBatch Size103104105106Iteration\nConstant\nDecay 1\nDecay 2\nDecay 3\nFigure 3. Number of iterations needed for SGD with\n(Constant), (Decay 1), (Decay 2), and (Decay 3) to\nachieve a test accuracy of 0 .6 versus batch size (ResNet-\n18 on CIFAR-100)\n212223242526272829210211212\nBatch Size106107SFO\nConstant\nDecay 1\nDecay 2\nDecay 3Figure 4. SFO complexity needed for SGD with\n(Constant), (Decay 1), (Decay 2), and (Decay 3) to\nachieve a test accuracy of 0 .6 versus batch size (ResNet-\n18 on CIFAR-100)\n212223242526272829210211212\nBatch Size103104105106Iteration\nSGD with Decay1\nmomentum\nAdam\nAdamW\nRMSProp\nFigure 5. Number of iterations needed for SGD with\n(Decay 1), momentum, Adam, AdamW, and RMSProp\nto achieve a test accuracy of 0 .6 versus batch size\n(ResNet-18 on CIFAR-100)\n212223242526272829210211212\nBatch Size106107SFO\nSGD with Decay1\nmomentum\nAdam\nAdamW\nRMSPropFigure 6. SFO complexity needed for SGD with (De-\ncay 1), momentum, Adam, AdamW, and RMSProp to\nachieve a test accuracy of 0 .6 versus batch size (ResNet-\n18 on CIFAR-100)\n10ResNet-18 on the CIFAR-100 dataset. These figures indicates that SGD with (Decay\n1)and a critical batch size ( b= 24) outperformed the other optimizers in the sense\nof minimizing the number of iterations and the SFO complexity. Figure 6 also indi-\ncates that the existing optimizers using constant learning rates had critical batch sizes\nminimizing the SFO complexities. In particular, AdamW using the critical batch size\nb⋆= 25performed well.\n4.2. Training Wide-ResNet on the CIFAR-10 and CIFAR-100 datasets\n212223242526272829210\nBatch Size104105Iteration\nConstant\nDecay 1\nDecay 2\nDecay 3\nFigure 7. Number of iterations needed for SGD with\n(Constant), (Decay 1), (Decay 2), and (Decay 3) to\nachieve a test accuracy of 0 .9 versus batch size (Wide-\nResNet-28-10 on CIFAR-10)\n212223242526272829210\nBatch Size106SFO\nConstant\nDecay 1\nDecay 2\nDecay 3Figure 8. SFO complexity needed for SGD with\n(Constant), (Decay 1), (Decay 2), and (Decay 3) to\nachieve a test accuracy of 0 .9 versus batch size (Wide-\nResNet-28-10 on CIFAR-10)\nNext, we trained Wide-ResNet-28 [30] on the CIFAR-10 and CIFAR-100 datasets.\nThe stopping condition of the optimizers was 200 epochs. Figures 7 and 8 show that\nthe number of iterations and the SFO complexity of SGD to achieve a test accuracy\nof 0.9 (CIFAR-10) versus batch size. Figures 7 and 8 indicate that the critical batch\nsize was b⋆= 24in each case of SGD using (Constant) –(Decay 3) .\n212223242526272829210\nBatch Size104105Iteration\nConstant\nDecay 1\nDecay 2\nDecay 3\nFigure 9. Number of iterations needed for SGD\nwith (Constant), (Decay 1), (Decay 2), and (Decay\n3) to achieve a test accuracy of 0 .6 versus batch size\n(WideResNet-28-12 on CIFAR-100)\n212223242526272829210\nBatch Size106SFO\nConstant\nDecay 1\nDecay 2\nDecay 3Figure 10. SFO complexity needed for SGD with\n(Constant), (Decay 1), (Decay 2), and (Decay 3)\nto achieve a test accuracy of 0 .6 versus batch size\n(WideResNet-28-12 on CIFAR-100)\nFigures 9 and 10 indicate that the number of iterations and the SFO complexity of\nSGD to achieve a test accuracy of 0.6 (CIFAR-100) versus batch size and show that a\ncritical batch size existed for (Constant) –(Decay 3) .\nAs in Figures 5 and 6, Figures 11 and 12 indicate that SGD using (Decay 3) and the\nexisting optimizers using constant learning rates had critical batch sizes minimizing\nthe SFO complexities.\n11212223242526272829210211\nBatch Size103104105106Iteration\nSGD with Decay3\nmomentum\nAdam\nAdamW\nRMSPropFigure 11. Number of iterations needed for SGD with\n(Decay 3), momentum, Adam, AdamW, and RMSProp\nto achieve a test accuracy of 0 .9 versus batch size\n(WideResNet-28-10 on CIFAR-10)\n212223242526272829210211\nBatch Size106SFO\nSGD with Decay3\nmomentum\nAdam\nAdamW\nRMSPropFigure 12. SFO complexity needed for SGD with\n(Decay 3), momentum, Adam, AdamW, and RMSProp\nto achieve a test accuracy of 0 .9 versus batch size\n(WideResNet-28-10 on CIFAR-10)\n4.3. Estimation of critical batch sizes\nTable 3. Measured (left) and estimated (right; bold) critical batch sizes (D1 and D3 stand for (Decay 1) and\n(Decay 3))\nResNet-18 Wide-ResNet-28\nCIFAR10 CIFAR100 CIFAR10 CIFAR100\nD1 24242223,24\nD3 2424 2424 22,2462412, 24\nWe estimated the critical batch sizes of (Decay 3) using Theorem 3.3 and measured\nthe critical batch sizes of (Decay 1) . From (4) and a=1\n4((Decay 1) ), we have that,\nfor training ResNet-18 on the CIFAR-10 dataset, b⋆= 24=(1−a)D2\na(1−2a)D1, i.e.,D2\nD1=8\n3.\nThen, the estimated critical batch size of SGD using (Decay 3) (a=3\n4) for training\nResNet-18 on the CIFAR-10 dataset is\nb⋆=2a2\n(1−a)(2a−1)D2\nD1=2a2\n(1−a)(2a−1)8\n3\n= 24∈(24,25),\nwhich implies that the estimated critical batch size b⋆= 24 is close to the measured\nsizeb= 24. We also found that the estimated critical batch sizes are close to the\nmeasured critical batch sizes (see Table 3).\n5. Conclusion and future work\nThis paper investigated the required number of iterations and SFO complexities for\nSGD using constant or decay learning rates to achieve an ϵ–approximation. Our theo-\nretical analyses indicated that the number of iterations needed for an ϵ–approximation\nis monotone decreasing and convex with respect to the batch size and the SFO com-\nplexity needed for an ϵ–approximation is convex with respect to the batch size. More-\nover, we showed that SGD using a critical batch size reduces the SFO complexity.\n12The numerical results indicated that SGD using the critical batch size performs better\nthan the existing optimizers in the sense of minimizing the SFO complexity. 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In: Advances in Neural Information Processing Systems; Vol. 31; 2018.\n[29] He K, Zhang X, Ren S, et al. Deep residual learning for image recognition. In: Proceedings\nof the IEEE conference on computer vision and pattern recognition; 2016. p. 770–778.\n[30] Zagoruyko S, Komodakis N. Wide residual networks. arXiv preprint arXiv:160507146.\n2016;.\n14Appendix A. Appendix\nA.1. Lemma\nFirst, we will prove the following lemma.\nLemma A.1. The sequence (θk)k∈Ngenerated by Algorithm 1 under (C1)–(C3) sat-\nisfies that, for all K≥1,\nK−1X\nk=0αk\u0012\n1−Lαk\n2\u0013\nE\u0002\n∥∇f(θk)∥2\u0003\n≤E[f(θ0)−f⋆] +Lσ2\n2bK−1X\nk=0α2\nk,\nwhere Estands for the total expectation.\nProof: Condition (C1) ( L-smoothness of f) implies that the descent lemma holds,\ni.e., for all k∈N,\nf(θk+1)≤f(θk) +⟨∇f(θk),θk+1−θk⟩+L\n2∥θk+1−θk∥2,\nwhich, together with θk+1:=θk−αk∇fBk(θk), implies that\nf(θk+1)≤f(θk)−αk⟨∇f(θk),∇fBk(θk)⟩+Lα2\nk\n2∥∇fBk(θk)∥2. (A1)\nCondition (C2) guarantees that\nEξk[∇fBk(θk)|θk] =∇f(θk) and Eξk\u0002\n∥∇fBk(θk)− ∇f(θk)∥2|θk\u0003\n≤σ2\nb. (A2)\nHence, we have\nEξk\u0002\n∥∇fBk(θk)∥2|θk\u0003\n=Eξk\u0002\n∥∇fBk(θk)− ∇f(θk) +∇f(θk)∥2|θk\u0003\n=Eξk\u0002\n∥∇fBk(θk)− ∇f(θk)∥2|θk\u0003\n+ 2Eξk[⟨∇fBk(θk)− ∇f(θk),∇f(θk)⟩|θk]\n+Eξk\u0002\n∥∇f(θk)∥2|θk\u0003\n≤σ2\nb+Eξk\u0002\n∥∇f(θk)∥2\u0003\n. (A3)\nTaking the expectation conditioned on θkon both sides of (A1), together with (A2)\nand (A3), guarantees that, for all k∈N,\nEξk[f(θk+1)|θk]≤f(θk)−αkEξk[⟨∇f(θk),∇fBk(θk)⟩|θk] +Lα2\nk\n2Eξk\u0002\n∥∇fBk(θk)∥2|θk\u0003\n≤f(θk)−αk∥∇f(θk)∥2+Lα2\nk\n2\u0012σ2\nb+∥∇f(θk)∥2\u0013\n.\n15Hence, taking the total expectation on both sides of the above inequality ensures that,\nfor all k∈N,\nαk\u0012\n1−Lαk\n2\u0013\nE\u0002\n∥∇f(θk)∥2\u0003\n≤E[f(θk)−f(θk+1)] +Lσ2α2\nk\n2b.\nLetK≥1. Summing the above inequality from k= 0 to k=K−1 ensures that\nK−1X\nk=0αk\u0012\n1−Lαk\n2\u0013\nE\u0002\n∥∇f(θk)∥2\u0003\n≤E[f(θ0)−f(θK)] +Lσ2\n2bK−1X\nk=0α2\nk,\nwhich, together with (C1) (the lower bound f⋆off), implies that the assertion in\nLemma A.1 holds.\nA.2. Proof of Theorem 3.1\n(Constant): Lemma A.1 with αk=αimplies that\nα\u0012\n1−Lα\n2\u0013K−1X\nk=0E\u0002\n∥∇f(θk)∥2\u0003\n≤E[f(θ0)−f⋆] +Lσ2α2K\n2b.\nSince α <2\nL, we have that\nmin\nk∈[0:K−1]E\u0002\n∥∇f(θk)∥2\u0003\n≤1\nKK−1X\nk=0E\u0002\n∥∇f(θk)∥2\u0003\n≤2(f(θ0)−f⋆)\n(2−Lα)α|{z }\nC11\nK+Lσ2α\n2−Lα|{z}\nC21\nb.\n(Decay): Since ( αk)k∈Nconverges to 0, there exists k0∈Nsuch that, for all k≥k0,\nαk<2\nL. We assume that k0= 0 (see Section 2.2.2). Lemma A.1 ensures that, for all\nK≥1,\nK−1X\nk=0αk\u0012\n1−Lαk\n2\u0013\nE\u0002\n∥∇f(θk)∥2\u0003\n≤E[f(θ0)−f⋆] +Lσ2\n2bK−1X\nk=0α2\nk,\nwhich, together with αk+1≤αk<2\nL(k∈N), implies that\nαK−1\u0012\n1−Lα0\n2\u0013K−1X\nk=0E\u0002\n∥∇f(θk)∥2\u0003\n≤E[f(θ0)−f⋆] +Lσ2\n2bK−1X\nk=0α2\nk.\nHence, we have that\nK−1X\nk=0E\u0002\n∥∇f(θk)∥2\u0003\n≤2(f(θ0)−f⋆)\n(2−Lα0)αK−1+Lσ2\nb(2−Lα0)αK−1K−1X\nk=0α2\nk,\n16which implies that\nmin\nk∈[0:K−1]E\u0002\n∥∇f(θk)∥2\u0003\n≤2(f(θ0)−f⋆)\n2−Lα01\nKαK−1+1\nbLσ2\n2−Lα01\nKαK−1K−1X\nk=0α2\nk.\nMeanwhile, we have that\nK−1X\nk=0α2\nk≤K−1X\nk=0Tα2\n(k+ 1)2a≤Tα2\u0012\n1 +ZK−1\n0dt\n(t+ 1)2a\u0013\n≤\n\nTα2\n1−2aK1−2a(Decay 1)\nTα2(1 + log K)(Decay 2)\n2aTα2\n2a−1(Decay 3)\nand\nαK−1=α\u0000\n⌊K−1\nT⌋+ 1\u0001a≥α\u0000K−1\nT+ 1\u0001a≥α\nKa.\nHere, we define\nD1:=2 (f(θ0)−f⋆)\nα(2−Lα)andD2:=Tα2Lσ2\n2−Lα.\nAccordingly, we have that\nmin\nk∈[0:K−1]E\u0002\n∥∇f(θk)∥2\u0003\n≤\n\nD1\nK1−a+D2\n(1−2a)Kab(Decay 1)\nD1√\nK+D2(1 + log K)√\nKb(Decay 2)\nD1\nK1−a+2aD2\n(2a−1)K1−ab(Decay 3)\nwhich, together with log K <√\nKand the condition on a, implies that\nmin\nk∈[0:K−1]E\u0002\n∥∇f(θk)∥2\u0003\n≤\n\nD1\nKa+D2\n(1−2a)Kab(Decay 1)\nD1√\nK+\u00121√\nK+ 1\u0013D2\nb(Decay 2)\nD1\nK1−a+2aD2\n(2a−1)K1−ab(Decay 3) .\nA.3. Proof of Theorem 3.2\n(i) Let us consider the case of (Constant) . We consider that the upper bound\nC1\nK+C2\nbin Theorem 3.1 is equal to ϵ2. This implies that K=C1b\nϵ2b−C2achieves an\nϵ–approximation. A discussion similar to the one showing that K=C1b\nϵ2b−C2is an ϵ–\napproximation ensures that the assertion in Theorem 3.2(i) is true.\n17(ii) It is sufficient to prove that K′=K′(b)<0 and K′′=K′′(b)>0 hold.\n(Constant): LetK=C1b\nϵ2b−C2. Then, we have that\nK′=C1(ϵ2b−C2)−ϵ2C1b\n(ϵ2b−C2)2=−C1C2\n(ϵ2b−C2)2<0,\nK′′=2ϵ2C1C2(ϵ2b−C2)\n(ϵ2b−C2)4=2ϵ2C1C2((C1\nK+C2\nb)b−C2)\n(ϵ2b−C2)4=2ϵ2C2\n1C2\n2\nK(ϵ2b−C2)4>0.\n(Decay 1): LetK= (1\nϵ2(D1+D2\n(1−2a)b))1\na. Then, we have that\nK′=1\na\u001a1\nϵ2\u0012\nD1+D2\n(1−2a)b\u0013\u001b 1\na−1\u0012\n−D2\nϵ2(1−2a)b2\u0013\n=−D2\naϵ2(1−2a)b2\u001a1\nϵ2\u0012\nD1+D2\n(1−2a)b\u0013\u001b 1−a\na\n<0,\nK′′=2D2\naϵ2(1−2a)b3\u001a1\nϵ2\u0012\nD1+D2\n(1−2a)b\u0013\u001b 1−a\na\n+2(1−a)D2\na2ϵ2(1−2a)b3\u001a1\nϵ2\u0012\nD1+D2\n(1−2a)b\u0013\u001b 1−a\na−1D2\nϵ2(1−2a)b2>0.\n(Decay 2): LetK= (bD1+D2\nbϵ2−D2)2. Then, we have that\nK′=2D1(bD1+D2)(bϵ2−D2)2−2ϵ2(bϵ2−D2)(bD1+D2)2\n(bϵ2−D2)4,\nwhich, together with bϵ2−D2>0 and D1> ϵ2, implies that\n(bϵ2−D2)3K′= 2D1(bD1+D2)(bϵ2−D2)−2ϵ2(bD1+D2)2\n= 2(bD1+D2)\b\nD1(bϵ2−D2)−ϵ2(bD1+D2)\t\n=−2(bD1+D2)(D1D2−ϵ2D2)\n=−2D2(bD1+D2)(D1−ϵ2)<0.\nMoreover,\nK′′=−2D1D2(D1−ϵ2)(bϵ2−D2)3+ 6D2ϵ2(bϵ2−D2)2(bD1+D2)(D1−ϵ2)\n(bϵ2−D2)6,\nwhich implies that\n(bϵ2−D2)4K′′=−2D1D2(D1−ϵ2)(bϵ2−D2) + 6D2ϵ2(bD1+D2)(D1−ϵ2)\n= 2D2(D1−ϵ2)\b\n−D1(bϵ2−D2) + 3ϵ2(bD1+D2)\t\n= 2D2(D1−ϵ2)(2D1ϵ2b+D1D2+ 3D2ϵ2)>0.\n18(Decay 3): LetK= (1\nϵ2(D1+2aD2\n(2a−1)b))1\n1−a. Then, we have that\nK′=1\n1−a\u001a1\nϵ2\u0012\nD1+2aD2\n(2a−1)b\u0013\u001b 1\n1−a−1\u0012\n−2aD2\nϵ2(2a−1)b2\u0013\n=−2aD2\nϵ2(1−a)(2a−1)b2\u001a1\nϵ2\u0012\nD1+2aD2\n(2a−1)b\u0013\u001b a\n1−a\n<0,\nK′′=4aD2\nϵ2(1−a)(2a−1)b3\u001a1\nϵ2\u0012\nD1+2aD2\n(2a−1)b\u0013\u001b a\n1−a\n+2a2D2\nϵ2(1−a)2(2a−1)b2\u001a1\nϵ2\u0012\nD1+2aD2\n(2a−1)b\u0013\u001b a\n1−a−12aD2\nϵ2(2a−1)b2>0.\nA.4. Proof of Theorem 3.3\n(Constant): LetN=C1b2\nϵ2b−C2. Then, we have that\nN′=2C1b(ϵ2b−C2)−ϵ2C1b2\n(ϵ2b−C2)2=C1b(ϵ2b−2C2)\n(ϵ2b−C2)2.\nIfN′= 0, we have that ϵ2b−2C2= 0, i.e., b=2C2\nϵ2. Moreover,\nN′′=(2ϵ2C1b−2C1C2)(ϵ2b−C2)2−2ϵ2(ϵ2b−C2)(ϵ2C1b2−2C1C2b)\n(ϵ2b−C2)4\n(ϵ2b−C2)3N′′= (2��2C1b−2C1C2)(ϵ2b−C2)−2ϵ2(ϵ2C1b2−2C1C2b)\n= 2C1C2\n2>0,\nwhich implies that Nis convex. Hence, there is a critical batch size b⋆=2C2\nϵ2>0 at\nwhich Nis minimized.\n(Decay 1): LetN=Kb. Then, we have that\nN′=K+bK′\n=\u001a1\nϵ2\u0012\nD1+D2\n(1−2a)b\u0013\u001b 1\na\n+1\na\u001a1\nϵ2\u0012\nD1+D2\n(1−2a)b\u0013\u001b 1\na−1\u0012\n−D2\nϵ2(1−2a)b2\u0013\nb\n=\u001a1\nϵ2\u0012\nD1+D2\n(1−2a)b\u0013\u001b 1\na−1\u001a1\nϵ2\u0012\nD1+D2\n(1−2a)b\u0013\n−D2\naϵ2(1−2a)b\u001b\n.\nIfN′= 0, we have that\n1\nϵ2\u0012\nD1+D2\n(1−2a)b\u0013\n−D2\naϵ2(1−2a)b= 0,i.e.,b=D2(a−1)\naD1(2a−1).\n19Moreover,\nN′′=K′+ (K′+bK′′) = 2 K′+bK′′\n=−2D2\naϵ2(1−2a)b2\u001a1\nϵ2\u0012\nD1+D2\n(1−2a)b\u0013\u001b 1−a\na\n+2D2\naϵ2(1−2a)b2\u001a1\nϵ2\u0012\nD1+D2\n(1−2a)b\u0013\u001b 1−a\na\n+2(1−a)D2\na2ϵ2(1−2a)b2\u001a1\nϵ2\u0012\nD1+D2\n(1−2a)b\u0013\u001b 1−a\na−1D2\nϵ2(1−2a)b2\n=2(1−a)D2\na2ϵ2(1−2a)b2\u001a1\nϵ2\u0012\nD1+D2\n(1−2a)b\u0013\u001b 1−a\na−1D2\nϵ2(1−2a)b2>0,\nwhich implies that Nis convex. Hence, there is a critical batch size b⋆=D2(a−1)\naD1(2a−1)>0.\n(Decay 2): LetN=bK. Then, we have that\nN′=K+bK′\n=(bD1−D2)2\n(bϵ2−D2)2−2D2b(bD1+D2)(D1−ϵ2)\n(bϵ2−D2)3\n=bD1+D2\n(bϵ2−D2)3{(bD1+D2)(bϵ2−D2)−2D2b(D1−ϵ2)}\n=bD1+D2\n(bϵ2−D2)3{D1ϵ2b2+ 3D2(ϵ2−D1)b−D2\n2}.\nIfN′= 0, we have that D1b+D2= 0, i.e., b=−D2\nD1<0. Moreover,\nN′′= 2K′+bK′′\n=−4D2(bD1+D2)(D1−ϵ2)\n(bϵ2−D2)3+2D2b(D1−ϵ2)(2D1ϵ2b+D1D2+ 3D2ϵ2)\n(bϵ2−D2)4\n=2D2(D1−ϵ2)\n(bϵ2−D2)4{−2(bD1+D2)(bϵ2−D2) +b(2D1ϵ2b+D1D2+ 3D2ϵ2)}\n=2D2(D1−ϵ2)\n(bϵ2−D2)4(3D1D2b+D2ϵ2b+ 2D2\n2)>0,\nwhich implies that Nis convex. We can check that N′(b)>0 for all b >D2\nϵ2.\n(Decay 3): LetN=bK. Then, we have that\nN′=K+bK′\n=\u001a1\nϵ2\u0012\nD1+2aD2\n(2a−1)b\u0013\u001b 1\n1−a\n−2aD2\nϵ2(1−a)(2a−1)b\u001a1\nϵ2\u0012\nD1+2aD2\n(2a−1)b\u0013\u001b 1\n1−a−1\n=\u001a1\nϵ2\u0012\nD1+2aD2\n(2a−1)b\u0013\u001b 1\n1−a−1\u001a1\nϵ2\u0012\nD1+2aD2\n(2a−1)b\u0013\n−2aD2\nϵ2(1−a)(2a−1)b\u001b\n.\nIfN′= 0, we have that\n1\nϵ2\u0012\nD1+2aD2\n(2a−1)b\u0013\n−2aD2\nϵ2(1−a)(2a−1)b= 0,i.e.,b=2a2D2\n(2a−1)(1−a)D1.\n20Moreover,\nN′′= 2K′+bK′′\n=−2aD2\nϵ2(1−a)(2a−1)b2\u001a1\nϵ2\u0012\nD1+2aD2\n(2a−1)b\u0013\u001b a\n1−a\n+2aD2\nϵ2(1−a)(2a−1)b2\u001a1\nϵ2\u0012\nD1+2aD2\n(2a−1)b\u0013\u001b a\n1−a\n+2a2D2\nϵ2(1−a)2(2a−1)b\u001a1\nϵ2\u0012\nD1+2aD2\n(2a−1)b\u0013\u001b 2a−1\n1−a 2aD2\nϵ2(2a−1)b2\n=2a2D2\nϵ2(1−a)2(2a−1)b(\n1\nϵ2(D1+2aD2\n(2a−1)b))2a−1\n1−a2aD2\nϵ2(2a−1)b2>0,\nwhich implies that Nis convex. Hence, there is a critical batch size b⋆=\n2a2D2\n(2a−1)(1−a)D1>0.\nA.5. Proof of Theorem 3.4\nUsing Kdefined in Theorem 3.2 leads to the iteration complexity. For example, SGD\nusing (Constant) satisfies N(b) =C1b2\nϵ2b−C2(Theorem 3.3). Using the critical batch size\nb⋆=2C2\nϵ2in (4) leads to\ninf\u001a\nN: min\nk∈[0:K−1]E[∥∇f(θk)∥]≤ϵ\u001b\n≤N(b⋆) =4C1C2\nϵ4,i.e.,Nϵ=O\u00121\nϵ4\u0013\n.\nA similar discussion, together with using Ndefined in Theorem 3.3 and the critical\nbatch size b⋆in (4), leads to the SFO complexities of (Decay 1) and(Decay 3) . Using\nNdefined in Theorem 3.3 and a batch size b=D2+1\nϵ2leads to the SFO complexity of\n(Decay 2) .\n21"    },    {        "title": "2402.15348v1.Tight_Approximation_and_Kernelization_Bounds_for_Vertex_Disjoint_Shortest_Paths.pdf",        "content": "arXiv:2402.15348v1  [cs.DS]  23 Feb 2024Tight Approximation and Kernelization Bounds for\nVertex-Disjoint Shortest Paths\nMatthias Bentert Fedor V. Fomin Petr A. Golovach\nAbstract\nWe examine the possibility of approximating Maximum Vertex-Disjoint Shortest Paths .\nIn this problem, the input is an edge-weighted (directed or u ndirected) n-vertex graph Galong\nwithkterminal pairs ( s1,t1),(s2,t2),...,(sk,tk). The task is to connect as many terminal pairs\nas possible by pairwise vertex-disjoint paths such that eac h path is a shortest path between the\nrespective terminals. Our work is anchored in the recent bre akthrough by Lochet [SODA ’21],\nwhich demonstrates the polynomial-time solvability of the problem for a fixed value of k.\nLochet’s result implies the existence of a polynomial-time ck-approximation for Maximum\nVertex-Disjoint Shortest Paths , where c≤1 is a constant. (One can guess1/ctermi-\nnal pairs to connect in kO(1/c)time and then utilize Lochet’s algorithm to compute the solu -\ntion innf(1/c)time.) Our first result suggests that this approximation alg orithm is, in a sense,\nthe best we can hope for. More precisely, assuming the gap-ET H, we exclude the existence of\nano(k)-approximations within f(k)·poly(n) time for any function fthat only depends on k.\nOur second result demonstrates the infeasibility of achiev ing an approximation ratio of n1/2−ε\nin polynomial time, unless P = NP. It is not difficult to show tha t a greedy algorithm selecting a\npath with the minimum number of arcs results in a ⌈√\nℓ⌉-approximation, where ℓis the number\nof edges in all the paths of an optimal solution. Since ℓ≤n, this underscores the tightness of\nthen1/2−ε-inapproximability bound.\nAdditionally, weestablishthat MaximumVertex-DisjointShortest Paths isfixed-parameter\ntractable when parameterized by ℓbut does not admit a polynomial kernel.\nOur hardness results hold for undirected graphs with unit we ights, while our positive results\nextend to scenarios where the input graph is directed and fea tures arbitrary (non-negative) edge\nweights.\n1 Introduction\nWe study a variant of the well-known Vertex-Disjoint Paths problem, where the input comprises\na (directed or undirected) graph Gandkterminal pairs as input. The task is to identify whether\npairwise vertex-disjoint paths can connect all terminals. Vertex-Disjoint Paths has long been\nestablished as NP-complete [ 16] and has played a pivotal role in the graph-minor project by Robert son\nand Seymour [ 23].\nEilam-Tzoreff [ 12] introduced a variant of Vertex-Disjoint Paths where all paths in the solu-\ntion must be shortest paths between the respective terminals. The parameterized comp lexity of this\nvariant, known as Vertex-Disjoint Shortest Paths , was recently resolved [ 2,19]: The problem,\nparameterized by k, is W[1]-hard and in XP for undirected graphs. On directed graphs, t he problem\nis NP-hard already for k= 2 if zero-weight edges are allowed. The problem is solvable in polynomia l\ntime for k= 2 for strictly positive edge weights [ 3]. It is NP-hard when kis part of the input, and the\ncomplexity for constant k >2 remains open.\nAn optimizationvariantof Vertex-Disjoint Shortest Paths , wherenotnecessarilyallterminal\npairs need to be connected, but at least pof them, is referred to as Maximum Vertex-Disjoint\nShortest Paths .\n1Maximum Vertex-Disjoint Shortest Paths\nInput: A graph G= (V,E), an edge-length function w:E→Q≥0, terminal\npairs (s1,t1),(s2,t2),...,(sk,tk) wheresi/\\⌉}atio\\slash=tifori∈[k], and integers pandℓ.\nQuestion: Is there a set S⊆[k] with|S| ≥psuch that there is a collection C={Pi}i∈Sof\npairwisevertex-disjointpathssatisfyingthefollowingconditions: f oreachi∈S,\npathPiis a shortest path from sitotiand the total amount of edges in all\npaths of Cis at most ℓ?\nA few remarksare in order. In the literature concerning Vertex-Disjoint Paths and its variants,\nit is common for paths in a solution to share a terminal. However, in our context, terminal pairs may\nindeed share a terminal, but the paths comprising a feasible solution m ust be vertex-disjoint, and this\nconstraint also applies to their endpoints.\nNote that Vertex-Disjoint Shortest Paths is a special case of Maximum Vertex-Disjoint\nShortest Paths withp=kandℓ=n. For the maximization version, we are not given pas input\nbut are instead asked to find a set Sthat is as large as possible. Slightly abusing notation, we do not\ndistinguish between these two variants and refer to both as Maximum Vertex-Disjoint Shortest\nPaths.\nIn the definition of Maximum Vertex-Disjoint Shortest Paths , we also incorporatethe upper\nboundℓon the number of edges in a solution. This parameter ℓproves to be very useful for approx-\nimation and parameterized algorithms. While parameterization by kyields strong hardness bounds\n(both in parameterized complexity and approximation), another na tural parameterization would be\nthe sum of path lengths. If we confine all edge weights to be positive integers, then ℓserves as a lower\nbound for the sum of path lengths. Since our hardness results app ly to unweighted graphs, studying ℓ\ninstead of the sum of path lengths does not weaken the negative re sults.\nFor the parameterized complexity of Maximum Vertex-Disjoint Shortest Paths , we note\nthat the results for Vertex-Disjoint Shortest Paths [2,19] for the parameterization by kdirectly\ntranslate for Maximum Vertex-Disjoint Shortest Paths parameterized by p. The problem is\nW[1]-hardasa generalizationof Vertex-Disjoint Shortest Paths , andto obtain an XP algorithm,\nit is sufficient to observe that in nO(p)time we can guess a set S⊆[k] of size pand apply the XP\nalgorithm for Vertex-Disjoint Shortest Paths for the selected set of terminal pairs.\nIn terms of approximations, we are not aware of any studies of Maximum Vertex-Disjoint\nShortest Paths . ForMaximum Vertex-Disjoint Paths , where the task is to connect the maxi-\nmum number of terminal pair by disjoint but not necessarily shortes t paths, there is a known O(√n)-\napproximation [ 18] and the best known lower bounds are 2Ω(√\nlog(n))andnΩ(1/(log log n)2), where the\nformer lower bound holds even if the input graph is an unweighted plan ar graph and the latter lower\nbound holds even if the input graph is an unweighted grid graph [ 8,9]. The best known approximation\nalgorithms for these two special cases are ˜O(n9/19) and˜O(n1/4), respectively.\nWhen requiring the solution paths to be edge-disjoint rather than v ertex-disjoint, there have been\nsome studies on relaxing the notion so that each edge appears in at m ostc >1 of the solution paths.\nThe integer cis called the congestion and the currently best known approximation algorithm achieves\na poly(log n)-approximation with c= 2 [7].\nOur results. We showthatcomputing a n1/2−ε-approximationsisNP-hardforany ε >0(Theorem 2).\nFor FPT-approximation, we show in Theorem 1that any ko(1)-approximation in f(k)·poly(n) time\nimplies FPT = W[1] and that one cannot o(k)-approximate Maximum Vertex-Disjoint Shortest\nPathsinf(k)·poly(n) time unless the gap-ETH fails. We complement the first lower bound b y\ndeveloping a ⌈√\nℓ⌉-approximation in Theorem 3. In Theorems 4and5, we show that Maximum\nVertex-Disjoint Shortest Paths is fixed-parameter tractable when parameterized by ℓ, but it\ndoes not admit a polynomial kernel. We mention that all of our hardne ss results hold for undirected\ngraphs with unit weights and all of our positive results hold even for d irected and edge-weighted input\ngraphs. Our results are summarized in Table 1.\nThe remainder of this work is organized as follows. We start in Section 2by introducing concepts\nand notation used throughout the paper. In Section 3, we present our results on approximation\nalgorithmsandlowerboundsfor Maximum Vertex-Disjoint Shortest Paths . Section 4isdevoted\nto the parameterized complexity of Maximum Vertex-Disjoint Shortest Paths with respect to ℓ,\nthat is, we show fixed-parameter tractability and exclude polynomia l kernels. We conclude with an\n2Table 1: Overview over results. New results are bold. All hardness r esults hold for unweighted and\nundirected graphs, while all new algorithmic results hold even for dire cted graphs with arbitrary non-\nnegative edge weights.\nExact Approximation\nno parameter NP-complete no n1/2−ε-approximation in poly (n)time\nk XP and W[1]-hard no o(k)-approximation in f (k)·poly(n)time\nℓ FPTandno poly kernel ⌈√\nℓ⌉-approximation\nopen problem in Section 5.\n2 Preliminaries\nFor a positive integer x, we denote by [ x] ={1,2,...,x}the set of all positive integers at most x. We\ndenote by G= (V,E) a graph and by nandmthe number of vertices and edges in G, respectively.\nThe graph Gis said to be k-partiteifVcan be partitioned into kdisjoint sets V1,V2,...,V ksuch\nthat each set Viinduces an independent set, that is, there is no edge {u,v} ∈Ewith{u,v} ⊆Vifor\nsomei∈[k]. Thedegreeof a vertex vis the number of edges in Ethat contain vas an endpoint and\nthemaximum degree of a graph is highest degree of any vertex in the graph.\nApathin a graph Gis a sequence ( v0,v1,...,v ℓ) of distinct vertices such that each pair ( vi−1,vi) is\nconnected by an edge in G. The length of a path is the sum of its edge lengths or simply the numbe rℓ\nof edges if the graph is unweighted. For two vertices v,w, we denote the length of a shortest v-w-path\ninGby dist G(v,w) or dist( v,w) if the graph Gis clear from the context. The first and last vertex v0\nandvℓare called the endsofP. We also say that Pis a path fromv0tovℓor av0-vℓ-path.\nWe assume the reader to be familiar with the big-O notation and basic c oncepts in computational\ncomplexity like NP-completeness and reductions. We refer to the te xtbook by Garey and Johnson [ 14]\nfor an introduction. Throughout this paper, we reduce from 3-Sat,Clique, andMulticolored\nClique, three of the most fundamental problems in theoretical compute r science. We state their\ndefinitions for the sake of completeness.\n3-Sat\nInput: A Boolean formula φin conjunctive normal form where each clause contains at\nmost three literals.\nQuestion: Isφsatisfiable?\nClique\nInput: An undirected graph Gand an integer k.\nQuestion: Is there a clique of size kinG, that is, a set of kvertices that are pairwise\nneighbors?\nMulticolored Clique\nInput: An undirected x-partite graph Gand an integer k≤x.\nQuestion: Is there a clique of size kinG?\nIt is more common to state Multicolored Clique forx=kand, in this case, the partitions of\nthe input graph are often modelled as colors and a clique is called multico lored as it contains exactly\none vertex from each color class. However, it is convenient for us t o allowk≤xand we call a clique\nin this context multicolored if it contains at most one vertex from each color class.\nFor a detailed introduction to parameterized complexity and kerneliz ation, we refer the reader to\nthe text books by Cygan et al. [ 10] and Fomin et al. [ 13]. Aparameterized problem Pis a language\ncontainingpairs( I,ρ) whereIisan instance ofan (unparameterized)problemand ρisan integercalled\ntheparameter . In this paper, the parameter will usually be either the number kof terminal pairs or\nthe upper bound ℓon the number of edges in a solution. A parameterized problem Pisfixed-parameter\ntractable if there exists an algorithm solving any instance ( I,ρ) ofPinf(ρ)·poly(|I|) time, where f\n3is some computable function only depending on ρ. The class XPcontains all parameterized problems\nwhich can be solved in |I|f(ρ)time, that is, in polynomial time if ρis constant. A parameterized\nproblem is said to admit a polynomial kernel , if there is a polynomial-time algorithm that given an\ninstance ( I,ρ) computes an equivalent instance ( I′,ρ′) (called the kernel) such that |I′|+ρ′are upper-\nbounded by a polynomial in ρ. It is known that any parameterized problem admitting a polynomial\nkernel is fixed-parameter tractable and each fixed-parameter t ractable problem is contained in XP.\nAn optimization problem does not ask whether a given instance belong s to a language or not.\nInstead, it asks for an optimal feasible solution (usually some kind of maximum or minimum). In\nthe case of Maximum Vertex-Disjoint Shortest Paths , the task is to compute a maximum size\nsubsetSof [k] such that there exist vertex-disjoint shortest paths Pifor alli∈S. Anα-approximation\nalgorithm for a maximization problem is a polynomial-time algorithm that f or any input returns a\nsolution of size at least OPT/αwhere OPT is the size of an optimal solution S∗. A parameterized α-\napproximation algorithm also returns a solution of size at least OPT/α, but its running time is allowed\nto bef(ρ)·poly(n), where ρis the parameter and fis some computable function only depending on ρ.\nIn this work, we always consider (unparameterized) approximation algorithms unless we specifically\nstate a parameterized running time.\nTo exclude an α-approximation for an optimization problem, one can use the framew orkofapproxi-\nmation-preserving reductions . A strict approximation-preserving reduction is a pair of algorithms —\ncalled the reduction algorithm and the solution-lifting algorithm —that both run in polynomial time\nand satisfy the following. The reduction algorithm takes as input an in stanceIof a problem Land\nproduces an instance I′of a problem L′. The solution-lifting algorithm takes any solution SofI′and\ntransforms it into a solution S∗ofIsuch that if Sis anα-approximationfor I′for some α≥1, thenS∗\nis anα-approximation for I. If a strict approximation-preserving reduction from LtoL′exists and L\nis hard to approximate within some value β, thenLis also hard to approximate within β.\nTheexponential time hypothesis (ETH) introduced by Impagliazzo and Paturi [ 15] states that there\nis someε >0 such that each (unparameterized) algorithm solving 3-Sattakes at least 2εn+o(n)time,\nwherenis the numberofvariablesin the input instance. Astrongerconject ure calledthe gap-ETH was\nindependently introduced by Dinur [ 11] and Manurangsi and Raghavendra [ 21]. It states that there\nexistε,δ >0 such that any (1+ ε)-approximation algorithm for Max 3-Sat1takes at least 2δn+o(n)\ntime.\n3 Approximation\nIn this section, we show that Maximum Vertex-Disjoint Shortest Paths cannot be o(k)-approxi-\nmated in f(k)·poly(n) time and not n1/2−ε-approximated in polynomial time. We complement the\nlatterresultbydevelopinga ⌈√\nℓ⌉-approximationalgorithmthatrunsinpolynomialtime. Westartwith\nareductionbasedon apreviousreductionbyBentertetal.[ 2]. Notehowever,thatthe knownreduction\nis not approximation-preserving. Moreover, our result is tight in th e sense that a k-approximation can\nbe computed in polynomial time by simply connecting any terminal pair b y a shortest path. A ck-\napproximation for any constant c≤1 can also be computed in polynomial time by guessing1\ncterminal\npairs to connect and then using the XP-time algorithm by Bentert et al. [2] to find a solution. Note\nthat since cis a constant, the XP-time algorithm for1\ncterminal pairs runs in polynomial time.\nTheorem 1. Unless FPT =W[1],Maximum Vertex-Disjoint Shortest Paths cannot be ko(1)-\napproximated in f(k)·poly(n)time, and assuming the gap-ETH, it cannot be o(k)-approximated\ninf(k)·poly(n)time. All of these results hold even for subcubic graphs with terminals of degree\nat most two.\nProof.We present a strict approximation-preservingreduction from Multicolored Clique toMax-\nimum Vertex-Disjoint Shortest Paths such that the maximum degree is three and each terminal\nvertex has degree two. Moreover,the maximum number OPT of ver tex-disjoint shortest paths between\nterminalpairswill be equalto the largestclique in the originalinstanc e. The theoremthen followsfrom\nthe fact that a f(k)·poly(n)-timeko(1)-approximationfor Cliquewould imply that FPT=W[1] [ 6,17],\nand af(k)·poly(n)-timeo(k)-approximation for Clique refutes the gap-ETH [ 5], and the fact that\n1Max 3-Sat is a generalization of 3-Satwhere the question is not whether the input formula is satisfi able but rather\nhow many clauses can be satisfied simultaneously.\n41,1\n2,1\n3,14,11,2\n2,2\n3,24,21,3\n2,3\n3,34,31,4\n2,4\n3,44,4s1\nt1s2\nt2s3\nt3s4\nt4s1\n1\ns4\n1\nt1\n1t4\n1u1,2\n1,1v1,2\n1,1\nu2,3\n3,1\nx1,3\n4,4\nFigure 1: An illustration of the reduction from Multicolored Clique toMaximum Vertex-\nDisjoint Shortest Paths .\nLeft side: Example instance for Multicolored Clique withk= 4 colors and n= 4 vertices per\ncolor. A multicolored clique is highlighted (by thick edges).\nRight side: The constructed instance with the four shortest paths corresp onding to the vertices of\nthe clique highlighted. Note that these paths are pairwise disjoint. T he dotted edges (incident to si\nandtivertices) indicate binary trees (where all leaves have distance ⌈log(ν)⌉from the root). Red edges\nindicate paths of length 2 νand blue edges indicate paths of length 2.\nthe textbook reduction from CliquetoMulticolored Clique only increases the number of vertices\nby a quadratic factor and does not change the size of a largest cliqu e in the graph.\nThe reduction is depicted in Figure 1and works as follows. Let G= (V,E) be ak-partite graph\n(or equivalently a graph colored with kcolors where all vertices of any color form an independent\nset) with νvertices of each color. Let Vi={vi\n1,vi\n2,...,vi\nν}be the set of vertices of color i∈[k]\ninG. We start with a terminal pair ( si,ti) for each color iand a pair of vertices ( sj\ni,tj\ni) for each\nvertexvi\nj∈Vi. Next for each color i, we add a binary tree of height ⌈log(ν)⌉with root siand leaves sj\ni\nforvi\nj∈Vi. Analogously, we add a binary tree of the same height with root tiand leaves ti\nj. Next,\nwe add a crossing gadget for each pair of vertices ( vi\nj,va\nb) withi < a. If{vi\nj,va\nb}/∈E, then the gadget\nconsists of four vertices ui,a\nj,b,vi,a\nj,b,xi,a\nj,b, andyi,a\nj,band edges {ui,a\nj,b,vi,a\nj,b}and{xi,a\nj,b,yi,a\nj,b}. If{vi\nj,va\nb} ∈E,\nthen the gadget consists of only two vertices ui,a\nj,bandvi,a\nj,band the edge {ui,a\nj,b,vi,a\nj,b}. For the sake of\nnotational convenience, we will in the latter case also denote ui,a\nj,bbyxi,a\nj,bandvi,a\nj,bbyyi,a\nj,b. To complete\nthe construction, we connect the different gadgets as follows. Fir st, we connect via paths of length\ntwovi,a\nj,bandui,a\nj,b+1for allb < νandyi,a\nj,bandxi,a\nj−1,bfor allj >1. Second, we connect via paths of\nlength two the vertices vi,a\nj,νtoui,a+1\nj,1for allj∈[ν] and all a < kandyi,a\n1,btoxi+1,a\nν,bfor allb∈[ν] and\nalli < a−1. Third, we connect also via paths of length two yi,i+1\n1,btoui+1,i+2\nb,1for alli < k−1 and\nallb∈[ν]. Next, we connect via paths of length 2 νeach vertex sj\nitox1,i\nν,jfor each i >1 andj∈[ν].\nSimilarly, vi,k\nj,νis connected to tj\nivia paths of length 2 ν. Finally, we connect s1\njwithu1,2\nj,1for allj∈[ν]\nandyk−2,k−1\n1,jwithtk\njfor allj∈[ν]. This concludes the construction.\nWe next prove that all shortest si-ti-paths are of the form\nsi−sj\ni−x1,i\nν,j−yi−1,i\n1,j−ui,i+1\nj,1−vi,k\nj,ν−tj\ni−ti (1)\nfor some j∈[ν] and where the s1-t1-paths go directly from sj\nitou1,2\nj,1and thesk-tk-paths go directly\nfromyk−1,k\n1,jtoti\nj. We say that the respective path is the jthcanonical path for color i.\nTo show the above claim, first note that the distance from sito any vertex sj\niis the same\nvaluex=⌈ν⌉for all pairs of indices iandj. Moreover, the same holds for tiandti,j, eachsi-ti-\npath contains at least one vertex sj\niand one vertex sj′\nifor some j,j′∈[ν], and all paths of the form\n5in Equation ( 1) are of length y= 2x+ 4ν+ 3(k−1)ν−2. We first show that each path si-ti-\npath of length at most ycontains an edge of the form yi,i+1\n1,btoui+1,i+2\nb,1. Consider the graph where\nall of these edges are removed. Note that due to the grid-like stru cture, the distance between si\nandxi′,a\nj,bfor any values i′≤i≤a,j, andbis at least x+ 2ν+3(i′−1)ν+ 3(ν−j) ifi=aand at\nleastx+2ν+3(i′−1)ν+3(a−i)ν+3(ν−j)+3bifi < a.2Hence, all shortest si-ti-paths use an edge\nof the form yi,i+1\n1,btoui+1,i+2\nb,1and the shortest path from sj\nito some vertex yi,i+1\n1,bis to the vertex yi,i+1\n1,j.\nNote that the other endpoint of the specified edge is ui,i+1\nj,1and the shortest path to tinow goes via tj\ni\nfor analogous reasons. Thus, all shortest si-ti-paths have the form ( 1).\nWe next prove that any set of pdisjoint shortest paths between terminal pairs ( si,ti) in the\nconstructed graph has a one-to-one correspondence to a multic olored clique of size pfor anyp. For\nthe first direction, assume that there is a set Pof disjoint shortest paths between pterminal pairs.\nLetS⊆[k] be the set of indices such that the paths in Pconnectsiandtifor each i∈S. Moreover,\nletjibe the index such that the shortest si-ti-path in Pis thejithcanonical path for ifor eachi∈S.\nNow consider the set K={vi\nji|i∈S}of vertices in G. Clearly Kcontains at most one vertex of\neach color and is of size pasSis of size p. It remains to show that Kinduces a clique in G. To this\nend, consider any two vertices vi\nji,vi′\nji′∈K. We assume without loss of generality that i < i′. By\nassumption, the jithcanonical path for iand theji′thcanonical path for i′are disjoint. This implies\nthatui,i′\nji,ji′/\\⌉}atio\\slash=xi,i′\nji,ji′as thejithcanonical path for icontains the former and the ji′thcanonical path\nfori′contains the latter. By construction, this means that {vi\nji,vi′\nji′} ∈E. Since the two vertices\nwere chosen arbitrarily, it follows that all vertices in Kare pairwise adjacent, that is, Kinduces a\nmulticolored clique of size p.\nFor the other direction assume that there is a multicolored clique C={vi1\nj1,vi2\nj2,...,vip\njp}of sizep\ninG. We will show that the jqthcanonical path for iqis vertex disjoint from the jrthcanonical path\nforirforallq/\\⌉}atio\\slash=r∈[p]. Letq,rbe two arbitrarydistinct indices in [ p] and let without lossofgenerality\nbeq < r. Note that the two mentioned paths can only overlap in vertices uiq,ir\njq,jr,viq,ir\njq,jr,ciq,ir\njq,jr,oryiq,ir\njq,jr\nand that the jqthcanonical path for iqonly contains vertices uiq,ir\njq,jrandviq,ir\njq,jrand thejrthcanonical\npath for ironly contains xiq,ir\njq,jrandyiq,ir\njq,jr. Moreover, since by assumption viq\njqandvir\njrare adjacent,\nit holds by construction that uiq,ir\njq,jr,viq,ir\njq,jr,xiq,ir\njq,jr, andyiq,ir\njq,jrare four distinct vertices. Thus, we found\nvertex disjoint paths between pdistinct terminal pairs. This concludes the proof of correctness.\nTo finish the proof, observe that the constructed instance has m aximum degree three, all terminal\nvertices are roots of binary trees and therefore have degree tw o, and that the construction can be\ncomputed in polynomial time.\nWe mention in passing that in graphs of maximum degree three with ter minal vertices of degree at\nmost two, two paths are vertex disjoint if and only if they are edge d isjoint. Hence, Theorem 1also\nholds for the edge-disjoint version of Maximum Vertex-Disjoint Shortest Paths .\nWe continue with an unparameterized lowerbound by establishing tha t computing a n1\n2−ε-approxi-\nmation is NP-hard. To the best of our knowledge, this is the first unp arameterized approximation\nlower bound for Maximum Vertex-Disjoint Shortest Paths and no similar bound is known for\nDisjoint Paths . We mention that the reduction is quite similar to the reduction in the p roof for\nTheorem 1.\nTheorem 2. Computing a n1/2−ε-approximation for any ε >0forMaximum Vertex-Disjoint\nShortest Paths is NP-hard.\nProof.It is known that computing a n1−ε-approximation for Cliqueis NP-hard [ 24]. We present an\napproximation-preservingreduction from CliquetoMaximum Vertex-Disjoint Shortest Paths\nbased on Theorem 1. We use basically the same reduction as in Theorem 1but we start from an\ninstance of Clique and have a separate terminal pair for each vertex in the graph. Mo reover, we do\nnot require the binary trees pending from the terminal vertices (t he terminal vertices now have degree\n2We mention that there are some pairs of vertices xi1,a1\nj1,b1andxi2,a2\nj2,b2, where the distance between the two is less\nthan 3(|i1−i2|+|a1−a2|)ν+ 3(|j1−j2|+|b1−b2|). An example is the pair ( x1,2\n1,1=u1,2\n1,1,x1,2\n2,2) in Figure 1with a\ndistance of 4. However, in all examples it holds that i1ν−j1/negationslash=i2ν−j2anda1ν+b1/negationslash=a2ν+b2such that the left side is\neither smaller in both inequalities or larger in both inequa lities. Hence, these pairs cannot be used as shortcuts as the y\nmove “down and left” instead of towards “down and right” in Fi gure1.\n6v1 v2 v3\nv4 v5 v6s1\nt1s2\nt2s3\nt3s4\nt4s5\nt5s6\nt6\nFigure 2: An illustration of the reduction from CliquetoMaximum Vertex-Disjoint Shortest\nPaths.\nLeft side: Example instance for Cliquewith a highlighted solution (by thick edges).\nRight side: The constructed instance with the four shortest paths corresp onding to the solution on\nthe left side highlighted. Note that each shortest si-ti-path uses exactly two of the diagonal edges.\none instead) and neither do we require long induced paths (red edge s in Figure 1). These are instead\npaths with one internal vertex. An illustrationof the modified reduc tion is given in Figure 2. Note that\nthe number of vertices and edges in the graph is at most 3 N2, whereNis the number of vertices in the\ninstance of Clique. Moreover, for each terminal pair ( si,ti), there is exactly one shortest si-ti-path\n(the path that moves horizontally in Figure 2until it reaches the main diagonal, then uses exactly two\nedges on the diagonal, and finally moves vertically to ti).\nWe next prove that for any p, there is a one-to-one correspondence between a set of pdisjoint\nshortest paths between terminal pairs ( si,ti) in the constructed graph and a clique of size pin the\ninput graph. For the first direction, assume that there is a set Pof disjoint shortest paths between p\nterminal pairs. Let S⊆[k] be the set of indices such that the paths in Pconnect siandtifor\neachi∈S. Now consider the set K={vi|i∈S}of vertices in G. Clearly Kcontains pvertices.\nIt remains to show that Kinduces a clique in G. To this end, consider any two vertices vi,vj∈K.\nWe assume without loss of generality that i < j. By assumption, the unique shortest si-ti-path and\nthe unique shortest sj-tj-path are vertex-disjoint. By the description of the shortest pa ths between\nterminal pairs and the fact that siis higher than sjandtiis to the left of tj, it holds that the two\nconsidered paths both visit the region that is to the right of sjand above ti. This implies that two\nedges must be crossing at this position, that is, there are four ver tices in the described region and not\nonly two. By construction, this means that {vi,vj} ∈E. Since the two vertices were chosen arbitrarily,\nit follows that all vertices in Kare pairwise adjacent, that is, Kinduces a clique of size pin the input\ngraph.\nFor the other direction assume that there is a clique C={vi1,vi2,...,v ip}of sizepin the input\ngraph. Wewillshowthattheuniqueshortest siq-tiq-pathisvertex-disjointfromtheuniqueshortest sir-\ntir-path for all q/\\⌉}atio\\slash=r∈[p]. Letq,rbe two arbitrary distinct indices in [ p] and let without loss of\ngenerality be q < r. Note that the two mentioned paths can only overlap in the region th at is to\nthe right of sirand above tiq. Moreover, since by assumption viqandvirare adjacent, it holds by\nconstruction that there are four distinct vertices in the describe d region and the two described paths\nare indeed vertex-disjoint. Thus, we found vertex disjoint paths between pdistinct terminal pairs.\nWe conclude by analyzing the approximation ratio. Note that we tech nically did not prove a\nstrict reduction for the factor n1−εas the number of vertices in the two instances are not the same.\nStill, any n1/2−ε-approximation for Maximum Vertex-Disjoint Shortest Paths corresponds to\na (3N2)1/2−ε=N1−ε′-approximation for Clique for some 0 < ε′<2εand therefore computing\nan1/2−ε-approximation for Maximum Vertex-Disjoint Shortest Paths is NP-hard.\nNote that the maximum degree of the constructed instance is again three and all terminal vertices\nare of degree one. Thus, Theorem 2also holds for the edge disjoint version of Maximum Vertex-\nDisjoint Shortest Paths .\n7We next show that this result is tight, that is, we show how to comput e a√n-approximation for\nMaximum Vertex-Disjoint Shortest Paths in polynomial time. We also show that the same\nalgorithm achieves a ⌈√\nℓ⌉-approximation. Note that we can always assume that ℓ < nas a set\nof vertex-disjoint paths is a forest and the number of edges in a fo rest is less than its number of\nvertices. We mention that this algorithm is basically identical to the be st known (unparameterized)\napproximation algorithm for Maximum Disjoint Paths [18].\nTheorem 3. There is a polynomial-time algorithm for Maximum Vertex-Disjoint Shortest\nPathson directed and weighted graphs that achieves an approximat ion factor of min{√n,⌈√\nℓ⌉}.\nProof.Let OPT be a maximum subset of terminal pairs that can be connecte d by shortest pairwise\nvertex-disjoint paths and let jbe the index of a terminal pair ( sj,tj) such that a shortest ( sj,tj)-path\ncontains a minimum number of arcs. We can compute the index jas well as a shortest sj-tj-path\nwith a minimum number of arcs by running a folklore modification of Dijks tra’s algorithm from each\nterminal vertex si.3Letℓjbe the number of arcs in the found path. Our algorithm iteratively pic ks\nthe shortest sj-tj-path using ℓjarcs, removes all involved vertices from the graph, recomputes t he\ndistance between all terminal pairs, removes all terminal pairs who se distance increased, updates the\nindexj, and recomputes ℓj. For the analysis of our algorithm, we assume that we know an optima l\nsolution and pretend to also remove a terminal pair ( si,ti) if the minimum number ℓiof arcs in a\nshortest si-ti-path increases. Moreover, we distinguish whether ℓj+1≤min(√n,⌈√\nℓ⌉) or not.\nWhileℓj+1≤min(√n,⌈√\nℓ⌉), note that we removedat most ℓj+1 terminals pairs in OPT. Hence,\nifℓj+1≤min(√n,⌈√\nℓ⌉) holds at every stage, then we connected at least |OPT|/min(√n,⌈√\nℓ⌉)terminal\npairs, that is, we found a min(√n,⌈√\nℓ⌉)-approximation.\nSo assume that at some point ℓj>min(√n,⌈√\nℓ⌉) and let xbe the number of terminal pairs\nthat we already connected by disjoint shortest paths. By the arg ument above, we have removed at\nmostxmin(√n,⌈√\nℓ⌉) terminal pairs from OPT thus far. We now make a case distinction wh ether\nor not√n≤ ⌈√\nℓ⌉. Ifℓj+ 1>⌈√\nℓ⌉ ≥√n, then we note that all remaining paths in OPT contain\nat least√nvertices each and since the paths are vertex-disjoint, there can be at most√npaths left\nin OPT. Hence, we can infer that |OPT| ≤(x+1)·√n. Consequently, even though we might remove\nall remaining terminal pairs in OPT by connecting sjandtj, this is still a√n-approximation (and\na⌈√\nℓ⌉-approximation as we assumed ⌈√\nℓ⌉ ≥√n).\nIfℓj+1>√n≥ ⌈√\nℓ⌉, then we note that all remaining paths in OPT contain at least ℓj>⌈√\nℓ⌉−1\nedges each. Moreover, since ℓjand⌈√\nℓ⌉are integers, each path contains at least ⌈√\nℓ⌉edges each.\nSince all paths in OPT contain by definition at most ℓedges combined, the number of paths in OPT is\nat most ℓ/⌈√\nℓ⌉≤ ⌈√\nℓ⌉. Hence, we can infer in that case that |OPT| ≤(x+1)·⌈√\nℓ⌉. Again, even if we\nremove all remaining terminal pairs in OPT by connecting sjandtj, this is still a ⌈√\nℓ⌉-approximation\n(and a√n-approximation as we assumed√n≥ ⌈√\nℓ⌉). This concludes the proof.\n4 Exact Algorithms\nIn this section, we show that Maximum Vertex-Disjoint Shortest Paths is fixed-parameter\ntractable when parameterized by ℓ, but it does not admit a polynomial kernel. The proof of our first\ntheorem uses the color-coding technique of Alon, Yuster, and Zwick [ 1]. Imagine we are searching for\nsome structure of size kin a graph. The idea of color coding is to color the vertices (or edges) of the\ninput graphwith aset of kcolorsandthen only searchforcolorfulsolutions, that is, struct uresin which\nall vertices have distinct colors. Of course, this might not yield an op timal solution, but by trying\nenough different random colorings, one can often get a constant e rror probability in f(k)·poly(n)\ntime. Using the following result by Naor, Schulman, and Srinivasan [ 22], this can also be turned\ninto a deterministic algorithm showing that the problem is fixed-param eter tractable. The result\nstates that for any n,k≥1, one can construct an ( n,k)-perfect hash family of size ekkO(logk)log(n)\ninekkO(logk)nlog(n) time. An ( n,k)-perfect hash family Fis a family of functions from [ n] to [k] such\nthat for every set S⊆[n] with|S| ≤k, there exists a function f∈ Fsuch that fcolors all vertices\ninSwith distinct colors.\n3The standard Dijkstra’s algorithm is modified by assigning t o each vertex a pair of labels: the distance from the\nterminal and the number of arcs in the corresponding path; th en the pairs of labels are compared lexicographically.\n8Theorem 4. Maximum Vertex-Disjoint Shortest Paths on weighted and directed graphs can\nbe solved in 2O(ℓ)poly(n)time.\nProof.Let (G,w,(s1,t1),...,(sk,tk),p,ℓ) be an instance of Maximum Vertex-Disjoint Shortest\nPaths. We assume that p≤ℓas we have a trivial no-instance otherwise. Notice that the total n umber\nof vertices in a (potential) solution with ppaths is at most ℓ+p. We use the color-coding technique\nof Alon, Yuster, and Zwick [ 1]. We color the vertices of Guniformly at random using p+ℓcolors (the\nset of colors is [ ℓ+p]) and observe that the probability that all the vertices in the paths in a solution\nhave distinct colors is at least(p+ℓ)!\n(p+ℓ)(p+ℓ)≥e−(p+ℓ). We say that a solution to the considered instance is\ncolorfulif distinct paths in the solution have no vertices of the same color. No te that we do not require\nthat the vertices within a path in the solution are colored by distinct c olors. The crucial observations\nare that any colorful solution is a solution and the probability of the e xistence of a colorful solution for\na yes-instance of Maximum Vertex-Disjoint Shortest Paths is at least e−(p+ℓ)as any solution\nin which all vertices receive distinct colors is a colorful solution.\nWe use dynamic programming over subsets of colors to find a colorfu l solution. More precisely, we\nfind the minimum number of arcs in a collection C={Pi}i∈Sofppairwise vertex-disjoint paths for\nsomeS⊆[k] satisfying the conditions: (i) for each i∈S, the path Piis a shortest path from sitoti\nand (ii) there are no vertices of distinct paths of the same color.\nFor a subset X⊆[p+ℓ] of colors and a positive integer r≤p, we denote by f[X,r] the minimum\ntotal number of arcs in rshortest paths connecting distinct terminal pairs such that the p aths contains\nonlyverticesofcolorsin Xandtherearenoverticesofdistinctpathsofthesamecolor. Wese tf[X,r] =\n∞if such a collection of rpaths does not exist.\nTo compute f, ifr= 1, then let W⊆Vbe the subset of vertices colored by the colors in X.\nWe use Dijkstra’s algorithm to find the set I⊆[k] of all indices i∈[k] such that the lengths of the\nshortest si-ti-paths in GandG[W] are the same. If I=∅, then we set f[X,1] =∞. Assume that\nthis is not the case. Then, we use the variant of Dijkstra’s algorithm mentioned in Theorem 3to find\nthe index i∈Iand a shortest si-ti-pathPinG[W] with a minimum number of arcs. Finally, we\nsetf[X,1] to be equal to the number of arcs in P.\nForr≥2, we compute f[X,r] for each X⊆[p+ℓ] using the recurrence relation\nf[X,r] = min\nY⊂X{f[X\\Y,r−1]+f[Y,1]}. (2)\nThe correctness of computing the values of f[X,1] follows from the description and the correctness\nof recurrence ( 2) follows from the condition that distinct paths should not have vert ices of the same\ncolor.\nWe compute the values f[X,r] in order of increasing r∈[p]. Since computing f[Y,1] for a given\nsetYof colors can be done in polynomial time, we can compute all values in ov erall 3p+ℓpoly(n) time.\nOnce all values f[X,r] arecomputed, we observethat acolorful solution exists if and on ly iff[S,p]≤ℓ.\nIf there is a colorful solution, then we conclude that ( G,w,(s1,t1),...,(sk,tk),p,ℓ) is a yes-instance\nofMaximum Vertex-Disjoint Shortest Paths . Otherwise, we discard the considered coloring\nand try another random coloring and iterate. If we fail to find a solu tion after executing N=⌈ep+ℓ⌉\niterations, we obtain that the probability that ( G,w,(s1,t1),...,(sk,tk),p,ℓ) is a yes-instance is at\nmost (1−1\nep+ℓ)ep+ℓ≤e−1. Thus, we return that ( G,w,(s1,t1),...,(sk,tk),p,ℓ) is a no-instance with\ntheerrorprobabilityupperboundedby e−1<1. Sincetherunningtimeineachiterationis3p+ℓpoly(n)\nandp≤ℓ, the total running time is in 2O(ℓ)poly(n).\nThe above algorithm can be derandomized using the results of Naor, Schulman, and Srinivasan [ 22]\nby replacing random colorings by prefect hash families. We refer to t he textbook by Cygan et al. [ 10]\nfor details on this common technique.\nThe FPT result of Theorem 4immediately raises the question about the existence of a poly-\nnominal kernel. To show that a parameterized problem Pdoes presumably not admit a polynomial\nkernel, one can use the framework of cross-compositions . Given an NP-hard problem L, a polynomial\nequivalence relation Ron the instances of Lis an equivalence relation such that (i) one can decide\nfor any two instances in polynomial time whether they belong to the s ame equivalence class, and\n(ii) for any finite set Sof instances, Rpartitions the set into at most max I∈Spoly(|I|) equivalence\nclasses. Given an NP-hard problem L, a parameterized problem P, and a polynomial equivalence\nrelation Ron the instances of L, an OR-cross-composition of LintoP(with respect to R) is an\n9s3\nu1\nu2\nt3s1\nt1s2\nt2\nFigure 3: An example of the construction in the proof of Proposition 1for the input formula\n(x1∨x2∨x3)∧(x1∨x2∨x3).\nalgorithm that takes qinstances I1,I2,...,I qofLbelonging to the same equivalence class of Rand\nconstructsin poly(/summationtextq\ni=1|Ii|) time an instance ( I,ρ) ofPsuchthat (i) ρis polynomiallyupper-bounded\nby max i∈[q]|Ii|+log(q), and (ii) ( I,ρ) is a yes-instanceof Pif and only if at least one of the instances Ii\nis a yes-instance of L. If a parameterized problem admits an OR-cross-composition, the n it does not\nadmit a polynomial kernel unless NP ⊆coNP/poly [ 4].\nIn order to exclude a polynomial kernel, we first show that a special case ofMaximum Vertex-\nDisjoint Shortest Paths remains NP-hard. We call this special case Layered Vertex-Disjoint\nShortest Paths and it is the special case of Vertex-Disjoint Shortest Paths where all edges\nhave weight 1 and the input graph is layered, that is, there is a partit ion of the vertices into (dis-\njoint) sets V1,V2,...,V λsuch that all edges {u,v}are between two consecutive layers, that is u∈Vi\nandv∈Vi+1oru∈Viandv∈Vi−1for some i∈[λ]. Moreover, each terminal pair ( si,ti) satisfies\nthatsi∈V1,ti∈Vλ, and each shortest path between the two terminals is monotone , that is, it\ncontains exactly one vertex of each layer. Layered Vertex-Disjoint Shortest Paths is formally\ndefined as follows.\nLayered Vertex-Disjoint Shortest Paths\nInput: Aλ-layered graph G= (V,E) with a λ-partition {V1,V2,...,V λ}of the ver-\ntex set, terminal pairs ( s1,t1),(s2,t2),...,(sk,tk) withsi∈V1,ti∈Vλ,\nand dist( si,ti) =λ−1 for alli∈[λ].\nQuestion: Is there a collection C={Pi}i∈[k]of pairwise vertex-disjoint paths such that Pi\nis ansi-ti-path of length λ−1 for all i∈[k]?\nWe now prove that Layered Vertex-Disjoint Shortest Paths is NP-complete.\nProposition 1. Layered Vertex-Disjoint Shortest Paths is NP-complete.\nProof.We focus on the NP-hardness as Layered Vertex-Disjoint Shortest Paths is a special\ncase ofVertex-Disjoint Shortest Paths and therefore clearly in NP. We reduce from 3-Sat.\nThe main part of the reduction is a selection gadget. The gadget con sists of a set Uofn+ 1 ver-\nticesu0,u1,...,u nand between each pair of consecutive vertices ui−1,ui, there are two paths with m\ninternal vertices each. Let the set of vertices be Vi={vi\n1,vi\n2,...,vi\nm}andWi={wi\n1,wi\n2,...,wi\nm}.\nThe set of edges in the selection gadget is\nE={{ui−1,vi\n1},{ui−1,wi\n1},{vi\nm,ui},{wi\nm,ui} |i∈[n]}\n∪{{vi\nj,vi\nj+1},{wi\nj,wi\nj+1} |i∈[n]∧j∈[m−1]}.\nThe constructed instance will have m+1 terminal pairs and is depicted in Figure 3. We set sm+1=u0\nandtm+1=unand we will ensure that any shortest sm+1-tm+1-path contains all vertices in Uand\nfor each i∈[n] either all vertices in Vior all vertices in Wi. These choices will correspond to setting\ntheithvariable to either true or false. Additionally, we have a terminal pair ( sj,tj) for each clause Cj.\n10There are (up to) three disjoint paths between sjandtj, each of which is of length n·(m+1). These\npaths correspondto which literal in the clause satisfies it. For each of these paths, let xibe the variable\ncorresponding to the path. If xiappears positively in Cj, then we identify the ( i−1)(m+1)+j+1st\nvertex in the path with wi\njand ifxiappears negatively, then we identify the vertex with vi\nj. Note that\nthe constructed instance is ( m+1)n-layered and that once any monotone path starting in sm+1leaves\nthe selection gadget, it cannot end in tm+1as any vertex outside the selection gadget has degree at\nmost two and at the end of these paths are only terminals t1,t2,...,tm.\nSince the construction clearly runs in polynomial time, we focus on th e proof of correctness. If\nthe input formula is satisfiable, then we connect all terminal pairs as follows. Let βbe a satisfying\nassignment. The terminal pair ( sm+1,tm+1) is connected by a path containing all vertices in Uand for\neachi∈[n], ifβassigns the ithvariable to true, then the path contains all vertices in Viand otherwise\nall vertices in Wi. For each clause Cj, letxijbe variable in Cjwhichβuses to satisfy Cj(if multiple\nsuch variables exist, we choose any one). By construction, there is a path associated with xijthat\nconnects sjandtjand only uses one vertex in Wiifxijappears positively in Cjand a vertex in Vi,\notherwise. Since eachvertexin ViandWiisonlyassociatedwith atmostonesuchpath, wecanconnect\nall terminal pairs. For the other direction assume that all m+1 terminal pairs can be connected by\ndisjoint shortest paths. As argued above, the sm+1-tm+1-path stays in the selection gadget. We define\na truth assignment by assigning the ithvariable to true if and only if the sm+1-tm+1-path contains the\nvertices in Vi. For each clause Cj, we look at the neighbor of sjin the solution. This vertex belongs\nto a path of degree-two vertices that at some point joins the selec tion gadget. By construction, the\nvertex where this happens is not used by the sm+1-tm+1-path, which guarantees that Cjis satisfied\nby the corresponding variable. Since all clauses are satisfied by the same assignment, the formula is\nsatisfiable and this concludes the proof.\nWith the NP-hardness of Layered Vertex-Disjoint Shortest Paths at hand, we can now\nshow that it does not admit a polynomial kernel when parameterized byℓby providing an OR-cross-\ncomposition from its unparameterized version to the version param eterized by ℓ.\nTheorem 5. Layered Vertex-Disjoint Shortest Paths parameterized by ℓ=k·(λ−1)does\nnot admit a polynomial kernel unless NP ⊆coNP/poly.\nProof.We present an OR-cross-compositionfrom Layered Vertex-Disjoint Shortest Paths into\nLayered Vertex-Disjoint Shortest Paths parameterizedby ℓ. Tothisend, assumewearegiven t\ninstances of Layered Vertex-Disjoint Shortest Paths all of which have the same number λof\nlayers and the same number kof terminal pairs. Moreover, we assume that tis some power of two.\nNote that we can pad the instance with at most ttrivial no-instances to reach an equivalent instance\nin which the number of instances is a power of two and the size of all ins tances combined has at most\ndoubled.\nThe main ingredient for our proof is a construction to merge two inst ances into one. The construc-\ntion is depicted in Figure 4. We first proof that the constructed instance is a yes-instance if and only\nif at least one of the original instances was a yes-instance. Afterw ards, we will show how to use this\nconstruction to get an OR-cross-composition for all tinstances.\nTo show that the construction works correctly, first assume tha t one of the two instances original\ninstancesisayes-instance. Sinceboth casesarecompletelysymme trical, assumethat thereareshortest\ndisjoint paths between all terminal pairs ( si\na,ti\na) for alla∈[k] inGi. Then, we can connect all terminal\npairs (sb,tb) by using the unique shortest paths between sbandsi\nband between ti\nbandtbfor allb∈[k]\ntogether with the solution paths inside Gi. Now assume that there is a solution in the constructed\ninstance, that is, there are pairwise vertex-disjoint shortest pa ths between all terminal pairs ( sb,tb)\nfor allb∈[k]. First assume that the s1-t1-path passes through Gi. Then, this path uses the unique\nshortest path from ti\n1toti. Note that this path blocks all paths between tj\nband vertices in Gjfor\nallb/\\⌉}atio\\slash= 1. Thus, all paths have to pass through the graph Gi. Note that the only possible way to route\nvertex-disjoint paths from all s-terminals to all siterminals and from all ti-terminals to all t-terminals\nis to connect satosi\naandti\natotafor alla∈[k]. This implies that there is a solution that contains\nvertex-disjoint shortest paths between si\naandti\nainGifor alla∈[k], that is, at least one of the two\noriginal instances is a yes-instance. The case where the s1-t1-path passes through Gjis analogous\nsince the only monotone path from s1to a vertex in Gjis the unique shortest s1-sj\n1-path and this path\nblocks all monotone paths from sato vertices in Gifor alla/\\⌉}atio\\slash= 1.\n11s1s2s3\n...sk\nsi\n1si\n2si\n3...si\nk\nGi\nti\n1ti\n2ti\n3...ti\nksj\n1sj\n2sj\n3...sj\nk\nGj\ntj\n1tj\n2tj\n3...tj\nk\nt1t2t3\n...tk\nFigure 4: The construction to merge two instances of Layered Vertex-Disjoint Shortest Paths\ninto one equivalent instance. The dotted edges can be read as regu lar edges for k= 4 and indicate\nwhere additional vertices and edges have to be added for more ter minal pairs. Note that the height of\na vertex in the drawing does not indicate its layer as dotted edges dis tort the picture.\nNote that the constructed graph is layered and that the number o f layers is λ+2k. Moreover, the\nsize of the new instance is in O(|Gi|+|Gj|+k2). To complete the reduction, we iteratively half the\nnumber of instances by partitioning all instances into arbitrary pair s and merge the two instances in a\npair into one instance. After log titerations, we are left with a single instance which is a yes-instance\nif and only if at least one of the toriginal instances is a yes-instance. The size of the instance is\ninO(/summationtext\ni∈[t]|Gi|+t·k2) which is clearly polynomial in/summationtext\ni∈[t]|Gi|as each instance contains at least k\nvertices. Moreover, the parameter ℓin the constructed instance is k·(λ−1) + 2klogt, which is\npolynomial in |Gi|+logtfor each graph GiasGicontains at least one vertex in each of the λlayers\nand at least kterminal vertices. Thus, all requirements of an OR-cross-compo sition are met and this\nconcludes the proof.\nNote that since Layered Vertex-Disjoint Shortest Paths is a special case of Maximum\nVertex-Disjoint Shortest Paths , Theorem 5also excludes polynomial kernels for Maximum\nVertex-Disjoint Shortest Paths parameterized by ℓ.\n5 Conclusion\nIn this paper, we studied Maximum Vertex-Disjoint Shortest Paths . We show that there is\nnon1/2−ε-approximation in polynomial time unless P = NP. Moreover, if FPT /\\⌉}atio\\slash=W[1] or assuming\nthe stronger gap-ETH, we show that there are no non-trivial app roximations for Maximum Vertex-\nDisjoint Shortest Paths inf(k)·poly(n) time. When parameterized by ℓ, there is simple ⌈√\nℓ⌉-\napproximation in polynomial time that matches the n1/2lower bound as ℓ < n. Finally, we showed\nthatMaximum Vertex-Disjoint Shortest Paths is fixed-parametertractablewhen parameterized\nbyℓ, but it does not admit a polynomial kernel.\n12A way to combine approximation algorithms and the theory of (polyno mial) kernels are lossy\nkernels[20]. Since the exact definition is quite technical and not relevant for th is work, we only give\nan intuitive description. An α-approximate kernel or lossy kernel for an optimization problem is a\npair of algorithms that run in polynomial time which are called pre-processing algorithm andsolution-\nlifting algorithm . The pre-processing algorithm takes as input an instance ( I,ρ) of a parameterized\nproblem Pand outputs an instance ( I′,ρ′) ofPsuch that |I′|+ρ′≤g(ρ) for some computable\nfunction g. The solution-lifting algorithm takes any solution Sof (I′,ρ′) and transforms it into a\nsolution S∗of (I,ρ) such that if Sis anc-approximation for ( I′,ρ′) for some γ≥1, thenS∗is\nanγ·α-approximation for ( I,ρ). If size of the kernel is g(ρ) and if gis constant or a polynomial,\nthen we call it a constant-size or polynomial-size α-approximate kernel, respectively. It is known\nthat a (decidable) parameterized problem admits a constant-size a pproximate α-kernel if and only if\nthe unparameterized problem associated with Pcan beα-approximated (in polynomial time) [ 20].\nMoreover, any (decidable) parameterize problem admits an α-approximate kernel (of arbitrary size) if\nand only if the problem can be α-approximated in f(ρ)·poly(|I|) time.\nIn terms of lossy kernelization, our results imply that there are no n on-trivial lossy kernels for the\nparameter k. For the parameter ℓ, Theorem 3implies a constant-size lossy kernel for α∈Ω(√\nℓ) and\nTheorem 4implies an f(ℓ)-size lossy kernels for any α≥1. This leaves the following gap which we\npose as an open problems.\nOpen Problem 1. Are there any poly(ℓ)-size lossy kernels for Maximum Vertex-Disjoint Short-\nest Paths withα∈o(√\nℓ)(or even constant α)?\nReferences\n[1] Noga Alon, Raphael Yuster, and Uri Zwick. Color-coding. Journal of the ACM , 42(4):844–856,\n1995.\n[2] Matthias Bentert, Andr´ e Nichterlein, Malte Renken, and Philipp Z schoche. Using a geometric\nlens to find k-disjoint shortest paths. SIAM Journal on Discrete Mathematics , 37(3):1674–1703,\n2023.\n[3] Kristof Berczi and Yusuke Kobayashi. The directed disjoint sho rtest paths problem. In Proceed-\nings of the 25th Annual European Symposium on Algorithms (ES A), pages 13:1–13:13. Schloss\nDagstuhl–Leibniz-Zentrum fuer Informatik, 2017.\n[4] Hans L. Bodlaender, Bart M. P. Jansen, and Stefan Kratsch. K ernelization lower bounds by\ncross-composition. SIAM Journal on Discrete Mathematics , 28(1):277–305, 2014.\n[5] Parinya Chalermsook, Marek Cygan, Guy Kortsarz, Bundit Laek hanukit, Pasin Manurangsi,\nDanupon Nanongkai, and Luca Trevisan. From Gap-ETH to FPT-inap proximability: Clique,\ndominating set, and more. In Proceedings of the 58th IEEE Annual Symposium on Foundation s\nof Computer Science (FOCS) , pages 743–754. IEEE Computer Society, 2017.\n[6] Yijia Chen, Yi Feng, Bundit Laekhanukit, and Yanlin Liu. Simple comb inatorial construction\nof the ko(1)-lower bound for approximating the parameterized k-clique. CoRR, abs/2304.07516,\n2023.\n[7] Julia Chuzhoy and Shi Li. A polylogarithmic approximation algorithm f or edge-disjoint paths\nwith congestion 2. Journal of the ACM , 63(5):45:1–45:51, 2016.\n[8] Julia Chuzhoy, David H. K. Kim, and Rachit Nimavat. Almost polynomia l hardness of node-\ndisjoint paths in grids. Theory of Computing , 17:1–57, 2021.\n[9] Julia Chuzhoy, David H. K. Kim, and Rachit Nimavat. New hardness r esults for routing on\ndisjoint paths. SIAM Journal on Computing , 51(2):17–189, 2022.\n[10] Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtan ov, D´ aniel Marx, Marcin\nPilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms . Springer, 2015.\n13[11] Irit Dinur. Mildly exponential reduction from gap 3SAT to polynom ial-gap label-cover. Electronic\nColloquium on Computational Complexity , TR16-128, 2016.\n[12] Tali Eilam-Tzoreff. The disjoint shortest paths problem. Discrete Applied Mathematics , 85(2):\n113–138, 1998.\n[13] FedorV.Fomin, DanielLokshtanov,SaketSaurabh,andMeira vZehavi. Kernelization . Cambridge\nUniversity Press, Cambridge, 2019.\n[14] M. R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of\nNP-Completeness . W. H. Freeman, 1979.\n[15] Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-sat.Journal on Computer\nand Syststem Sciences , 62(2):367–375, 2001.\n[16] Richard M. Karp. On the computational complexity of combinato rial problems. Networks , 5(1):\n45–68, 1975.\n[17] Karthik C. S. and Subhash Khot. Almost polynomial factor inapp roximability for parameterized\nk-clique. In Proceedings of the 37th Computational Complexity Conferen ce (CCC) , pages 6:1–6:21.\nSchloss Dagstuhl - Leibniz-Zentrum f¨ ur Informatik, 2022.\n[18] Jon M. Kleinberg. Approximation algorithms for disjoint paths problems . PhD thesis, Mas-\nsachusetts Institute of Technology, 1996.\n[19] William Lochet. A polynomial time algorithm for the k-disjoint shortest paths problem. In\nProceedings of the 32nd ACM-SIAM Symposium on Discrete Algo rithms (SODA) , pages 169–178.\nSIAM, 2021.\n[20] Daniel Lokshtanov, Fahad Panolan, M. S. Ramanujan, and Sak et Saurabh. Lossy kernelization.\nIn Hamed Hatami, Pierre McKenzie, and Valerie King, editors, Proceedings of the 49th Annual\nACM SIGACT Symposium on Theory of Computing (STOC) , pages 224–237. ACM, 2017.\n[21] Pasin Manurangsi and Prasad Raghavendra. A birthday repet ition theorem and complexity of\napproximating dense csps. In Proceedings of the 44th International Colloquium on Automa ta,\nLanguages, and Programming (ICALP) , pages 78:1–78:15. Schloss Dagstuhl - Leibniz-Zentrum\nf¨ ur Informatik, 2017.\n[22] Moni Naor, Leonard J. Schulman, and Aravind Srinivasan. Splitt ers and near-optimal deran-\ndomization. In Proceedings of the 36th Annual Symposium on Foundations of C omputer Science\n(FOCS), pages 182–191. IEEE Computer Society, 1995.\n[23] Neil Robertson and Paul D. Seymour. Graph minors. XIII: The disjoint paths problem. Journal\nof Combinatorial Theory. Series B , 63(1):65–110, 1995.\n[24] DavidZuckerman. Lineardegreeextractorsandtheinapprox imabilityofmaxcliqueandchromatic\nnumber. Theory of Computing , 3(1):103–128, 2007.\n14"    },    {        "title": "2402.15419v1.Uncertainty_Quantification_in_Atomistic_Simulations_of_Silicon_using_Interatomic_Potentials.pdf",        "content": "Uncertainty Quantification in Atomistic Simulations of Silicon using\nInteratomic Potentials\nI. R. Best,1T. J. Sullivan,2, 1and J. R. Kermode1\n1)School of Engineering, University of Warwick, Coventry, CV4 7AL, United Kingdom\n2)Warwick Mathematics Institute, University of Warwick, Coventry, CV4 7AL,\nUnited Kingdom\n(Dated: 26 February 2024)\nAtomistic simulations often rely on interatomic potentials to access greater time- and length- scales than those\naccessible to first principles methods such as density functional theory (DFT). However, since a parameterised\npotential typically cannot reproduce the true potential energy surface of a given system, we should expect a\ndecrease in accuracy and increase in error in quantities of interest calculated from simulations. Quantifying\nthe uncertainty on the outputs of atomistic simulations is thus an important, necessary step so that there is\nconfidence in results and available metrics to explore improvements in said simulations. Here, we address this\nresearch question by forming ensembles of Atomic Cluster Expansion (ACE) potentials, and using Conformal\nPrediction with DFT training data to provide meaningful, calibrated error bars on several quantities of interest\nfor silicon: the bulk modulus, elastic constants, relaxed vacancy formation energy, and the vacancy migration\nbarrier. We evaluate the effects on uncertainty bounds using a range of different potentials and training sets.\nI. INTRODUCTION\nAn interatomic potential (IP) is any function V(x)\nwhich maps atomic positions xto the potential energy\nsurface (PES) of a given material. They are constructed\nwith the goal of efficiently computing energies and forces\non atoms, but at the cost of reduced accuracy when com-\npared to reference quantum-mechanical methods such as\nDensity Functional Theory (DFT). This loss of accuracy\nis regrettable but necessary in the computation of ex-\npensive Quantities of Interest (QoIs), which would oth-\nerwise be inaccessible due to the unfeasible scaling of\nthese ab-initio techniques with system size or simulation\ntime. Historically, IPs with fixed functional forms were\ntuned on a per-species and/or per-QoI basis1,2with the\nspecific form of the potential usually inspired by physical\nor chemical intuition. While these IPs are usually suc-\ncessful on the quantities and chemical systems on which\nthey are trained, they typically have limited transferabil-\nity to new species or quantities since the parameters and\nform of the model are highly specific, and targeting an-\nother QoI or material requires either re-tuning the model\nparameters, or building a new model from scratch.\nThe advent of Machine Learning Interatomic Po-\ntentials (MLIPs), trained on DFT or other high-\naccuracy data, explores the construction of systemati-\ncally improvable approximations to the PES of a given\nmaterial. These potentials3–7promise accuracy and\ntransferability8between different QoIs for an appropri-\nately chosen and trained potential. However, it remains\na difficult task to identify where a MLIP is extrapolating\noutside of observed training data, since the PES land-\nscape is complex for even simple materials. Accurate\nestimation of the PES and transferability between QoIs\nis only likely in regions where a model is close to, or\ninterpolating between, training data.\nMLIPs are frequently leveraged within atomistic simu-\nlations to take advantage of the aforementioned accuracyand transferability, coupled with the availability of ab-\ninitio materials science databases9,10on which to train.\nA high-accuracy input dataset from a more fundamental\nlevel of theory is used to train a potential, and this data\nis typically split into training and test sets - the perfor-\nmance of an MLIP is often evaluated using metrics like\ntrain and test error. These measures can be useful in\ndiagnosing features of a trained MLIP, an example being\nto characterise the level of over-fitting to training data;\nnevertheless, these error metrics should not be treated\nas uncertainty measures. Even a machine learned po-\ntential which achieves low training and test errors can\nperform poorly when extrapolating outside of previously\nobserved configurations. These error metrics are also cal-\nculated on quantities already in the training set, and not\non the QoIs which the model will eventually be used to\npredict: there is no clear connection or conversion be-\ntween these metrics and an uncertainty in a given QoI.\nWhilst we might hope that a model with low train and\ntest errors will perform well on a range of different QoIs,\nthis is not a guarantee since our train and test datasets\ndo not cover all possible atomic configurations.\nUncertainty quantification (UQ) in the context of\natomistic simulation must therefore account for several\ndifferent sources of uncertainty - for example, noise on\nunderlying training data, parametric uncertainty, and\nmodel form error - and combine these different sources\nin a statistically meaningful way. Previous work in-\nspired by Bayesian frameworks has constructed ensem-\nbles of fixed form IPs by sampling coefficients from a cost\nlandscape11,12, weighting different possible choices of co-\nefficients with a fictitious temperature to assess the para-\nmetric uncertainty in fixed form and machine-learned\nIPs, and analysing the ‘sloppiness’ in model parameters\nthrough different statistical lenses13. For MLIPs, several\ndifferent frameworks have been used to estimate uncer-\ntainties. As a non-exhaustive list of examples: Gaus-\nsian Process Regression based UQ via cross validation14arXiv:2402.15419v1  [cond-mat.mtrl-sci]  23 Feb 20242\nand subsampling15, uncertainties using ACE potentials\nto drive active learning of datasets16,17, ensembles with\nneural networks18(NNs), dropout NNs with a Bayesian\ninterpretation19, and conformal prediction with NNs on\nquantities within the training set as a comparison to\nother methods20and to calibrate uncertainties in build-\ning potentials with MD21.\nIn an attempt to build upon these previous approaches\nto UQ in atomistic simulation, in this paper we aim to\nprovide robust uncertainty estimates on a range of QoIs\ncalculated with a variety of different Atomic Cluster Ex-\npansion (ACE)6potentials for pure silicon, trained on a\nreference DFT dataset22. The choice of potential family\nto ACE is made to take advantage of the linear struc-\nture of the IP, which is advantageous both for evaluation\nspeed23and to enable efficient statistical sampling - it\nis conceptually easier to place priors on coefficients of\na linear model, rather than considering classical IPs or\nsome differently architectured MLIP, where there may\nexist some hierarchy of importance in coefficients.\nWe setup our linear ACE basis inside a Bayesian in-\nverse problem, which after accounting for parametric un-\ncertainty and noise within our training data, provides a\nposterior distribution of coefficients from which we can\nform an ensemble of ACE potentials - here the speed of\nenergy and force evaluations becomes important, since\nif we desire statistical significance we require many eval-\nuations of a given QoI, which can translate to an ex-\ntremely large amount of atomic observation calculations.\nWe push samples from the ensemble through the calcu-\nlation of a QoI, and calibrate the observed uncertainties\nusing conformal prediction24. This workflow can be read-\nily translated to other classes of MLIPs, different QoIs\nand different sources of training data.\nII. METHODOLOGY\nA. ACE Potentials\nThe Atomic Cluster Expansion exploits a body order\nexpansion to construct potentials of a desired correlation\norder ν(= body order −1) which are naturally invari-\nant to translations, rotations and permutations of same\nspecies atoms. An outline of the ACE construction is\nprovided below; for more details, see e.g.6,25,26.\nThe core of an ACE model is the projection of the\natomic density ρ\nρ(r) =X\nj̸=iδ(r−rji) (1)\nonto the one-particle basis φv\nφv(rij) =Rnl(rij)Ym\nl(ˆr) (2)\nwhere v= (nlm) is a multi-index of the principal, or-\nbital and magnetic quantum numbers, rij=rijˆris the\ndistance vector of magnitude rijand direction ˆ r, andRnl(rij) and Ym\nl(ˆr) are the radial functions and the\nspherical harmonics respectively. These φare evaluated\nfor the N(i) neighbours of a given atom iwithin some\ncutoff region. To then construct a description/basis of\nthe atomic environment which is invariant to permuta-\ntions, we perform a summation of all the φwithin this\ncutoff\nAiv=X\nj∈N(i)φv(rij). (3)\nand then form the product A-basis\nAi,v=νY\nγAivγ (4)\nwithv= (v1, v2, ..., v ν) and νthe correlation order, or\nequivalently the ( ν+ 1) body order of the expansion.\nPerforming a body-order expansion of the site energy\nVi(which can also be done equivalently with other quan-\ntities) with this Abasis leads to\nVi=V(0)\ni+X\nv˜c(1)\nvAiv+v1≥v2X\nv1v2˜c(2)\nv1v2Aiv1Aiv2\n+v1≥v2≥v3X\nv1v2v3˜c(3)\nv1v2v3Aiv1Aiv2Aiv3+... .(5)\nIn order for this basis to be rotationally invariant, it\nis necessary to average over the three-dimensional rota-\ntions and remove those basis functions which are zero or\nlinearly dependent, i.e. those encoding the same infor-\nmation, which leads to the Bbasis\nBi,v=X\nv′Cvv′Ai,v′ (6)\nwithCvv′the sparse Clebsch-Gordon coefficient matrix.\nAn energy evaluation using ACE is then given by\nV(c,R) =X\nici,vBi,v(R) =c·B(R) (7)\nwhere R= (r1,r2, ...,rn) are the positions of the atoms\n(and, in practice, the unit cell of the structure Rand\nnumbering of the species of atom for each ri, omitted\nhere for clarity). The linear nature of the potential is\nnow obvious, and the parameters care the ones that\nmust be determined by fitting to data.\nThe important properties of the potential for this\npaper are that we have a flexible, accurate6,27and\nefficient23,25MLIP, and once we specify νand the maxi-\nmum total polynomial degree pmax(controlling the num-\nber of coefficients in our potential, by picking out all basis\nfunctions such that pmax≥Pν\ni=1ni+li, see25,26), as well\nas relevant cutoffs and weights on different observation\ntypes, then atomic properties - energies, forces and virial\nstresses - are linear combinations of the potential coeffi-\ncients basis functions (and relevant derivatives). These3\natomic properties can then be fed into any more complex\nQoI evaluation and propagated through a UQ procedure.\nWe utilise the ACEpotentials.jl Julia package28for\nconstruction of all potentials and evaluations of atomic\nproperties. The potentials in this paper use a cubic pair\npotential term for both the short range repulsive be-\nhaviour and long distance behaviour up to final (smooth)\ncutoff of 3 .0r0(where r0≈2.35˚A is an estimate of\nthe nearest neighbour distance in silicon), and a many\nbody ACE part (of order pmax) for the intermediate\n(0.8r0−2.0r0) regime. We also set the correlation or-\nderν= 3, i.e. we include up to 4-body terms in the\nexpansion, and set the weights on our differently labelled\ntraining data as per the supplementary information of\nRef. 23. This set-up is not intended to be well optimised\nfor the problems considered below; this methodology can\nbe applied to any linear potential constructed in a similar\nmanner, and is not specific to these design choices.\nB. Bayesian Inverse Problems\nBegin with some generic data ( x,y), both of length\nN. We can view this data as atomic positions of differ-\nent structures x= (R1,R2, ...,RN), and the correspond-\ning DFT energies, forces and virials y= (y1,y2, ...,yN)\nwhere yiconsists of a single energy, six virial stresses and\n3nforces with nthe number of atoms in Ri.\nUtilising the linear nature of the ACE potentials de-\nscribed previously, we proceed from Equation 7 and set\nup our data problem in the following way:\ny=V(c,x) +ϵ,ϵ∼ N(0, β−1I), (8)\nwhere we assume that the potential Vis capable of re-\nproducing our targets y(the ‘ground truth’) which are\ncontaminated with Gaussian, zero mean noise ϵ, with\nhomoscedastic precision β, representing the noise we es-\ntimate on our data.\nWe wish to find a distribution describing the param-\neters (or coefficients) cof the model Vthrough a basic\nBayesian inverse problem. Utilising Bayes’ rule,\nP(c|y, α, β ) =P(y|x,c, β)P(c|α)\nP(y), (9)\nour posterior distribution of weights given data,\nP(c|y, α, β ), is the product of the likelihood of observing\nthe data P(y|x,c, β) and the prior for our parameters\nP(c|α) given some assumed precisions on parameters α\nand data β, normalised by the evidence P(y). Here we\nassume a single shared precision hyperparameter αfor\nall the coefficients in the model.\nTo proceed from Equation 9, we must either sample\nfrom our posterior P(c|y, α, β ) using e.g. MCMC, or have\nan analytical form for the posterior distribution. Forcing\nanalyticity, while simpler mathematically, is not the most\ngeneral approach, however this is not expected to be a\nproblem for most atomistic systems; as the amount oftraining data is typically quite large, and therefore the\nposterior will be less sensitive to the chosen prior.\nAlso of note here is the divergence between the con-\nstruction of the relatively simple potentials within this\npaper and the current state of the art for ACE poten-\ntials (as well as the discussion on the construction choices\nin Section II A). Construction of state of the art ACE\npotentials26utilise smoothness priors to impose smooth-\nness on the potential fit - one possible choice to perform\nthis regularisation in the fitting procedure is to specify\nthe prior distribution P(c|α), and can be compared to\nperforming Bayesian ridge regression / regularised least\nsquares - however this is only one possibility: for more\ndetails, refer to the recent overview of ACEpotentials.jl.26\nWe can write the likelihood of our model as a multi-\nvariate normal distribution\nP(y|x,c, β) =N(y|V(c,x), β−1I), (10)\nand we choose a conjugate prior for the parameters\nP(c|α) =N(0, α−1I), (11)\nsuch that we can construct our posterior analytically:\nP(c|y, α, β ) =N(c|µ,S), (12)\nwhere\nS= (αI+βΦTΦ)−1,µ=βSΦTy. (13)\nHere, the mean vector and covariance matrix are pa-\nrameterised in terms of our target data y, precision hy-\nperparameters αandβ, and a design matrix\nΦ =\nB1(R1)B2(R1)... B M(R1)\nB1(R2)B2(R2)... B M(R2)\n............\nB1(RN)B2(RN)... B M(RN)\n, (14)\nthe entries of which evaluate the jthbasis function on the\nithdata point.\nC. Hyperparameters and optimisations\n1. Hyperparameters and the Evidence function\nWe have introduced two hyperparameters into the in-\nverse problem: a single noise precision βwhich estimates\nthe noise inherent in our training data y, and a single\nweight precision αwhich estimates our confidence in the\npossible model parameter values.\nA single, homoscedastic noise precision value βas-\nsumes that our training data are independent draws from\nthe same underlying distribution. While the training\ndata in our case ( x,y) are atomic configurations and\nDFT observations from the same database, they are not\ndrawn from the same distribution - ycontains related4\nbut markedly different quantities (total energies, forces\non atoms, and virial stress components). The single βas-\nsumption is an extreme simplification - introducing het-\neroscedasticity between different types of quantities con-\ntained within y, or further having different βfor different\natomic configurations x, would be a far more accurate\napproach at the expense of a higher dimensional hyper-\nparameter space.\nSimilarly, the single weight precision value αassumes\nsimilar confidence in all coefficients cin the chosen model.\nThis is unlikely to be true, especially in the construction\nof ACE potentials in this paper as we have pair potential\nand many-body terms included in the linear basis. An ex-\ntension which would follow naturally through Equations\n9-12 is the promotion of αfrom scalar to vector-valued,\nwhich would allow more flexibility in the weight poste-\nrior (12) and further analysis (for example, in the form\nof Automatic Relevance Detection, ARD), again at the\ncost of increased complexity of the inverse problem and\nsubsequent hyperparameter optimisations and sampling.\nIn a fully Bayesian treatment, these hyperparameters\nshould be sampled over by setting suitable hyperpriors,\nand then sampling the entire Bayesian posterior. For\nsimplicity, we instead optimise αandβsuch that they\nmaximise the (log-) evidence29or log-marginal likelihood:\nthe likelihood of observing our data ygiven a model de-\nfined by hyperparameters αandβ. This is given by\nlnP(y|α, β) =M\n2lnα+N\n2lnβ−E(c)\n−1\n2ln|S−1| −N\n2ln 2π,(15)\nwhere\nE(c) =β\n2||y−Φc||2+α\n2cTc. (16)\nNote the trade-off in Equation 16 between reproduc-\ning the training data and the penalisation for complicated\nmodels with many parameters. For simple models with\nsmall coefficient vector c(length M), agreement between\ntargets yand prediction Φ cwill likely be poor, and the\nevidence will be low. For larger Mthis agreement will be\nbetter but the model evidence is penalised for the added\ncomplexity, due to the increase in size of cand the inner\nproduct cTc. This is one method of analysing the level of\nover-fitting when comparing different models: the value\nof lnP(y|α, β) for increasing size of some model Vshould\nhave a maximum value, past which the increasing perfor-\nmance on the training set can no longer compensate for\nthe increasing complexity of the model.\n2. Optimisation with Particle Swarm\nNow we must choose an algorithm to maximise Equa-\ntion 15 with respect to αandβ. There exist iterative\nupdate schemes29,30to perform this, but they scale andperform poorly as Φ becomes larger - the hyperparame-\nter space becomes more complex, and individual evidence\nevaluations become slower as the corresponding matrices\nin Equation 15 grow in size. A gradient based approach\nlike an L-BFGS performs more efficiently, but is sensitive\nto initial point chosen during optimisation and can fail if\nit reaches a very unlikely position in the hyperparameter\nlandscape. This is made worse since we are generating\nmany different potentials throughout this paper - either\nin terms of varying types and amounts of training data,\nand/or vastly different numbers of coefficients. It be-\ncomes difficult to make a meaningful ansatz for sensible\ninitial precision hyperparameter values which can span\nseveral orders of magnitude for different models.\nTo circumvent these problems, we utilise a particle\nswarm optimisation (PSO) algorithm. This global op-\ntimisation scheme initialises a population of particles\nwith random velocities across the hyperparameter space\n(logα,logβ), where we operate in a log- space since pre-\ncisions must be positively valued and can vary over many\norders of magnitude. At each step, each particle is accel-\nerated towards the position of the global best value for\nthe (log-) evidence, and the position of the best value the\nindividual has observed.\nBy tuning these relative accelerations - as well as other\nparameters of the optimisation like the number and in-\nertia of the particles in the swarm, and the number of\niterations - the algorithm can balance local and global\nexploration of the hyperparameter landscape and rela-\ntively quickly converge to a sensible result for αandβ.\nIndividual iteration steps for each particle do not require\ngradient evaluations, however the calculation of the (log-\n) evidence (15) must occur for each member of the swarm\nat each iteration step.\nWe do not have a guarantee of convergence to a lo-\ncal or global optimum using PSO, but in application the\nswarm regularly converges for the precision hyperparam-\neters αandβfor a range of different potentials and train-\ning data for a swarm of 50 particles allowed to proceed\nfor 200 iterations, with equal weightings for local and\nglobal exploration and an overall inertia weight of 0.9\nto prevent divergence of swarm members. On the occa-\nsions for which PSO fails/diverges, the values observed\nforα,βand the evidence are easily identified as unphysi-\ncal, and PSO can usually be re-run without changing the\nalgorithm settings.\nD. Conformal Predictions\nIf we perform our parameter estimation as a Bayesian\ninverse problem as in Section II B, we operate under the\nassumptions that the noise in our data and the prior and\nposterior distributions for our coefficients are normally\ndistributed, and that the model we are using is the cor-\nrect one, i.e., there is no model form error. While the\ndistributional assumptions might be allowable, there will\nalmost always be some amount of model form error for5\nan IP attempting to match DFT training and test data\n(e.g. due to limited body order and cutoff distances),\nand there are other sources of uncertainty which we will\nnot explicitly account for in this workflow (for instance,\nalgorithmic uncertainty in calculating specific QoIs).\nFurthermore, if we wish to transform our weight poste-\nriorP(c|y, α, β ) into a distribution of output quantities\nynewof the same type as y(so in our case, containing\nenergies, forces and virial stresses, on the new atomic\nconfigurations xnew), and from these calculate a given\nQoI, we can do so using a predictive posterior distribu-\ntion given by29\nP(ynew|y, α, β ) =Z\nP(ynew|x,c, β)P(c|y, α, β ) dc,\n(17)\nwhich for our normal likelihood in Equation 10 and\nweight posterior in Equation 12 can be evaluated ana-\nlytically:\nP(ynew|y, α, β ) =N(µTB(xnew),Σ) (18)\nwhere\nΣ =1\nβ+B(xnew)TSB(xnew). (19)\nThis predictive posterior would in principle give the\ndistributions in energies, forces and virial stresses which\ncan be fed through a QoI calculation. However for atom-\nistic simulations it is in general quite difficult to form,\nsince we require foreknowledge of the atomic configura-\ntions xnew- for QoIs involving relaxations or dynamics\nunder a given potential, this is not possible.\nInstead, we are left with sampling from our weight pos-\nterior from Equation 12 and pushing the corresponding\npotentials through a QoI simulation. This results in tight\noutput QoI distributions, with very small predicted un-\ncertainties, since we are not performing the integral in\nEquation 17. The uncertainties we could calculate by\ndoing this here are not the ones from a true Bayesian in-\nverse problem, since we only propagate potential param-\neter vectors cfrom the weight posterior forward through\nthe model. Nevertheless, the small uncertainties we pre-\ndict from the forward propagation of ensemble members\ncan still be useful, since the spread of predictions still has\nsome of the correct qualitative behaviour we would like\nour uncertainty to capture more rigourously.\nWe therefore aim to calibrate these uncertainties by\nperforming conformal prediction,24,31which seeks to con-\nstruct a distribution free, frequentist prediction set C\nsuch that the probability the true result lies within C\nis close to a set desired coverage 1 −ζ, i.e.\n1−ζ≤P(ynew∈ C(xnew))≤1−ζ+1\nn+ 1,(20)\nwith nthe length of the calibration set and ( xnew,ynew)\nare points in a new set to be predicted from our model.\nThe requirements for such a conformal procedure are\na model V(c,x) which outputs some mean prediction µiand some heuristic uncertainty measure σifor an input i,\nas well as data ( x,y) split into training, calibration and\ntest subsets. The only change compared to the procedure\nabove is to set aside ndata points for the calibration of\nour error bars. To actually perform the calibration, we\nfollow the procedure outlined in24:\n1. Define a score function s(µi, σi, ycal,i) where larger\nscores encode worse agreement between the model\nprediction and the ‘true’ value in the calibration\nset, in our case\nsi=abs(µi−ycal,i)\nσi, (21)\nand compute the scores on the calibration set\ns={si}n\ni=1. (22)\n2. Compute ˆ q, the multiplicative factor by which we\nwill scale our uncertainties, from the list of scores\ns\nˆq= quantile( qval,s), qval=&\n(n+ 1)(1 −ζ)\nn'\n.(23)\n3. Form prediction sets on values in the\nnew/prediction set:\nC= [µnew−ˆqσnew, µnew+ ˆqσnew]. (24)\nNote that when reporting values with uncertainties\nwithin this paper, we will often report them in the form\nµnew±ˆqσnewas a short form for a bound like Equation\n24.\nThe form of Equation 20 is a strong condition for a pre-\ndiction set C, however (as discussed elsewhere24) much\nof the usefulness of Cis tied to the chosen score function\ns(µi, σi, ycal,i) and the make-up and size nof the cali-\nbration set: a poor choice of score function, or a small\ncalibration set, would result in a prediction set which in\nprinciple agrees with Equation 20, but in practice would\nnot be a useful measure of the true uncertainty of the\nquantities in the prediction set of the trained model V.\nIn this paper we will use the score function in Equation\n21 with the heuristic uncertainty σbeing the standard\ndeviation of the ensemble QoI prediction; refining this\nchoice is an area of interesting further research into the\nchoice of uncertainty measure σ(e.g. standard deviation,\ninterquartile range) and form of the score function, and\nthe effects on the subsequent UQ.\nWe can verify that the conformal approach we employ\nhas the correct coverage Con these atomic properties\nby shuffling the calibration and prediction/test sets for\na number of trials R, and plotting the distribution of\nthe empirical coverage. As Rincreases, we expect the\nmean empirical coverage to approach 1 −ζ, and the dis-\ntribution of empirical coverages to approach the analytic\nBeta-Binomial distribution24\nCanalytic ∼1\nntestBetaBin( ntest, n+ 1−l, l), (25)6\nwhere ntestis the length of the test set and\nl=⌊(n+ 1)ζ⌋. (26)\nFor two simple ACE models of 9 and 81 parameters,\nboth trained on 300 bulk configurations, calibrated on\n39 bulk configurations and a test set of 98 bulk config-\nurations, we plot the distribution of empirical coverages\nover 1000 trials in the insets of Figure 1, and can follow\nthis through by plotting uncertainties on these atomistic\nquantities, e.g. for the energy per atom of configurations\nin the calibration and test sets in Figure 1 - note that our\ncoverage target includes energies, forces and virials, and\ndoes not guarantee coverage on solely e.g. energies (the\ncoverage of energies within Figure 1 are 88% and 83% re-\nspectively on the 137 total structures). We can conclude\nthat our conformal procedure is working as intended for\nthe atomistic quantities in the training and calibration\nsets - we can place error bounds on energies, forces and\nvirial stresses of the same form as Equation 24, with a\ncoverage which we specify beforehand.\nIt is important to make clear the practical differences\nbetween an ideal application of conformal prediction and\nthe method we will develop here. A pure conformal ap-\nproach calibrates the uncertainties of data on which the\nmodel is trained and calibrated , which in our case are to-\ntal energies, forces on atoms, and virial stresses.\nWe extend the method above by assuming we can also\nscale the uncertainty in a given QoI prediction from a\nsampled posterior by the same ˆ qvalue and obtain a sen-\nsible estimate of uncertainty - scaling the QoI spread of a\nmodel formed as in Equation 12 (a ‘base’ Bayesian model\nwhich accounts for parametric uncertainty and noise pre-\ndicted on the data) by the performance of said model\non the output of atomistic properties on which it was\ntrained, leading to a frequentist prediction set for a QoI.\nWe are also relaxing / ignoring the assumption that the\npoints on which we make predictions ( xnew,ynew) are\ndrawn from the same distribution as those in the training\nand calibration sets: this is already difficult in atomistic\ndatasets in general due to the different types of observa-\ntion present (e.g. energies, forces, virials, whilst related\nwill be drawn from different underlying distributions)\nand different types of configurations (e.g. bulk, mono-\nvacancy, interstitials). In the case of differently labelled\ndata, we also have the problem that a random split is not\nguaranteed to place members of a given label in all the\nsubsets - train, calibrate and test - of the original data.\nIn this paper we assume that members from each label\nare present in all the subsets, but there can be edge cases\nwhere there would be unexpected behaviour either in the\ntrained potential or in the calibration of uncertainties,\nespecially for a label with fewer members. In situations\nof data scarcity, whether from expense of simulation or\notherwise, it may become necessary to explicitly control\nthe split location of some data points.\nIt is the hope that this two stage UQ procedure leads\nto robust and interpretable error bounds on QoIs, and\nthat this procedure can be adapted across a wide rangeof applications e.g. other forms of MLIPs, different for-\nmulations of the inverse problem (‘base’ models), other\nmeasures of heuristic uncertainty or score functions in the\nconformal procedure, and varied QoIs for different atomic\nsystems. In application in this paper, following the pre-\nscription outlined in Equations 21-24 with the same score\nfunction s, we set our desired coverage 1 −ζ= 0.95\nand split the available data such that approximately the\ntrain/calibrate/test split is 72 : 8 : 20; as well as using\nan ensemble size of 100 members to both push through\nsimulation and to calibrate our error bars. These choices\nare to a certain extent arbitrary: for example, the 95%\ncoverage choice is due to a traditional assumption when\nreporting uncertainties; to assume a normal distribution\nand report plus-minus two standard deviations around\nthe mean. We would like the train/calibrate/test split to\ncontain data from different labels, yet we do not want to\nsacrifice too much training data for calibration or testing.\nThe ensemble size is seen to be sufficient for the appli-\ncations presented here, but for more realistic potentials\ntrained on more diverse datasets and more challenging\nQoIs, a rigorous test of convergence would be necessary\nto establish a minimum number of samples to balance be-\ntween statistical need and computational effort, as well as\nidentfiying any QoI difficulty dependence - i.e. for more\ndifficult QoIs, are more ensemble members required to\nobtain a useful heuristic uncertainty and accurate ‘mean’\nvalue.\nIII. RESULTS\nTo review the workflow, for a given QoI, and model\nVwith dataset ( x,y), we train an ACE potential on a\nsubset of the data ( xtrain,ytrain), optimise the precision\nhyperparameters αandβto maximise the evidence and\nform an ensemble of potentials drawn from our poste-\nrior distribution for coefficients. The potentials defined\nby these coefficients are pushed through the QoI simula-\ntion to obtain a mean prediction µand heuristic estimate\nof uncertainty σ(in our case, the standard deviation).\nWe then calculate the ˆ qvalue using our calibration set\n(xcal,ycal) of energy, force and virial stress observations,\nand form a prediction set Con the predicted QoI.\nWe will evaluate the uncertainty in the results of four\ndifferent QoIs of increasing complexity, investigating the\neffect of increasing the amount of data available for train-\ning and calibration, and increasing the complexity/size of\nthe potential basis.\nA. Bulk modulus\nWe begin by estimating the uncertainty in predictions\nof the bulk modulus Bof cubic silicon. The calculation\nof this QoI is simple as it does not require any geometry\nrelaxations or dynamics, we simply shrink or expand the\nunit cell around the minimum, and perform a cubic fit the7\n163.1\n163.0\n162.9\n162.8\n162.7\nDFT Energy/atom (eV)163.2\n163.1\n163.0\n162.9\n162.8\n162.7\n162.6\nACE Energy/atom (eV)\nPredicted Energy/atom0.930.940.950.960.97\nCoverage020406080100120Density\n Empirical\n Analytic\n163.1\n163.0\n162.9\n162.8\n162.7\nDFT Energy/atom (eV)163.1\n163.0\n162.9\n162.8\n162.7\n162.6\nACE Energy/atom (eV)\nPredicted Energy/atom0.930.940.950.960.97\nCoverage0255075100Density\n Empirical\n Analytic\nFIG. 1: The energy per atom with conformal uncertainty bounds evaluated on the calibration and test sets for two\ndifferently sized potentials of 9 and 81 parameters, both trained on 300 bulk diamond configurations of silicon. The\ninset histogram compares the empirical observed conformal coverage for 1000 shuffles of the calibration and test sets\nwith the analytic coverage distribution.\nbulk modulus to the curvature of the calculated energy-\nvolume curve. This property is relatively straightforward\nto achieve a result with only bulk configurations, and\nfor this reason it is the only QoI for which we will also\nevaluate a matrix of different models trained on varying\namounts of input DFT data.\nUtilising a dataset of 489 diamond-cubic Si configura-\ntions with DFT energy, force and virial observations, we\nalways calibrate on the same calibration set of 39 config-\nurations. The training set varies in size from 25 to 350\nDFT configurations. The linear potentials vary in size\nfrom pmax= 3−24, or 9 −4330 parameters.\nAn overview contour plot of the (data, complexity, un-\ncertainty) space is presented in Figure 2, where we show\nthe half bound of size ˆ qσpredacross the entire space - see\nTable I for selected predictions and uncertainties in B.\nAs expected, in the low data, simple potential regime,\nthe uncertainty in our predictions of Bare large. Train-\ning potentials on more data (slicing Figure 2 from a point\non the y−axis) generally improves the prediction and de-\ncreases the uncertainty in B, but with diminishing im-\nprovements past ∼150 atomic configurations. Similarly\nfor a large enough training set, increasing the size of the\npotential (by slicing upwards from a given number of con-\nfigurations on the x−axis) also improves the prediction\nand decreases the uncertainty in B. However for small\ntraining datasets we see some overfit, leading to uncer-\ntainties that can grow rapidly, and for large datasets the\npredictions can become overconfident, as represented by\nthe white crosses in Figure 2.The reference DFT value, which was not included in\nthe training or calibration sets, is contained within the\nuncertainty bounds of a majority of the space of Figure 2,\nand for those for which the predictions do not agree, the\ndiscrepancy is usually a small amount. This conservative\nbehaviour is a desirable feature: if we have constructed\nour model such that the variation in ensemble members\ncaptures the behaviour of the underlying uncertainty in\na QoI, and conformalized the uncertainty correctly, the\nprediction bounds should contain the true values of the\nenergies, forces and virials of the new structures used in\nthe QoI calculation with probability close to 1 −ζ- and\ntherefore we expect that the uncertainty in our QoI is\nalso a good estimate of the true error.\nWe can decompose the conformal uncertainty of our\npredictions of Bfrom Figure 2 into the components ˆ q\nand the ensemble standard deviation σ; this is presented\nin Figure 3. Regardless of the size of the basis, the value\nof the ensemble σforBbecomes very small once the\npotentials are trained on more than 150 configurations.\nAdding more of the same type of configuration past this\npoint does not lead to a large change in the value of the\nparameters - the weight posterior (Equation 12) becomes\nvery narrow, and the ensemble members become very\nsimilar. The largest σvalues occur for potentials with\nlarge basis sets (high maximum polynomial degree) and\nfew training configurations, which we have seen the effect\nof previously in Figure 2.\nThe conformal ˆ qscale factor gives the largest correc-\ntions to simple potentials trained on large amounts of8\n50100150200250300350\nTraining Configurations4681012141618202224MaximumPolynomialDegreepmax\n20406080\nBulkmodulusuncertaintyq(GPa)\nFIG. 2: Conformal uncertainty ˆ qσbound for predictions\nof bulk modulus of Si, for ACE potentials with\npmax= 3−24 (between 9 and 4330 parameters) and\ntrained on between 25 and 350 DFT structures of bulk\nSi, with those structures ranging in size from 2 to 128\natoms. White crosses indicate points where the DFT\nreference value lies outside the predicted range.\ndata. Since we become more certain of our hyperpa-\nrameters αandβas we train on more data for a fixed\nbasis, and the posterior distribution of weights becomes\nnarrower, so there is less variation in predictions on the\natomistic quantities (energies, forces and virials) con-\ntained in the calibration set. For the simple IPs, where\npredictions are likely to be less good, the corresponding\nscores (Equation 21) will be larger and hence ˆ qmust be-\ncome larger.\nWe can focus on the top right region of the heatmaps\nfrom Figures 2 and 3 - models trained on larger amounts\nof data and with more parameters to fit - to observe the\neffect of overfitting / overconfidence in our uncertainties.\nIn this region, since our ensemble members do not vary\ngreatly, we rely on the ˆ qcorrective factor - calculated\nby assessing performance on the quantities the poten-\ntials are trained on - to capture the QoI uncertainty. We\ncan pick out individual points whose predictions agree\nwith DFT (e.g. (#Configs , pmax) = (150 ,22),(325,21))\nand attribute those to a more fair assessment of the ˆ q\nfactor for those potentials’; nevertheless these points are\noutliers. Increasing the number of ensemble members, as-\nsessing the performance of the chosen score function, and\ntraining on a more diverse dataset in this region would\nall be suitable next steps to rectify these disagreements\nand improve the uncertainty estimates.B. Elastic constants\nMoving up a step in QoI complexity from calculation\nof bulk moduli, we now focus on calculating elastic con-\nstants C11,C12andC44of bulk silicon. These quantities\nare more complex because they require the chosen po-\ntential to perform optimisations of the atomic positions\nand the crystal unit cell for the initial bulk configura-\ntion, and of atomic positions for strained configurations.\nUsing five configurations, with strain ranging between\n±0.025, are then used in a linear regression scheme to\ncalculate C11,C12andC44. These quantities are related\nto the bulk modulus for cubic crystals, and we therefore\nexpect that for the suitably complex potentials trained\non enough data from Figure 2, we should be able to cap-\ntureC11andC12fairly confidently, and account for C44\nwithin uncertainty.\nSince we have already constructed the potentials, built\nthe design matrix Φ and vector of DFT observations y,\nand performed the hyperparameter optimisations for α\nandβfor a wide range of potentials trained only on\nbulk configurations in our bulk modulus results above, we\ncould perform a similar analysis here - the potentials, op-\ntimised hyperparameters and ˆ qvalues are QoI-agnostic,\nwhich is a desirable feature for transferable potentials.\nHowever, we choose to focus on the ‘data’ axis of Figure\n2, fixing the size of the potentials to pmax= [4,12,19] cor-\nresponding to [13 ,214,1432] parameters, so that we can\nvisualize the uncertainties and predictions of the three\nelastic constants for a varying amount of training data.\nAs we might expect due to the relationship between the\nbulk modulus and elastic constants for cubic crystals, we\ncapture C11andC12across almost the entire range of\ntraining configurations used for each potential; the same\nis also true for C44- this is shown in Figure 4, and selected\nvalues are also tabulated in Table II.\nThe predictions for the elastic constants seem to have\ntwo distinct regions of behaviour: a region of fewer train-\ning configurations where we can infer that the uncer-\ntainty is caused by lack of data, and a remaining region of\nincreasing training configurations where we infer that the\nuncertainty is dominated by lack of complexity in the po-\ntential. Interestingly, comparing Figure 4 to Figure 2, the\nposition of this transition changes depending on the basis\nsize: for pmax= 4, the top row of Figure 4, the transition\noccurs at approximately 150 training configurations. For\nthe more complex potentials, the transition occurs sooner\nat approximately 75 training configurations. To investi-\ngate this further we decompose the uncertainty into ˆ q\nand ensemble standard deviations for C11,C12andC44,\nshown in Figure 5. We can identify that in the regions\nthat are training data limited, the ensemble σfor all\ntheCijdecrease quickly as we increase the amount of\ntraining data; this decrease appears to occur faster as\nthe complexity of the potential increases. In the region\nwhere we are limited by the complexity of the IP, the\nensemble standard deviations are decreasing slowly, with\nthe corresponding increasing ˆ qvalues being driven by in-9\n50100150200250300350\nTraining Configurations4681012141618202224MaximumPolynomialDegreepmax\n50100150200\nConformalrescalefactorq\n50100150200250300350\nTraining Configurations4681012141618202224MaximumPolynomialDegreepmax\n3691215\nBulkmodulusEnsemble(GPa)\nFIG. 3: Left: conformal rescale factor ˆ qvalues for various potentials and training data used to calculate bulk\nmodulus in Figure 2. Right: ensemble standard deviation σforBacross the same space. The product of these two\nheatmaps gives Figure 2. Note the different colourbar scales for the two subplots.\ncreasingly narrow posterior weight distributions, as the\nmodel becomes more certain of the parameter values.\nNot investigated here is the effect on the uncertainty\nestimate from the random splits into training and calibra-\ntion sets, which may alter the mean prediction, ensemble\nstandard deviation, and the multiplicative ˆ qfactor. From\nthe 489 total bulk structure data points, the number of\npossible ways to choose bpoints is given by the binomial\ncoefficient489Cb- for btaking values of training config-\nurations as in Figure 4, the number of combinations are\nincredibly large. An interesting place for further work\nwould be quantifying this effect; we might expect that\nthe fine structure we see in ˆ qandσin Figure 5, when\naveraged over different train/calibration/test splits is ef-\nfectively noise, leaving the underlying trend.\nC. Vacancy formation energy\nTo increase the QoI difficulty once again, we target a\nquantity which requires geometry optimisations, but also\nneeds more varied training data. The calculation of a\nvacancy formation energy Evacfits this description since\nthe IP must accurately represent both the bulk crystal\nand the monovacancy configurations.\nThis requires the generation of new potentials and cal-\nculation of hyperparameters (see Methodology), since we\nnow train and calibrate on the bulk configurations used\npreviously as well as monovacancy and divacancy config-\nurations. In a similar, perpendicular vein to the previ-ous section, we restrict our analysis this time to a fixed\namount of training data whilst allowing the number of\nparameters in the potential to vary. We train on a fixed\nrandom selection of 500 structures and calibrate on a sep-\narate 62, with 156 structures held back as a test set. As\noutlined in the discussion on this application of conformal\npredictions, with this new train/calibrate/test split we\ndo not explicitly guarantee that we have vacancy struc-\ntures in the training and calibration sets, but this is not\nexpected to be a problem for this size of subsets of data.\nResults for the vacancy formation energy are shown in\nFigure 6. The simpler potentials with fewer available pa-\nrameters to be trained on the data give predictions that\nare both far from the reference result, and with large\nuncertainty bounds. Note that prediction sets which in-\nclude negative values for Evacdo not truly observe neg-\native values - instead the conformal scaling is large since\nthe model is very simple and does not perform well (i.e.\ntraining, calibration and test errors will be large).\nAspmaxis allowed to increase, the mean prediction\nof the ensemble generally comes closer to the reference\nvalue, and the uncertainty bounds shrink to a minimum\nvalue of ±0.036 eV at pmax= 26. The conformal uncer-\ntainty routine gives coverage for the DFT result across\n(almost) the entire range of potentials - only for the more\ncomplex IPs with many parameters do we lose this cov-\nerage as we overfit to the training data.\nTo investigate the structure of these uncertainty\nbounds, we again decompose the uncertainty into the\nˆqfactor (evaluated on atomic properties of a calibra-\ntion set), and the ensemble heuristic uncertainty for the10\n010020030050100150200C11(GPa)\n01002003006080100120C12(GPa)pmax=4,13parameters\n010020030020\n020406080100C44(GPa)\n0100200300110120130140150160C11(GPa)\n01002003004050607080C12(GPa)pmax=12,214parameters\nDFT\n0100200300606570758085C44(GPa)\n0100200300100120140160C11(GPa)\n0100200300\nTraining Configurations30405060708090C12(GPa)pmax=19,1432parameters\n0100200300606570758085C44(GPa)\nFIG. 4: Mean prediction and conformal uncertainty bounds for elastic constants C11,C12andC44of cubic Si, for\nthree potentials with pmax= [4,12,19] trained on varying numbers of atomic structures. The DFT reference\nprediction (dashed line) is captured within uncertainty across almost entire range for each potential.\nspread of Evacpredictions; this is presented in Figure 7.\nWe observe that both ˆ qand the ensemble standard de-\nviation σinitially decrease as the model gains more pa-\nrameters, since the increased flexibility allows for better\nrepresentation of the calibration set and for more sensible\nrelaxations and energy/force evaluations to be done on\nboth the bulk and monovacancy structures during evalu-\nation of Evac. This decrease slows as we continue increas-ing the number of parameters, and the σvalue stagnates\nto a roughly constant value at approximately pmax= 15,\nsuggesting that we have crossed from a region where we\nare limited by our model to a region where we are limited\nby the available training data. For ˆ qwe see the rate of\ndecrease slow as pmaxincreases, suggesting that the per-\nformance on the calibration set (see Equations 21-23) is\ncontinuing to improve as the basis becomes larger, even11\n100 200 3000.51.01.52.0Cijensemble(GPa)\npmax=4\nc11\nc12\nc44\n5075100125150175\n100 200 300\nTraining Configurations0.51.01.52.02.5pmax=12\n152025303540\n100 200 30012345pmax=19\n101520253035\nConformalrescalefactorq\nq\nFIG. 5: Ensemble standard deviations for elastic constants C11,C12andC44on the left y−axes, and ˆ qfactors on\nthe right y−axes, for three potentials, pmax= [4,12,19], trained on varying amounts of DFT atomic structures.\nNote the different scales for the three subplots. The three products of the curves from each of the subplots above\ncorrespond to half the uncertainty bounds presented in Figure 4.\n468101214161820222426\nMaximumPolynomialDegreepmax1\n012345VacancyformationenergyEvac(eV)\nN=500Trainingconfigurations\nDFT\n16 18 20 22 24 263.553.603.653.703.75\nFIG. 6: Mean prediction and conformal uncertainty\nbounds for relaxed vacancy formation energy of Si, for\npotentials of increasing size trained on the same dataset\nof 500 bulk, mono- and di-vacancy DFT structures, and\ncalibrated on 62 similar structures. The dashed line is\nthe DFT reference result22,EDFT\nvac= 3.67 eV.\nif this improvement is not represented in σ.\nEven though the performance on our calibration set\nsuggests that we could increase the size of our IP further\n468101214161820222426\nMaximumPolynomialDegreepmax0.0020.0040.0060.008Evacensemble(eV)\nN=500Trainingconfigurations\n50100150200250300\nConformalrescalefactorq\nFIG. 7: Ensemble standard deviations (left axis) for\nvacancy formation energy and ˆ qfactor (right axis) for\npotentials ranging from pmax= 3−26, (number of\nparameters 9–6,435) trained on 500 bulk, mono- and\ndi-vacancy DFT structures, and calibrated on 62 similar\nstructures. The product of the two curves produces half\nthe uncertainty bound presented in Figure 6.\nwithout risk of overfitting, the behaviour of the heuristic\nensemble σimplies that we require more training data.12\nAn interesting possible idea for determining whether to\ngather more training data or increase the size of a single\nIP would be to vary the amount of training data and size\nof the given potential, mapping the heuristic uncertain-\ntiesσfor some QoI(s) and the calculated ˆ qvalues, and\nthen inspecting similar graphs to Figures 5 and 7.\nD. Vacancy migration\nWe target a vacancy migration in silicon as our final\nQoI. This property is once again a step up in terms of dif-\nficulty, as it requires the calculation of an energy barrier\nEband the energies along the path as the configuration\nmoves from the initial to the final state. This is done\nusing the Nudged Elastic Band (NEB) technique - we set\nup a series of images between the initial and final states,\nand by minimising the energy between them whilst main-\ntaining the spacing between the images via spring forces,\nthe minimum energy path is produced. This is performed\nusing the ASE32NEB class, with 11 total images inclu-\nsive of the initial and final vacancies.\nAs QoIs become more complex, the chosen potential is\nmore likely to be extrapolating outside of environments\nit has been trained and calibrated on. This means that\nwe inherently have less intuition a priori for which types\nof data we need to include during training to obtain sen-\nsible predictions in target QoIs. This logic also applies\nto the calculation of uncertainties: if we calibrate uncer-\ntainties on some configurations, but during the process\nof a QoI calculation we have needed to extrapolate to un-\nseen regions in the PES, we cannot then be certain that\nthe calibration of uncertainties will be meaningful. This\nhighlights the need for either carefully curated datasets,\nor some automated active learning scheme based on UQ:\nboth prediction and uncertainty are dependent on the\ntraining and calibration sets being (at least close to) rep-\nresentative of the new points on which we predict in the\ncourse of a simulation.\nUsing one of the potentials from the previous section\n(pmax= 20, trained on the same data) we obtain an en-\nsemble of minimum energy paths for a single vacancy\nmigration in Figure 8, showing the central prediction as\nwell as the conformal uncertainty bound. Since the cali-\nbration set is fixed (and hence so is the value for ˆ q), the\nchanges in the uncertainty across the path are driven by\nthe spread of the ensemble members. At the initial and\nfinal points - which correspond to mono-vacancy configu-\nrations - we are very close to the DFT result and the un-\ncertainty is small, which is expected since the bulk struc-\nture and vacancies are well represented in the dataset.\nThe model is most uncertain at the saddle point, which\nmatches our intuition since our training and calibration\nsets do not include these configurations: the potential is\nbeing forced to extrapolate. We can see that all mem-\nbers of the ensemble find an energy barrier, and whilst\nthe uncertainty bound show a possible minimum at the\nsaddle point, this is not truly observed and is a symptomof the large spread of predictions there - this is similar\nto poor models in Figure 6 having uncertainty bounds\nwhich ‘predict’ negative vacancy formation energies.\n0.00.20.40.60.81.0\nReaction coordinate3.63.73.83.9Energy (eV)pmax=20,Bulk,vacancies,divacancies\nDFT\nGAP2018\nACE ensemble\nACE prediction\nFIG. 8: Single vacancy migration in silicon for an ACE\npotential with pmax= 20 (1821 parameters) trained on\n500 bulk, mono- and di-vacancy DFT structures,\nshowing samples drawn from the weight posterior (grey\ncurves), the mean prediction (black curve) and\nconformalized uncertainty (grey ribbon), with the DFT\nreference curve (dashed blue) and the Si GAP\nprediction22(trained on the full dataset) for\ncomparison.\nThe DFT result gives a barrier height EDFT\nb= 0.30 eV\nand a vacancy formation energy (which correspond to the\ninitial and final images of the paths) of EDFT\nvac= 3.63 eV,\nwhereas the ACE model predicts Eb= 0.12±0.12 eV\nandEvac= 3.63±0.06 eV. For such a relatively simple\npotential, trained and calibrated on a cut-down portion\nof the full DFT dataset, both the prediction of the en-\nergy barrier and the fact that our uncertainty bound is\nclose to the DFT across the predicted, symmetric mini-\nmum energy path is quite impressive. The full Si GAP\npotential22also struggles to reproduce the DFT barrier\nhere, predicting a barrier height EGAP\nb = 0.22 eV.\nIV. CONCLUSION\nWe have explored a new framework for quantifying the\nuncertainties in different QoIs for pure silicon, for a range\nof different size of ACE IPs and amounts of training data.\nThis new methodology combines a base, Bayesian for-\nward propagation method incorporating parametric un-\ncertainty and uncertainty in the training data, with a13\nfrequentist conformal prediction to generate robust pre-\ndiction sets on physical quantities outside of the training\nand calibration set. By comparison to reference DFT re-\nsults for a range of QoIs, we observe good coverage for\nmany of the constructed potentials. We further investi-\ngate the structure of these error bars by decomposition\ninto the constituent quantities; the ensemble standard\ndeviation and the conformal rescale factor.\nV. SOFTWARE AVAILABILITY\nCode and data supporting this work will be made avail-\nable on publication.\nACKNOWLEDGMENTS\nI.B. is supported by a studentship within the UK\nEngineering and Physical Sciences Research Council-\nsupported Centre (EPSRC) for Doctoral Train- ing\nin Modelling of Heterogeneous Systems, Grant No.\nEP/S022848/1. J.R.K. acknowledges funding from the\nLeverhulme Trust under grant RPG-2017-191 and the\nNOMAD Centre of Excellence funded by the European\nCommission under grant agreement 951786. Comput-\ning facilities were provided by the Scientific Computing\nResearch Technology Platform at the University of War-\nwick.\nThe authors acknowledge insightful discussions with\nRyan S. Elliot and Mark Transtrum in the context of the\npartnership between the HetSys CDT and OpenKIM and\nwith Ralf Drautz and Yury Lysogorskiy in the context of\nthe partnership between HetSys and ICAMS.\nVI. REFERENCES\n1F. H. Stillinger and T. A. Weber, “Computer simulation of lo-\ncal order in condensed phases of silicon,” Physical Review B 31\n(1985).\n2M. S. Daw and M. I. Baskes, “Embedded-atom method: Deriva-\ntion and application to impurities, surfaces, and other defects in\nmetals,” Physical Review B 29(1984).\n3A. P. Bart´ ok, M. C. Payne, R. Kondor, and G. Cs´ anyi, “Gaussian\napproximation potentials: The accuracy of quantum mechanics,\nwithout the electrons,” Physical review letters 104(2010).\n4A. P. Thompson, L. P. Swiler, C. R. Trott, S. M. Foiles, and\nG. J. 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Drautz, “Atomic Cluster Expansion for Quantum-Accurate\nLarge-Scale Simulations of Carbon,” Journal of Chemical The-14\nory and Computation (2023).\n28https://github.com/ACEsuit/ACEpotentials.jl (2023).\n29C. M. Bishop, Pattern Recognition and Machine Learning\n(Springer-Verlag, 2006).\n30D. J. C. MacKay, “Bayesian Interpolation,” Neural Computation\n4(1992).\n31V. Vovk, A. Gammerman, and G. Shafer, Algorithmic learning\nin a random world (Springer, 2005).\n32A. H. Larsen, J. J. Mortensen, J. Blomqvist, I. E. Castelli,\nR. Christensen, M. Du lak, J. Friis, M. N. Groves, B. Hammer,\nC. Hargus, E. D. Hermes, P. C. Jennings, P. B. Jensen, J. Ker-\nmode, J. R. Kitchin, E. L. Kolsbjerg, J. Kubal, K. Kaasbjerg,\nS. Lysgaard, J. B. Maronsson, T. Maxson, T. Olsen, L. Pastewka,\nA. Peterson, C. Rostgaard, J. Schiøtz, O. Sch¨ utt, M. Strange,\nK. S. Thygesen, T. Vegge, L. Vilhelmsen, M. Walter, Z. Zeng,\nand K. W. Jacobsen, “The atomic simulation environment—a\nPython library for working with atoms,” Journal of Physics: Con-\ndensed Matter 29(2017).Appendix15\npmax\\# Configs 50 150 250 350\n4 93.42±37.9 89.54±13.08 90.24±18.01 91.06±18.18\n9 85.44±96.6 88.01±3.81 87.44±4.13 87.66±3.97\n13 85.02±71.84 88.58±1.11 88.26±1.47 88.47±1.28\n19 75.75±99.89 88.77±0.65 89.09±0.95 88.33±1.05\n24 59.27±72.75 89.44±0.47 88.75±0.57 91.3±1.02\nTABLE I: Selected predictions for Si bulk modulus with uncertainties from Figure 2. All values are reported in units\nof GPa, and bold values are those which do not agree with the DFT reference result22ofBDFT= 88.6 GPa.\npmax\\# Configs 25 125 225 325\n4109.85 ±73.18\n76.6±28.89\n44.61±50.13119.42 ±80.57\n81.88±23.65\n43.89±63.24114.07 ±47.25\n78.71±21.87\n40.98±50.81111.29 ±53.71\n81.62±21.93\n37.45±57.71\n12139.66 ±27.75\n63.37±20.72\n72.81±11.33143.44 ±12.07\n60.94±6.67\n71.13±9.58144.39 ±12.02\n60.57±7.48\n72.92±11.06144.39 ±9.92\n60.34±6.04\n73.25±10.1\n19139.34 ±37.75\n59.53±29.43\n71.57±11.72151.65 ±10.04\n58.79±5.38\n73.54±5.95150.57 ±7.9\n58.34±4.61\n73.82±6.44148.53 ±7.91\n59.19±4.76\n72.7±6.31\nTABLE II: Selected predictions for Si Cijwith uncertainties from Figure 4, with values listed in the order C11,C12,\nC44in each cell. All values are reported in units of GPa, and bold values are those which do not agree with the DFT\nreference results22,C11= 153 .3 GPa, C12= 56.3 GPa, C44= 72.2 GPa."    },    {        "title": "2402.15442v1.GROS__A_General_Robust_Aggregation_Strategy.pdf",        "content": "GROS: A General Robust Aggregation\nStrategy.\nAlejandro Cholaquidis1, Emilien Joly2and Leonardo Moreno3\n1Centro de Matem´ atica, Facultad de Ciencias, Universidad de la\nRep´ ublica, Uruguay\n2Centro de Investigaci´ on en Matem´ aticas, CIMAT, M´ exico.\n3Departamento de M´ etodos Cuantitativos, Facultad de Ciencias\nEcon´ omicas y de Administraci´ on, Universidad de la Rep´ ublica, Uruguay.\nAbstract\nA new, very general, robust procedure for combining estimators in\nmetric spaces is introduced GROS. The method is reminiscent of the\nwell-known median of means, as described in [18]. Initially, the sample\nis divided into Kgroups. Subsequently, an estimator is computed\nfor each group. Finally, these Kestimators are combined using a\nrobust procedure. We prove that this estimator is sub-Gaussian and\nwe get its break-down point, in the sense of Donoho. The robust\nprocedure involves a minimization problem on a general metric space,\nbut we show that the same (up to a constant) sub-Gaussianity is\nobtained if the minimization is taken over the sample, making GROS\nfeasible in practice. The performance of GROS is evaluated through\nfive simulation studies: the first one focuses on classification using k-\nmeans, the second one on the multi-armed bandit problem, the third\none on the regression problem. The fourth one is the set estimation\nproblem under a noisy model. Lastly, we apply GROS to get a robust\npersistent diagram.\nKeywords: Bandits, Median of means, Robustness, Sub-Gaussian estimator,\nTopological data analysis\n1 Introduction\nThe problem of combining estimators has been extensively studied in statis-\ntics. There are recent proposals that merge regression estimators (see, for\ninstance, [5]), classifiers (see [12]), and density estimators (see [11]), among\nothers. In these scenarios, the aim is to merge the estimators to generate\none that, at least asymptotically, surpasses the best of the group. In other\n1arXiv:2402.15442v1  [math.ST]  23 Feb 2024instances, the aim is to derive a robust estimator. This is the case of the\nwell-known median of means (MOM) estimator, which aims to derive a ro-\nbust estimator of the mean of a random variable. In the MOM the data, ℵn\n(an i.i.d. sample of a random variable X), is first randomly partitioned into\nKgroups. Subsequently, the mean of each group is computed. The MOM\nestimator is then the median of these Kmeans. If the variance of the data\nis assumed to be finite, this estimator is sub-Gaussian. For further details,\nwe refer to [18], [24], and the references therein. For the case of random vec-\ntors, the so-called median of means tournament is introduced in [29], where\nis proved to be sub-Gaussian. In [36] it is proved that the median of means\ntournament has break-down point ⌊(K−1)/2⌋/n, where ⌊x⌋denotes the floor\nofxandnis the sample size.\nThe use of these estimators has proven to be valuable in varied statistical\nscenarios, such as in machine learning, see [28]. In these contexts, it is ad-\nvisable to consider estimators that, without removing outliers, do not reduce\ntheir precision. Robust statistics point in this direction, see [30] and [2].\nFollowing this idea of dividing into Kgroups, calculating the estimator\nin each group, and then combining them, we will introduce a new way to\ncombine the estimators in order to obtain, in a very general framework,\na new one that, under the assumption that the estimators by group are\nindependent, turns out to be sub-Gaussian (see Theorem 2 below). This new\nstrategy, in what follows: GROS, has breakdown point ⌈K/2⌉/n, where ⌈x⌉\ndenotes the ceiling of x, see Section 2. The only assumption we make is that\nthe original sample comes from a random variable with finite variance and\nthat the space where the group estimators take their values is a separable\nand complete metric space.\nWhile the combined estimator requires solving a minimization problem\nin a metric space, we prove that if it is minimized on the sample of the group\nestimators, an appropriate candidate is obtained. We also determine how\nmuch is lost by this choice, see Section 3.\nDue to the immense generality of GROS , it can be applied to various areas\nof statistics where robustness plays a key role. Furthermore, the space in\nwhich the estimators reside doesn’t need to be a metric space: a pseudometric\nsuffices. This permits the consideration of estimators in the space of bounded\nsubsets of Rdequipped with the Hausdorff distance or the measure distance,\nwhich is the case when the object to be estimated is, for instance, a set. This\nspace will be used in our fourth simulation example, see subsection 4.4.\nWe have chosen to present five problems to demonstrate its performance,\n2comparing it with techniques explicitly crafted for these specific issues. Some\nof these techniques were already designed to yield robust estimators. Specif-\nically, we treat:\n•The traditional clustering problem;\n•The multi-armed bandit problem with heavy-tailed rewards;\n•Regression in the presence of noisy data;\n•The estimation of a convex set when dealing with a noisy sample;\n•An application to topological data analysis.\nIt is worth noting that the fourth problem cannot be successfully treated\nusing conventional methods such as convex hulls or r-convex hulls, as we will\nsee.\nAs expected, in all cases the performance of GROS is noticeably better\nthan the proposals that do not consider the presence of outliers, see Section\n4. Moreover, the performance is good compared to methods that do consider\nthe presence of outliers, even surpassing some in certain cases.\nThe structure of the paper is outlined as follows: Section 2 introduces\nGROS within a broad context and examines its robustness properties. Sec-\ntion 3 presents a modification of GROS to simplify its computational aspects.\nIn Section 4, the application of GROS across five problems is discussed. The\npaper concludes with Section 5, where the findings and implications of the\nstudy are elaborated.\n2 Robust aggregation of weakly convergent\nestimators\nIn this section, we define and study a new proposal of a robust estimator\nbased on the aggregation of estimators. We assume given a sample ℵn=\n{X1, . . . , X n}of i.i.d. random elements with common distribution P. Let µ\nbe a certain characteristic of P. We assume that µbelongs to a complete and\nseparable metric space Mendowed with a metric d. As we mentioned in the\nintroduction, all the results in this paper remains true if dis a pseudo metric.\nIn the context of robust estimation, one goal is to obtain sub-Gaussian type\ninequalities for the deviation of an estimator bµfrom µ. A common way of\n3defining a robust, distribution-free estimator is to make Kdisjoint groups out\nofℵn, hence, to create a collection of Kindependent estimators µ1, . . . , µ K∈\nM. We define the robust aggregation of µ1, . . . , µ K, GROS, by\nµ∗= argmin\nν∈Mmin\nI:|I|>K\n2max\nj∈Id(µj, ν). (1)\nThe minimization is taken over all the possible subsets Iof{1, . . . , K }that\ncontain at least ⌊K/2⌋+ 1 indices. However, it is easy to see that it is\nenough to minimize over all possible subsets Iwhose cardinality, |I|, fulfills\n|I|=⌊K/2⌋+ 1. Observe that, for any ν∈ M ,\nmin\nI:|I|>K\n2max\nj∈Id(µj, ν) =:d(ν, ν(⌊K/2⌋+1)−NN),\nwhere ν(⌊K/2⌋+1)−NNdenotes the ( ⌊K/2⌋+1)-nearest neighbor of νinµ1, . . . , µ K.\nThis last quantity is a measure of the depth of νinside the set µ1, . . . , µ K.\nThen, µ∗is the point with the least depth from all the candidates ν∈ M .\nIn full generality, the set of minimizers of (1) may not be unique. In that\ncase, we still denote by µ∗one of the minimizers arbitrarily chosen. Since we\nassumed that Mis a Polish space, it is easy to see that µ∗̸=∅.\nRemark. A natural generalization of µ∗is to define, for q∈[1/2,1],\nµ∗\nq= argmin\nν∈Mmin\nI:|I|>Kqmax\nj∈Id(µj, ν).\nAll the results we present are for q= 1/2 but they remain true for any\nq∈[1/2,1].\nAs we said in the Introduction, we aim to combine the estimators µ1, . . . , µ k\nin a robust way. More precisely, let us recall the definition of finite-sample\nbreakdown point introduced by Donoho (see [19]).\nDefinition 1. Letx={x1, . . . , x n}be a dataset, θan unknown parameter\nlying in a metric space Θ, andbθn=bθn(x)an estimator based on x. LetXp\nbe the set of all datasets yof size nhaving n−pelements in common with\nx:\nXp={y:|y|=nand|x∩y|=n−p}.\nThen, the breakdown point of bθnatxisϵ∗\nn(bθn,x) =p∗/n,where\np∗= max {p≥0 :∀y∈ X p,bθn(y)is bounded and\nbounded away from the boundary ∂Θ,if∂Θ̸=∅}.\n4From (1) it follows easily that the finite-sample breakdown point of µ∗is\n⌈K/2⌉/n, which is the same order obtained in [36] for the MOM aggregation\nstrategy mentioned in the Introduction.\nThe following lemma states that if there exists an ηfor which at least\nK/2 of the µiare at a distance at most tfrom η, then any minimum in (1) is\nat a distance at most 2 tfrom η. This, as we will see, implies the robustness\nand sub-Gaussianity of the estimator (1). Let us write [ K] ={1, . . . , K }.\nLemma 1. Assume that there exist an η∈ M and an I⊂[K]with|I|> K/ 2\nsuch that for all j∈I,d(µj, η)≤t. Then, d(µ∗, η)≤2t.\nProof. By hypothesis, there exists a set Iof cardinality at least K/2 such\nthat max j∈Id(µj, η)≤t. Since µ∗is a minimizer of (1), there exists a set I0\n(a priori different from I) with |I0|> K/ 2 such that max j∈I0d(µj, µ∗)≤t.\nNow, note that |I|+|I0|> K and so there exists j∈I∩I0such that\nd(µ∗, η)≤d(µ∗, µj) +d(µj, η)≤2t, which concludes the proof.\nLemma 1 can be applied when η=µ, in which case if a group of at\nleast K/2 estimators is reasonably close to the objective µ, then µ∗itself\nis reasonably close. Such an estimator is robust to outliers since this effect\nwill not be altered by the bad behavior of up to K/2−1 estimators. This\nlemma is a technical fact that will allow us to use the so called binomial\nargument. Indeed, assume that µis such that for any 0 < p < 1/2, there\nexists t=t(n, K) such that for all k∈1, . . . , K ,P(d(µk, µ)> t)≤p. Since\nthe estimators µ1, . . . , µ Kare independent,\nP(d(µ∗, µ)>2t)≤P(∃I:|I| ≥ ⌊K/2⌋and∀i∈I, d(µi, µ)> t)\n=P KX\nk=1I{d(µk,µ)>t}≥ ⌊K/2⌋!\n≤P(BK,p≥ ⌊K/2⌋)\n≤e−2(⌊K/2⌋−Kp)2\nK , (2)\nwhere BK,pdenotes a random variable with binomial distribution, with pa-\nrameters Kandp. Let us assume that the µiare identically distributed.\nThen, if we choose p= 1/4 and K=⌈8 log( δ−1)⌉, using the fact that\n⌊⌈x⌉/2⌋ ≤ ⌈ x⌉/2, we get from (2) together with Markov’s inequality\nP\u0010\nd(µ∗, µ)>4p\nEd2(µ1, µ)\u0011\n≤δ.\n5Lastly, we have proved the following theorem:\nTheorem 2. Letℵn={X1, . . . , X n}be an i.i.d. sample from a distribution\nP. Let µ∈ M a certain characteristic of Pwhere (M, d)is a metric space.\nLetδ >0andK=⌈8 log( δ−1)⌉. We split ℵnintoKdisjoint groups (as-\nsume that nis large enough to guarantee that n/K =ℓ∈N), and create K\nestimators µ1, . . . , µ Kofµ. Let µ∗be the aggregation defined by (1). Then,\nP\u0010\nd(µ∗, µ)>4p\nEd2(µ1, µ)\u0011\n≤δ. (3)\nRemark. In Theorem 2, the restriction that nis large enough to guarantee\nn/K =ℓ∈Nis purely technical, to ensure that the µ1, . . . , µ Kare identically\ndistributed and then to get the clean expression (3). If this is not the case,\nthat is, the groups are unbalanced, the estimators µ1, . . . , µ Kare independent\nbut not necessarily identically distributed, and the obtained bound is\nP\u0012\nd(µ∗, µ)>4q\nmax\ni=1,...,KEd2(µi, µ)\u0013\n≤δ. (4)\n2.1 Mis-specification of the set M\nLemma 1 uses the fact that ηis inside the set M. Now, assume given a imper-\nfect set fMthat is mis-specified in the sense that d(η,fM) = infν∈fMd(η, ν) =\nϵ >0. The true parameter of interest does not belong to the set of features\nfM. The minimization is then given by\neµ= argmin\nν∈fMmin\nI:|I|>K\n2max\nj∈Id(µj, ν). (5)\nLemma 3. Assume that there exist an η∈ M and an I⊂[K]with|I|> K/ 2\nsuch that for all j∈I,d(µj, η)≤t. Then, d(eµ, η)<2t+ϵ.\nProof. By the triangle inequality ∀δ >0, there exists an ηδ∈fMsuch that\nmax\nj∈Id(µj, ηδ)≤t+ϵ+δ.\nSo there exists a set I0of cardinality |I0|> K/ 2, such that max j∈I0d(µj,eµ)≤\nt+ϵ+δ.\nSince|I|+|I0|> K, there exists j0∈I∩I0. Then, d(eµ, η)≤d(eµ, µ j0) +\nd(µj0, η)≤2t+ϵ+δ. Since this holds for all δ >0, it follows that d(eµ, η)<\n2t+ϵ.\n6The following corollary is a direct consequence of Lemma 3.\nCorollary 2.1. Letℵn={X1, . . . , X n}be an i.i.d. sample from a distri-\nbution P. Let µ∈ M be a certain characteristic of P, where (M, d)is a\nmetric space. We assume given a set fM ⊂ M (possibly random) such that\nd(η,fM) =ϵ >0. Let δ >0andK=⌈8 log( δ−1)⌉. We construct the K\ndisjoint groups and Kestimators µ1, . . . , µ Kofµas in Theorem 2. Then,\nP\u0010\nd(eµ, η)>4p\nEd2(µ1, µ) +ϵ\u0011\n≤δ. (6)\n3 Computational aspects\nEquation (1) supposes that one is able to find minimizers of a complex func-\ntional on the metric space M, which is often an unfeasible problem. To\nsimplify that task, one can restrict the minimization to the set of estimators\nµ1, . . . , µ K. That is, we find the index j∗such that\nj∗= argmin\nj=1,...,Kmin\nI:|I|>K/2max\ni∈Id(µi, µj). (7)\nThe next lemma and theorem state that µj∗has the same sub-Gaussian type\nbound (up to a constant) as µ∗.\nLemma 4. Assume that there exists an I⊂[K]such that |I|> K/ 2, and\nfor all j∈I,d(µ, µ j)≤t. Then d(µj∗, µ)≤3t.\nProof. LetI⊂[K] be such that |I|> K/ 2, and for all j∈I,d(µ, µ j)≤t,\nwe have d(µi, µj)≤d(µi, µ) +d(µ, µ j)≤2tfor all i, j∈I. Then, there\nexists an I0with cardinality at least K/2 such that d(µj∗, µi)≤2tfor all\ni∈I0. Since |I|+|I0|> K, there exists j0∈I∩I0. Lastly, d(µj∗, µ)≤\nd(µj∗, µj0) +d(µj0, µ)≤3t.\nBy means of Lemma 4, it is possible to give a practical version of Theorem\n2.\nTheorem 5. Assume the hypotheses of Theorem 2. Let µj∗∈ {µ1, . . . , µ K}\ndefined by the optimization (7). Then,\nP\u0010\nd(µj∗, µ)>6p\nEd2(µ1, µ)\u0011\n≤δ. (8)\n7Note that Equation (8) is the same as Equation (3) except that it has the\nconstant 6 in the right-hand side. This shows that the practical version of\nthe estimator, µj∗, has essentially the same rate of convergence as µ∗, but it\ncan be deteriorated by a constant factor.\n4 Some applications of GROS\n4.1 Classification by k-means\nOne of the most popular procedures for determining clusters in a dataset is\nthek-means method. Although the precursors of this algorithm were Mac-\nQueen in 1967, see [31], and Hartigan in 1978, see [23]. Pollard in 1981 [34]\nproved the strong consistency of the method and in 1982, in [35], determined\nits asymptotic distribution.\nGiven k, the k-means clustering procedure partitions a set {x1, . . . , x n} ⊂\nRdintokgroups as follows: first kcluster centres ajare chosen, in such a\nway as to minimise\nWn=1\nnnX\nl=1min\n1≤j≤k∥xl−aj∥2.\nThen it assigns each xlto its nearest cluster centre. In this way, each centre\nacquires a subset Clas its associated cluster. The mean of the points in\nClmust equal al, otherwise Wncould be decreased by replacing alwith the\ncluster mean in the first instance, and then reassigning some of the x’s to\ntheir new centres. This criterion is then equivalent to that of minimising the\nsum of squares between clusters.\nThe standard k-means algorithm starts from a set a(1)\n1, . . . , a(1)\nk,and alter-\nnates, up to a stopping criterion, two steps: Assignment step: Assigns each\nobservation xjto the cluster whose centre is the closest one. Update step:\nRecalculate means (centroids) for the observations assigned to each cluster,\nby averaging the observations in each cluster.\nThese k-centres are not robust to the presence of outliers, nor to the\ndistribution’s possession of heavy tails. There are several proposals in the\nliterature that seek to make the k-means algorithm more robust. An example\nis the k-medoids algorithm (called PAM), see [25, 26]. In this algorithm, in\nthe second step, one chooses that point in the cluster which minimises the\n8sum of the distances to the remaining observations. These points are called\nthe medoids.\nAnother proposal for a robust version of k-means, TClust, is developed\nin [16]. It is based on an α > 0 trimming of the data. This trimming\nis self-determined by the data and aims to mitigate the impact of extreme\ndata.\nWe propose a simple modification to the second step of the k-means\nalgorithm, which will be referred to as RobustkM . The idea is very simple:\ninstead of calculating the centroids by taking the arithmetic mean in each\ngroup, the centroids are determined using (7).\n4.1.1 Simulations\nTo evaluate the performance of this proposal, we run a small simulation\nstudy. In all cases, the data consist of an i.i.d. sample X1, . . . , X n, whose\ncommon distribution, FX, is given by the mixture of three bi-variate Student\ndistributions. More precisely,\nFX(x) = 0 .45T(x, µ 1, ν,Σ1) + 0.45T(x, µ 2, ν,Σ2) + 0.1T(x, µ 3, ν,Σ3),(9)\nwhere T(x, µ, ν, Σ) denotes the bi-variate cumulative Student distribution\nfunction with mean µ∈R2, variance and covariance matrix Σ, and ν= 2\ndegrees of freedom.\nIn the examples, we chose µ1= (6,0),µ2= (−6,0),µ3= (0,6), Σ 1=\nΣ2=\u00003 0\n0 3\u0001\nand Σ 3=\u00004 1\n1 9\u0001\n.\nFigure 1 shows a simulation with n= 1000 points. It can be seen that the\ndispersion of the third group makes the clustering problem more difficult.\nLetπbe a permutation of the set of labels {1,2,3}. Denote by C(x)∈\n{1,2,3}the true (unknown) label, and bC(x)∈ {1,2,3}the label assigned by\nthe algorithm to observation x. Then the classification error is given by\nmin\nπ1\nnnX\ni=11n\nC(xi)̸=π(bC(xi))o. (10)\nFigure 2 shows the performance of RobustkM (with K= 10), k-means,\nPAM, and TClust (with α= 0.01), over 1000 replications. In TClust, the\ntrimmed data (at the end of the algorithm) are assigned to the nearest cen-\ntres. This toy example shows that the proposed algorithm is a competitive\nalternative to other methods that “robustify” k-means.\n9Figure 1: Simulation of 1000 observations of the multivariate Student mixture\n(9). Observations are colored according to the component of the mixture\nwhich the data comes from.\n4.2 Bandits\nThe bandit problem was first proposed by Thompson in 1933, and has been\nrecently been gaining increasing attention. There have been several adapta-\ntions to various frameworks, as evidenced by comprehensive surveys, such as\n[7] and [9], which cover a broad spectrum of applications.\nAccording to [27], a bandit problem can be described as a sequential\ngame played between a learner and an environment, spanning over a number\nof rounds, T, which is a positive integer. In each round, denoted by t, the\nlearner first chooses an action (denoted by At), from a defined set of actions\n(denoted by A). Following this, the environment unveils a reward Xt, which\nis a real number. Given the action At, the reward Xtis assumed to be\nindependent of the past.\nIn the stochastic L-armed bandit problem, there are Lpossible actions.\nThe rewards are derived from a set of distributions denoted by P1through PL\n(not depending on t). The aim of the learner is to identify an arm, denoted\nbyi∗, whose distribution yields the highest mean, denoted by E(Pi∗). This\n10Figure 2: Box plot of classification errors, according to (10), of K-means,\nTClust, PAM and RobustKM over 1000 replicates.\nis equivalent to minimizing the regret\nRT=TE(Pi∗)−TX\nt=1E(PAt).\nA classic example of such an algorithm is the well-known “upper confi-\ndence bands” (UCB). Let us recall that, in the UCB, the first choice A1is\nmade at random, and for t∈ {1, . . . , T −1},\nAt+1= argmax\nj∈{1,...,L}X1I{A1=j}+···+XtI{At=j}\nNt,j+s\nlog(t)\nNt,j(11)\nwhere XiI{A1=j}is the reward of arm jat time iif this arm is chosen at\nthat time and Nt,j=Pt\ns=1I{As=j}is the number of times that the arm jwas\nchosen.\n4.2.1 Heavy-tailed bandits\nTo get a logarithmic regret for the UCB (i.e., RT/log(T)→C), it is usually\nassumed that the distributions Piare sub-Gaussian. This can be weakened\n11to require the distributions to have finite moment generating function, see\n[3]. To overcome this limitation, instead of just estimating the mean at each\nstep tof the UCB, in [8] it is proposed to use a robust estimator, bµi,t, of\nE(Pi), that fulfills the following assumption:\nAssumption 1 Letϵ∈(0,1] be a positive parameter, and let c, vbe a\npositive constant. Let X1, . . . , X nbe i.i.d. random variables with finite\nmean µ. Suppose that for all δ∈(0,1) there exists an estimator bµ=bµ(n, δ)\nsuch that, with probability at least 1 −δ,\nbµ≤µ+v1/(1+ϵ)\u0012clog(1/δ)\nn\u0013ϵ/(1+ϵ)\n(12)\nand also, with probability at least 1 −δ,\nµ≤bµ+v1/(1+ϵ)\u0012clog(1/δ)\nn\u0013ϵ/(1+ϵ)\n. (13)\nIn that case we say that bµfulfills Assumption 1.\nRemark. From (3), our robust proposal µ∗, applied on each arm, and based\nonK=⌈8 log(1 /δ)⌉groups, fulfills Assumption 1, for ϵ= 1, v=V(Pi),\n(V(Pi) being the variance of the distribution Pi), and c= 16. In that case,\nfor an arm i= 1, . . . , L ,n=Nt,i/K.\n[8] proposes the following robust variant of the UCB: given ϵ∈(0,1], for\narmi, define bµi,s,tas the estimator bµ(s, t−2) based on the first sobserved\nvalues Xi,1, . . . , X i,sof the reward of arm i. Define the index\nBi,s,t=bµi,s,t+v1/(1+ϵ)\u0012clog(t2)\ns\u0013ϵ/(1+ϵ)\nfors, t≥1 and Bi,0,t= +∞. Then, at time tdraw an arm maximizing\nBi,Ni,t−1,t.\nWe propose the following algorithm. First we choose the arms at ran-\ndom from t= 1, . . . , t 0, where t0guarantees that for all i= 1, . . . , L ,\nNt0,i/⌈8 log( t2\n0)⌉ ≥1. We compute, for each arm, the estimator (1), denoted\nbyµ∗\nt0,i, where we split the Nt0,iobservations into K=⌈8 log( t2\n0)⌉groups,\nand compute the mean of each group. Define the index\nBi,Nt0,i,t0=µ∗\nt0,i+ 4q\nbV(Pi)\u0012log(t2\n0)\nNt0,i\u00131/2\n,\n12wherebV(Pi) is any consistent estimation of V(Pi), or an upper bound. At\ntime t0+ 1, we choose the arm that maximize Bi,Nt0,i,t0, at time t0+ 2 we\nchoose the arm that maximize Bi,Nt0+1,i,t0+1, and so on.\nProposition 1 in [8] proves that this algorithm attains logaritmic regret.\nMore precisely,\nRT≤X\ni:∆i>0\u0012\n32\u0012V(Pi)\n∆i\u0013\nlog(T) + 5∆ i\u0013\n,\nwhere ∆ i=E(Pi∗)−E(Pi).\n4.2.2 Simulations\nAs a toy example, let us consider the classical two-armed bandit problem\nand rewards given by Xt|At=j∼µj+S(3),forj= 1,2, where S(3) is a\nrandom variable following Student’s distribution with 3 degrees of freedom,\nµ1= 7, and µ2= 8. In our algorithm, indicated by RUCB in Figure 3, the\nfirstt0= 40 arms are chosen at random.\nThe results are shown in Figure 3, where the red dotted horizontal line\n(y= 8) is the maximum expected gain. The dashed lines corresponds to the\nmean rewards of the UCB (orange) and the RUCB (blue) respectively. As\ncan be seen, it takes the RUCB algorithm about 120 steps to outperform\nthe UCB algorithm, and the difference grows larger as the number of steps\nincreases.\n4.3 Robust regression\nA current topic of interest in statistics is that of regression models in the\npresence of noise. It is known that a small fraction of outliers can cause\nsevere biases in classical regression estimators. These classical models gener-\nally assume additive residuals in the model with finite second moment (e.g.,\nGaussian). An extensive review of robust outlier estimation methods for\nnonparametric regression models is provided in [38].\nWe tackle the problem of estimating the function m:X →R, from an\ni.i.d. sample of a random element ( X, Y)∈ X × R, that satisfies the model\nY=m(X) +ϵ\n13Figure 3: Cumulative gains over 500 replications, for t= 1, . . . , 750. The\nred dotted horizontal line ( y= 8) is the maximum expected gain. The black\ndotted vertical line ( x= 40) indicates the number of random warm-up runs\nin the RUCB algorithm. The dashed lines depict the mean reward of the\nUCB (orange) and RUCB (blue) algorithms.\nwhere ϵis a noise term such that E[ϵ|X] = 0. To keep the following discussion\nfairly simple, we will only cover the case X=R. In the context where very\nlittle information is known about m, a general estimator is often given by\nthe kernel estimator\nbm(x) =Pn\ni=1Kh(Xi−x)YiPn\ni=1Kh(Xi−x), (14)\nwhere Kh(x) =h−1ϕ(h−1x) and h >0 is some bandwidth parameter. The\nfunction ϕis non-negative and such thatR\nRϕ(x)dx= 1. In general, the\ncontinuity of the function mis enough to get the consistency of the kernel\nestimator bmand a light tailed behavior for ϵgives sub-exponential deviation\nbounds for the estimator around its mean value m. Nevertheless, in various\nconcrete settings, one can face heavy tailed distributions for ϵin such a way\nthat the estimator proposed in (14) becomes highly unstable. Indeed, the\npresence of (virtually) one outlier is enough to drive the estimator towards\n14values very distant from m. One natural way to measure the quality of the\nestimator bmis to consider the L2distance\nd2(bm, m ) =E\u0002\n|bm(X)−m(X)|2\u00031/2.\nIn this context, it is possible to introduce the robust version of the estimator\nby considering GROS with the distance d2. In practice, the distance d2\nis actually intractable. To overcome this difficulty, it is common to use a\ndiscretization on a mesh for the space Xand approximate the integral by\na Riemann sum. This estimator verifies a bound as in Theorem 2 for the\nassociated distance. The estimator m∗is, in the sense of Definition (1), the\nbest candidate in the class of the estimator m1, . . . , m Kconstructed on the\ndisjoint groups G1, . . . , G Kof data points. These estimators m1, . . . , m Kare\ndefined by\nmj(x) =P\ni∈GjKh(Xi−x)YiP\ni∈GjKh(Xi−x),\nso that these estimators are independent. Following subsection 2.1, the fol-\nlowing estimator for bmcan be proposed:\nbm= argmin\nm∈fMmin\nI:|I|>K\n2max\nj∈Id2(mj, m),\nwhere the ( mj)K\nj=1are the kernel estimates of mon each of the Kgroups,\nand the set fMdenotes the set of functions that are piece-wise equal to\none of the mi. The difference from the naive definition is that the set of\nminimization does not end with an estimator of the form mi∗, which allows\navoiding choosing a function that may be a good fit in some regions of the\nspace but sensitive to outliers in other parts.\nIn order to fully use the result of Equation (4), we cite a result that gives\nan upper bound for the mean squared error for any of the estimators mj\npreviously defined. In chapter 5 of [22, Theorem 5.2], we get that if mis\nλ-Lipschitz, and that Var( Y|X=x)≤σ2for all x∈ X, then\nE\u0002\n∥m1−m∥2\u0003\n≤c\u0012σ2+ supx∈Xm(x)2\n(n/K)hd\u0013\n+λ2h2.\nThis induces a choice of hof the form\nh=c′\u0012K\nn\u0013 1\nd+2\n.\n15The upper bound then takes the form of\nE\u0002\n∥m1−m∥2\u0003\n≤c′′\u0012\nσ2+ sup\nx∈Xm(x)2\u00132/d+2\n×\u0010n\nK\u0011−2/d+2\n,\nwhich can be directly plugged into the main bound of the theorem.\n4.3.1 Simulations\nWe compare, by means of simulations, the performance of GROS with some\nclassical and robust regression alternatives proposed in the literature.\nIn this example we consider a sample X1, . . . , X 1000with uniform distri-\nbution on [0 ,5]. Let m(x) = 4 sin(3 x) and suppose that the centered noise\nfollow the skew-normal Student distribution, see [21, 4], whose density is\ndefined as follows: denote by t(x, κ, ν ) the density of the non-standardized\nStudent’s distribution with location κand and νdegrees of freedom, and\nwith cumulative distribution function T(x, κ, ν ). Then, the density of the\nskew-normal Student is\nf(x;κ, ν, σ, ξ ) =1\nσt(x/σ, κ, ν )T\u0012ξx\nσ, κ, ν\u0013\n,\nwhere σ >0 denotes the scale parameter. The slant (or skewness parameter)\nisξ.\nWe will write NW for the non-parametric Nadaraya–Watson kernel re-\ngression estimator (14), see [32, 41]. As robust estimators, we consider the\nproposals developed in [33] (ONL estimator) and [6] (SBMB estimator). Both\nestimators are implemented in the Rlanguage: the first in the fields library\nand the other in the RBF library. Our estimator in this context will be\ncalled RANW (Robust Aggregation for Naradaya–Watson). For the latter,\nK= 12 was considered. Figure 4 shows the estimated regressions in each\nscenario and in red the true function f. It is clear that the NW estimator\nperforms poorly in all cases.\nThe performance of each estimator is measured by the average distance\nd2(bm, m ) over 1000 replicates, considering κ= 0,ν= 3 (note that the noises\nhave heavy tails) and varying the parameters σandξ. For the bandwidth\nparameter h, we choose 0 .2 in all cases.\nFigure 4 shows box plots of the errors for four choices of the parame-\nters. As can be seen, when the noise is asymmetric (i.e., ξ̸= 1), the best-\nperforming estimators are RANW and ONL, depending on the value of the\n16scale parameter. Note that when the distribution of the noise is symmetric\n(i.e., ξ= 1), SBMB and RANW perform the best.\n(a)σ= 9, ν= 3, ξ= 1\n (b)σ= 9, ν= 3, ξ= 9\n(c)σ= 16, ν= 3, ξ= 1\n (d)σ= 16, ν= 3, ξ= 9\nFigure 4: Box plot of classification errors (according to L2 distance) in 1000\nreplicates. The different scenarios are obtained in the skew-normal Student\ndistribution with σ∈ {9,16}andξ∈ {1,9}, fixed ν= 3 and κ= 0.\n4.4 Robust set estimation\nSet estimation involves determining a set, or a function of that set, based\non a random sample of points. This set could represent various things, such\nas the support of a probability distribution (see, for instance, [37, 10]), its\nboundary (see, for instance [17], the surface area of the boundary (see, for\ninstance [1]), to name a few.\nWithin this framework, shape constraints are usually imposed, convexity\nbeing the most well-known. However, it can be restrictive for some appli-\n17cations, such as, for instance, if the set is the home-range of a species (see\n[13], [14], [15] and references therein). Usually, the available data is an i.i.d.\nsample of a random vector whose support is the unknown set. For classic\nestimators such as the convex hull, r-convex hull, or cone-convex hull, any\nnoise in the sample, no matter how small, drastically changes the estimators.\nThis becomes even more pronounced in the case of convexity.\n4.4.1 Simulations\nWe show the performance of our robust proposal under a noisy model, where\nthe aim is to estimate a convex set. More precisely, let D(r, R) be the uniform\ndistribution on the ring in R2with inner radius rand outer radius R. We\nsimulated 2000 i.i.d. observations of the mixture\n(1−λ)D(0,1) +λD(1,1.25).\nThe aim is to estimate D(0,1) from this sample. We have chosen λ= 0.01\nas the proportion of noise. The estimator, referred to as RChull, is the one\nproposed in Section 3, and we considered the Hausdorff distance between\ncompact sets to measure the discrepancy between D(0,1) and the estimator.\nTo build our estimator, we first split the original sample at random into\nK= 20 disjoint groups of points of size 100, and compute the convex hull\non each group. Then, we select from the Khulls H1, . . . , H 20, the hull Hj∗,\nwith j∗as in (7).\nTo gain an insight into the improvement resulting from using this robust\nprocedure, we show on the left of Figure 6 the set to be estimated, the\nclassical estimator (the convex hull (Chull) of the whole sample) and the 20\nhulls of the subsamples. The right panel shows the convex hull of the whole\nsample, together with Hj∗.\nTo evaluate the performance of GROS , we run 100 replicates and estimate\nthe set by both methods (the classical convex hull and our robust proposal)\nin each replicate. We calculate the Hausdorff distance of the estimated sets\nRChull and Chull from the set D(0,1). In Figure 7 box plots of these distances\nare shown. The RChull estimator outperforms the classical Chull estimator,\nbut as can be seen, it has larger variability.\n184.5 Robust persistent diagram\nThe robustness of persistent homology to perturbations in data measured by\nthe Hausdorff distance is well established. However, it has a high sensitivity\nto outliers, as discussed in [39, 40].\nIn this section, we introduce a robust persistence diagram, which we call\ntheRobust Wasserstein Estimator , using (7).\nThe measure of dissimilarity between two persistence diagrams P1and\nP2is quantified by the 1-Wasserstein distance W1(P1, P2). This distance\nquantifies the cost associated with achieving the optimal alignment of points\nbetween the two diagrams, as detailed in [20, p. 202].\n4.5.1 Simulations\nThe example examines uniformly simulated data on S1consisting of 600\npoints (baseline sample). It explores two potential scenarios of sample dis-\ntortion, as depicted in Figure 8.\n•Scenario 1: Local perturbation. The original sample is perturbed\nusing Gaussian noise centered at each sample point, with a standard\ndeviation matrix 0 .05×Id.\n•Scenario 2: Group of Outliers: We randomly selected 90% of the per-\nturbed sample as defined in Scenario 1. The remaining 10% are derived\nfrom a Mat´ ern cluster process within the square region [ −0.5,0.5]2.\nThis process is characterized by an intensity of 3 for the Poisson pro-\ncess of cluster centers, a scale of 0 .25, and an average of 20 points per\ncluster.\nThe persistence diagrams for the previously described tree samples were\ncomputed. Figure 9 displays these diagrams, where Dgm ,Dgm 1, and Dgm 2\nrepresent the persistence diagrams of the baseline sample, the locally per-\nturbed sample (Scenario 1), and the sample with outliers (Scenario 2), re-\nspectively. It is evident that the persistence diagram is more distorted in\nscenarios with groups of outliers than it is with those with local perturba-\ntions.\nTo evaluate the performance of our robust proposal (7), the sample is\ndivided into K= 6 distinct groups. The robust persistence diagrams for Sce-\nnarios 1 and 2 are labeled as RDgm 1andRDgm 2, respectively. We computed\nW1(Dgm 1, Dgm ),W1(Dgm 2, Dgm ),W1(RDgm 1, Dgm ), and W1(RDgm 2, Dgm ).\n19A total of 100 independent iterations of this experiment were conducted,\nand box plots of the respective distances are shown in Figure 10. The results\nindicate that in both scenarios, the robust estimates of the persistence dia-\ngrams show improved performance, being closer to the baseline diagram in\nterms of the Wasserstein distance.\n5 Concluding remarks\nWe have demonstrated through simulations that GROS , which is applicable\nto a broad range of problems, significantly improves upon the non-robust,\nproblem-specific solutions for each of the five examples treated. It is also\ncompetitive with robust solutions designed for each specific case, even show-\ning some improvements. GROS proposal’s flexibility makes it applicable to\na wide variety of problems, including those already presented, as well as\nany other scenario where robustness plays a crucial role. Furthermore, more\nexamples using only pseudo-distances may be of interest for future research.\nReferences\n[1] Catherine Aaron, A. Cholaquidis, and R. Fraiman. Estimation of surface\narea. Electron. J. 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Sankhy¯ a: The Indian\nJournal of Statistics, Series A , 26(4) 359–372, 1964.\n24(a)σ= 9, ν= 3, ξ= 1\n (b)σ= 9, ν= 3, ξ= 9\n(c)σ= 16, ν= 3, ξ= 1\n (d)σ= 16, ν= 3, ξ= 9\nFigure 5: Regression functions estimated with the RANW (orange), NW\n(black), ONL (light blue) and SBMB (blue) in one replicate. The true func-\ntion is shown in red. The different scenarios are obtained in the skew-normal\nStudent distribution with σ∈ {9,16}andξ∈ {1,9}, fixed µ= 0 and ν= 3.\n25Figure 6: In the blue dotted line the boundary of the ball of radius 1. The\nsample is shown as solid black points. Outside this ball the sample gener-\nated from D(1,1.25). On the left there are shown the 20 convex hulls of\nthe selected subsamples (of size 100). On the right the convex hullf of the\nwhole sample (Chull) and the robust estimator based on the 20-convex hulls,\n(RChull).\nFigure 7\n26−1.0−0.50.00.51.0\n−1.0 −0.5 0.0 0.5 1.0\nxy(a) Baseline sample\n−1.0−0.50.00.51.0\n−1.0 −0.5 0.0 0.5 1.0\nxy (b) Scenario 1: Local pertur-\nbation\n−1.0−0.50.00.51.0\n−1.0 −0.5 0.0 0.5 1.0\nxy(c) Scenario 2: Groups of out-\nliers\nFigure 8: (a) Baseline sample of 600 points uniformly distributed on S1. (b)\nLocally perturbed sample as described in Scenario 1. (c) Sample perturbed\nin accordance with Scenario 2.\n  \n012345012345\nBirthDeath\n(a)Dgm\n  \n012345012345\nBirthDeath (b)Dgm 1\n  \n012345012345\nBirthDeath (c)Dgm 2\nFigure 9: (a): Persistence diagram of the baseline sample. (b) and (c):\nPersistence diagrams for the samples perturbed according to Scenarios 1 and\n2, respectively.\n270.51.01.52.0\nScenario 1 Scenario 2\nPersistence_DiagramsDistance to DgmMethod\nRobust_Wasserstein\nWassersteinFigure 10: Box plots illustrating the distances between the persistence dia-\ngrams of perturbed samples and the baseline sample diagram Dgm , as well\nas the distances between the robust persistence diagrams and the baseline\nsample diagram.\n28"    },    {        "title": "2402.15457v1.Profile_cut_off_phenomenon_for_the_ergodic_Feller_root_process.pdf",        "content": "arXiv:2402.15457v1  [math.PR]  23 Feb 2024PROFILE CUT-OFF PHENOMENON FOR THE ERGODIC FELLER\nROOT PROCESS\nGERARDO BARRERA AND LILIANA ESQUIVEL\nAbstract. The present manuscript is devoted to the study of the convergen ce to equi-\nlibrium as the noise intensity ε >0 tends to zero for ergodic random systems out of\nequilibrium of the type\ndXε\nt(x) = (b−aXε\nt(x))dt+ε/radicalbig\nXε\nt(x)dBt, Xε\n0(x) =x, t/greaterorequalslant0,\nwherex/greaterorequalslant0,a>0 andb>0 are constants, and ( Bt)t/greaterorequalslant0is a one dimensional standard\nBrownian motion. More precisely, we show the strongest notion of a symptotic profile\ncut-off phenomenon in the total variation distance and in the renor malized Wasserstein\ndistance when εtends to zero with explicit cut-off time, explicit time window, and\nexplicit profile function. In addition, asymptotics of the so-called mix ing times are given\nexplicitly.\n1.Introduction\nThe concept of the asymptotic cut-off phenomenon in the total va riation distance\nwas coined by D. Aldous and P. Diaconis in the context of Markov chain s models of\ncard-shuffling, Ehrenfests’ urn and random transpositions, see [1, 32]. Very generally,\nit refers to the asymptotically abrupt dynamical transition (it is also known as dras-\ntic/sharp/step/switch convergence) of the current state out of equilibrium of the Markov\nchain to its dynamic equilibrium in a cut-off window around the so-called m ixing time\nas a function of the deck size. In other words, the distance to equ ilibrium behaves as\nan approximate step function that remains close to its maximal value (diameter) for a\nwhile, and then suddenly drops to zero as the running time paramete r reaches a critical\nthreshold, which corresponds to the mixing time. Since the state sp ace of card-shuffling\nMarkov chains models is finite, the preceding convergence to equilibr ium is naturally\nmeasured in terms of the total variation distance. This dynamical p hase transition has\nbeen observed in a broad spectrum of random models. For instance , in the discrete state\nspace the asymptotic cut-off phenomenon has been showed for sh uffling cards Markov\ndynamics [1, 32], random walks on the n-dimensional hypercube [56], birth and death\nMarkov chains [19, 74], sparse Markov chains [25], Glauber dynamics [3 3], SEP dynam-\nics [40, 52], SSEP dynamics [41], TASEP dynamics [34], random walks in ran dom regular\ngraphs [60], mean-zero field zero-range process [63], averaging p rocesses [71], and sam-\npling chains [7, 51]. For more general Markov processes taking value s in continuous\nstate-spaces there are relatively few results showing the asympt otic cut-off phenomenon.\nThey include linear SDEs driven by Brownian motion and more general L ´ evy perturba-\ntions [8, 18, 9, 17, 20, 51], non-linear over-damped and under-dam ped Langevin dynamics\ndriven by Brownian motion and more general L´ evy perturbations [ 15, 16, 12, 6, 10, 55],\nKey words and phrases. Affine processes, Asymptotic cut-off phenomenon, Brownian motio n, CIR\nmodel, Convergence to equilibrium, Decoupling, Fourier transform, Mixing times, Sharp transition,\nSquare-root diffusion, Thermalization, Total variation distance, W asserstein distance .\n12 GERARDO BARRERA AND LILIANA ESQUIVEL\nlinear heat and wave SPDEs driven by L´ evy noise [14], multivariate geo metric Brown-\nian motion [11], Dyson–Ornstein–Uhlenbeck process [26], the biased a djacent walk on\nthe simplex [50], Brownian motion on families of compact Riemannian manif olds [62].\nRecently, the asymptotic cut-off has been studied in: coagulation- fragmentation sys-\ntems [65], machine learning for neural networks [4], quantum Markov chains [46], open\nquadratic fermionic systems [76], chemical kinetics [22], viscous ener gy shell model [13],\nand random quantum circuits [67].\nIn the present manuscript, we are interested inthe asymptotic cu t-off phenomenon with\ncomplexity positive parameter εin the weak noise limit ε→0+for the uniquely ergodic\nrandomdynamics Xε,x:= (Xε\nt(x))t/greaterorequalslant0given by the unique strong solution of the stochastic\ndifferential equation (for short SDE)\n(1.1) d Xε\nt(x) = (b−aXε\nt(x))dt+ε/radicalbig\nXε\nt(x)dBt, Xε\n0(x) =x, t/greaterorequalslant0, x/greaterorequalslant0,\nwhereB:= (Bt)t/greaterorequalslant0is a one dimensional standard Brownian motion, and aandbare\npositive constants. The so-called Fokker–Planck equation for (1.1 ) is studied in 1951\nby W. Feller in [36] to study singular diffusion problems. It is then refer red as Feller’s\nsquare root process. The model (1.1) is introduced in 1985 by J. Co x, J. Ingersoll and\nS. Ross to describe the evolution of instantaneous interest rate a t timet,Xε\nt(x), with\nmean reversion of the interest rate towards the long-term value b/awith the speed of\nadjustment a. Nowadays, it is known as a Cox–Ingersoll–Ross (CIR) process, se e [29].\nIn Section 1.5 in [26] it is shown that cut-off phenomenon holds true, a s the number of\nsampling increases, for a particular CIR model (Bessel process), which can be written as\na square of an Ornstein–Uhlenbeck process. Furthermore, singu lar SDEs of the type (1.1)\nemerge in the model of evolution of chemical reactions and populatio n dynamics as diffu-\nsion approximations of res-caled Markov jump processes, see for instance [5, 35, 70] and\nthe references therein. The so-called fluid limit of (1.1) (i.e. the dete rministic ordinary\ndifferential equation obtained as ε→0+) may be interpreted as a mean-field approxi-\nmation and the stochastic term ε/radicalbig\nXε\nt(x)dBtmay be interpreted as the demographic or\ninternal noise.\nBy Theorem 5.1 in [39] or Theorem 3.2 in [43] we have that (1.1) has a uniq ue strong\nsolution in a filtered complete probability space (Ω ,F,(Ft)t/greaterorequalslant0,P) satisfying the usual\nconditions, where the standard Brownian motion Bis defined and denote by Ethe ex-\npectation with respect to the probability measure P. We stress that the density of the\nmarginalXε\nt(x) can be expressed in terms of Bessel functions, see [29, 36]. Sinc e we are\ninterested in the limit ε→0+, without loss of generality we can assume that the so-called\nergodic Feller regime holds true, i.e, ε∈(0,√\n2b). In the aforementioned ergodic Feller\nregime, the randomdynamics given by (1.1) is uniquely ergodic anditsin variant probabil-\nity measure µεpossesses a Gamma distribution with parameters2b/ε2and2a/ε2. Since (1.1)\nbelongs to the class of affine diffusions, the marginal Xε\nt(x) converges in distribution to\nXε\n∞asttends to infinity. Moreover, the law of Xε\n∞does not depend on the initial datum\nx. The preceding convergence can be improved to be valid in the total variation distance.\nFor more details, we refer to [29, 36, 38, 45].\nWe stress that in general, convergence in distribution does not imply convergence in\ntotal variation distance. For instance, the celebrated DeMoivre– Laplace Central Limit\nTheorem does not hold true in the total variation distance, nevert heless, it holds true\nfor other distances such as the Wasserstein, see Theorem 1 in [23]. Even for absolutely\ncontinuous (smooth) sequence of distributions, the convergenc e in distribution does not\nimply convergence in total variation distance, see for instance Fig.1 in [27] or Lemma 1.17CUT-OFF PHENOMENON IN THE TOTAL VARIATION AND WASSERSTEIN D ISTANCES 3\nin [10]. In fact, the convergence in the total variation distance is ve ry strong and hence it\ncannotbeexpected fromjusttheconvergence indistribution with outadditional structure.\nLet (Xn)n∈Nbe a sequence of random variables defined in a probability space (Ω ,F,P).\nIn addition, let Xbe a random variable defined in (Ω ,F,P). Assume that for each\nn∈N, the distribution of the random variable Xnis absolutely continuous with respect\nto the Lebesgue measure on Rand its density fnis unimodal. Moreover, assume that the\nrandom variable Xhas also a unimodal density fwith respect to the Lebesgue measure\nonR. Then Theorem 2.3 (Ibragimov) in [30] or Lemma 3 in [42] yield that the following\nstatements are equivalent:\n(i) Therandomsequence ( Xn)n∈Nconverges indistributionto Xasntendstoinfinity.\n(ii) The random sequence ( Xn)n∈Nconverges in the total variation distance to Xasn\ntends to infinity.\nNevertheless, showing that the distribution of a random variable ha s a unimodal density\nis in general a difficult task, see Khinchin’s Theorem in Theorem 4.5.1 of [6 1]. Without\nassuming that for each n∈N,fnandfare unimodal, in [24, 75] there are sufficient\nconditions on ( fn)n∈Nso that (i) is equivalent to (ii). However, such conditions requires\ntoverify thatthe sequence ( fn)n∈Nisequicontinuous, which typically isa challenging task.\nBearing all this in mind, we are interested in the quantitative behavior of the conver-\ngence to equilibrium of the out of equilibrium marginals of the SDE (1.1) in the total\nvariation distance and in the Wasserstein distance. The quantitativ e analysis of the con-\nvergence to equilibrium in a suitable distance or divergence of interes t for ergodic Markov\nprocesses, in particular SDEs, is typically a difficult task. By an analyt ical approach\nit typically requires a refined knowledge of the spectrum (roughly sp eaking, eigenvalues\nand their eigenfunctions) for the so-called generator of the SDE, which is an infinitely\ndimensional operator. In addition, Poincar´ e type inequalities, Log -Sobolev type inequal-\nities, relative entropy inequalities, Lyapunov functions, Kantorov ich potentials estimates\nand/or Talagrand’s transportation inequality are needed to hold tr ue in order to bound\nthecarr´ e duchamp in theBakry– ´Emery theory, see [54]. Wenote that forshort times, the\nlaw of the random dynamics starting from a deterministic initial datum is very far from\nthe invariant probability measure and hence estimates using the afo rementioned tech-\nniques are far to be optimal. Alternative approaches are offered by the so-called Stein’s\nmethod [28, 66, 49, 57] and probabilistic couplings methods [59]. Nev ertheless, in general\nonly upper bounds for such convergence can be obtained.\nHistorically, the exponential quantitative analysis of the converge nce to equilibrium\nfor Markov processes provides upper bounds of the type Ce−δtfor a suitable distance\nbetween the law of the dynamics (marginal) at time tand its corresponding invariant\nprobability measure. The positive ergodicity constants (clearly not unique) Candδ\nmay depend on the complexity parameter (here denoted by ε), the initial datum of the\ndynamics and/or the dimension of the underlying state space. Ther efore, generically, it\nis a difficult task to carry out the dependence with respect to the co mplexity parameter,\ninitial datum and/or dimension, see [64]. Numerical computations of e rgodicity constants\nfor stochastic dynamics have been studied in [58]. The constant −δis related with the\nspectral gap of the generator of the Markov process. As a cons equence, the computation\nand/or estimation of such constants aretypically done case by cas e. In addition, when the\ndistance can be written in dual-formulation (duality), it is possible to o btain lower bounds\nby a cunning choice of a distinguish statistic (observable), see for in stance the Wilson4 GERARDO BARRERA AND LILIANA ESQUIVEL\nmethod for Markov chains given in Proposition 7.7 of [56]. Neverthele ss, in general, we\nstress that lower bounds for the convergence to equilibrium are ty pically out of reach.\nThe objective of this manuscript is two-fold.\n(1) Provide a robust technique to show convergence to equilibrium in the total vari-\nation distance and in the Wasserstein distance, see Subsection 1.1 f or the precise\ndefinition of the distances. We remark that the Parseval–Plancher el–Fourier based\napproach used in this paper to study the asymptotic cut-off pheno menon in total\nvariation distance for (1.1) is robust and it is not based on explicit for mulas for its\nmarginals and/or the aforementioned analytical methods or inequa lities. It can be\nextended to the class of the so-called affine processes driven by ge neral perturba-\ntions whenever uniform integrability when ε→0+of the spatial Fourier transform\nis valid, see Proposition A.3 and Proposition A.4 in Appendix A for furthe rs de-\ntails. In particular, after identifying the Central Limit Theorem whe nε→0+for\nthe invariant probability measure, that is, after a suitable shift and scaling, we\nobtain the strong notion of asymptotic profile cut-off phenomenon whenε→0+\nwith explicit cut-off time, explicit time window, and explicit profile functio n in\nterms of the classical error function, see Theorem 1.3 below. For t he Wasserstein\ndistance, the same holds true whenever uniform integrability when ε→0+of\nthe corresponding p-th moments is valid. In addition, the profile function has an\nexplicit exponential shape, see Theorem 1.7 below.\n(2) Provide explicit asymptotics of the so-called mixing times in the tot al variation\ndistanceandintheWassersteindistance, seeCorollary1.5andCoro llary1.9below.\nWe emphasize that the asymptotics of the mixing times do not rely on e xplicit\nknowledge of optimal ergodicity constants aforementioned, which are generically\na difficult task.\nThe rest of the manuscript is organized as follows. In Section 1.1, we review the\nnecessary background about the Wasserstein distance and the t otal variation distance.\nWe then rigorous introduce the asymptotic cut-off phenomenon an d its relation with\nthe mixing times. In Section 1.2, we present the main results of the ma nuscript see\nTheorem 1.3 and Theorem 1.7 there. They give the profile cut-off phe nomenon for (1.1)\nin the total variation distance and a renormalized Wasserstein dista nce, respectively. In\naddition, explicit mixing time asymptotics are presented. In Section 2 detailed proofs\nof Theorem 1.3 and Theorem 1.7 are presented. More precisely, The orem 1.3 is showed\nin Subsection 2.1, and Theorem 1.7 is proved in Subsection 2.1.3. Finally, we provide\nan appendix with auxiliary results that have been used throughout t he manuscript. The\nappendix is divided in five sections. More specifically, in Appendix A we br iefly recall\ngeneral basic properties of the total variation distance. Then we introduce the Parseval–\nPlancherel–Fourier approach for total variation convergence. I n Appendix B we show\na useful scaling property for the Gamma distribution and prove a loc al central limit\ntheorem for the stationary probability measure of (1.1). In Appen dix C we recall the\nconvergencetoequilibrium(inthetotalvariationdistanceandinthe Wassersteindistance)\nfor the evolution of (1.1). In addition, we recall explicit formulas for the characteristic\nfunction and moment generating function of (1.1). In Appendix D we present asymptotic\nexpansions thatarecrucial intheproofsofthemainresults. Last lyinAppendix E,mixing\ntimes equivalence of asymptotic profile cut-off is presented.\n1.1.Preliminaries and asymptotic cut-off phenomenon. In this section, we review\nthe necessary background about Wasserstein distance and the t otal variation distance.CUT-OFF PHENOMENON IN THE TOTAL VARIATION AND WASSERSTEIN D ISTANCES 5\nWe point out that the weak convergence in the space of probability m easures with finite\np-moment can be metrized by the Wasserstein distance. In addition, it is a natural way\nto compare the discrepancy between the laws of two random elemen ts, even for degener-\nate cases, where one random element is derived from the other by a small perturbation.\nNowadays, it is an important and well-studied mathematical concept in its own right.\nMoreover, it has been ubiquitous in applications such as optimal tran sport theory, prob-\nability theory, partial differential equations, machine learning, for further details and\nproperties of the Wasserstein distance, we refer to the monogra phs [68, 77, 3, 37, 69].\nWe now define the asymptotic cut-off phenomenon in a general sett ing. Inwhat follows,\nwe recall the definition of coupling between two probability measures and then define the\nWasserstein distance and the total variation distance. Let ( H,d) be a Polish space, a\ncomplete separable metric space, equipped with its Borel σ-algebraB(H) and let M1(H)\nbe the set of probability measures defined in the measurable space ( H,B(H)). Given\nν1∈M1(H) andν2∈M1(H), we say that a probability measure Π defined in the product\nmeasurable space ( H×H,B(H)⊗B(H)) is a coupling (a.k.a. transportation plan or joint\ndistribution) between ν1andν2if and only if Π( B×H) =ν1(B) and Π(H×B) =ν2(B)\nfor any measurable set B∈ B(H). We then denote by C(ν1,ν2) the set of all couplings\nbetweenν1andν2. We note that the product measure ν1⊗ν2is a coupling between ν1\nandν2and hence C(ν1,ν2)/\\e}atio\\slash=∅.\nFor anyp >0 we denote by M1,pthe set of Borel probability measures on ( H,B(H))\nwith finite p-th moment, that is, M1,p⊂M1and for some u0∈Hit follows that/integraltext\nHd(u,u0)pν(du)<∞for allν∈M1,p(the choice of the point u0is irrelevant). Then\nthe Wasserstein distance of order p>0,Wp:M1,p×M1,p→[0,∞), is defined as\nWp(ν1,ν2) := inf\nΠ∈C(ν1,ν2)/parenleftbigg/integraldisplay\nH×Hd(u,v)pΠ(du,dv)/parenrightbigg1∧(1/p)\n, (1.2)\nwhere for any real numbers aandbwe denote the minimum between aandbbya∧b.\nRoughly speaking, (1.2) measures how expensive is the transporta tion of the measure ν1\ninto the measure ν2with respect to the (Monge–Kantorovich) cost function c(u,v) :=\nd(u,v)p,u,v∈H. We also define the total variation distance d TV:M1×M1→[0,1] by\ndTV(ν1,ν2) := inf\nΠ∈C(ν1,ν2)/integraldisplay\nH×H1{u/ne}ationslash=v}Π(du,dv),\nwhere1Udenotes the indicator function of a given set U⊂H.\nLetX1andX2be two random elements defined on the probability space (Ω ,F,P) with\nfinitep-th moment and taking values on H. The Wasserstein distance of order p>0 be-\ntweenX1andX2is defined by Wp(X1,X2) :=Wp(PX1,PX2), where PX1andPX2are the\npush-forward probability measures PX1(B) :=P(X1∈B) andPX2(B) :=P(X2∈B) for\nanyB∈ B(H). For convenience, we write Wp(X1,X2) in place of Wp(X1,X2). Similarly,\nwe write d TV(X1,X2) instead of d TV(PX1,PX2). Moreover, for the sake of intuitive rea-\nsoning and in a conscious abuse of notation we write Wp(X1,ν2) and d TV(X1,ν2) instead\nofWp(X1,X2) and d TV(X1,X2), respectively, where ν2:=PX2. In other words,\nWp(ν1,ν2) = inf\n(X1,X2)(EP[d(X1,X2)p])1∧(1/p)\nand\ndTV(ν1,ν2) = inf\n(X1,X2)P(X1/\\e}atio\\slash=X2),6 GERARDO BARRERA AND LILIANA ESQUIVEL\nwhere the infimum is taken over all pairs ( X1,X2) satisfying PXj=νj,j= 1,2 andEP\ndenotes the expectation with respect to P. While by definition, the Wasserstein distance\northetotal variationdistance areminimizers ofanexpected valueo f agiven cost function,\nthey are always bounded above by evaluating the expectation of th e cost function in any\ncoupling (for instance, the synchronous coupling), however, lowe r bounds are typically\nhard to establish.\nInthesequel, following[11,8]weembedourresultsinthecontextof cut-offphenomenon\nin a general framework. The complexity parameter of the model is d enoted byε>0. Let\n(Hε,dε) be a Polish space equipped with its corresponding Borel σ-algebraB(Hε) and let\nM1(Hε) be the set of probability measures defined on B(Hε). Let/tildewideM1(Hε)⊂M1(Hε) be\nequipped with a distance or divergence distε:/tildewideM1(Hε)×/tildewideM1(Hε)→[0,∞], and set\nDiamε:= sup\nν1,ν2∈/tildewideM1(Hε)distε(ν1,ν2)\nand assume that\n(1.3) lim\nε→0+Diamε=:Diam∈(0,∞].\nWe then consider Xε,xε:= (Xε\nt(xε))t/greaterorequalslant0a random dynamical system (including determinis-\ntic dynamical system) in continuous time and taking values in Hε, wherexε∈Hεdenotes\ntheinitial datum of the process Xε,xε. Inaddition, we assume that for any t/greaterorequalslant0 thelaw of\nXε\nt(xε), here denoted by LawXε\nt(xε), belongs to/tildewideM1(Hε) and the existence of νε∈/tildewideM1(Hε)\nsuch that\nlim\nt→∞distε(LawXε\nt(xε),νε) = 0.\nSince we have already fixed the distance distεfor the convergence to the limiting law νε,\nwe now define the asymptotic cut-off phenomenon.\nDefinition 1.1 (Asymptotic cut-off phenomenon) .We say that the family of random dy-\nnamical systems ( Xε,xε,νε,distε)ε>0exhibits:\n(i)asymptotic pre-cut-off at (tε,xε)ε>0with pre-cut-off time tε:=tε,xεiftε→ ∞, as\nε→0+and there are constants 0 <C∗<C∗<∞such that\nlim\nε→0+distε(LawXε\nδ·tε(xε),νε) =/braceleftBigg\nDiam forδ=C∗,\n0 for δ=C∗.\n(ii)asymptotic cut-off at (tε,xε)ε>0with cut-off time tε:=tε,xεiftε→ ∞, asε→0+\nand\n(1.4) lim\nε→0+distε(LawXε\nδ·tε(xε),νε) =/braceleftBigg\nDiam forδ∈(0,1),\n0 for δ∈(1,∞).\n(iii)asymptotic window cut-off at ((tε,xε,ωε,xε))ε>0with cut-off time tε:=tε,xε>0 and\ntime window ωε:=ωε,xε>0 iftε→ ∞, asε→0+, lim\nε→0+ωε\ntε= 0, and the limits\nG∗(r) := liminf\nε→0+distε(LawXε\ntε+r·ωε,νε)\nand\nG∗(r) := limsup\nε→0+distε(LawXε\ntε+r·ωε,νε)CUT-OFF PHENOMENON IN THE TOTAL VARIATION AND WASSERSTEIN D ISTANCES 7\nsatisfy\nlim\nr→−∞G∗(r) =Diamand lim\nr→∞G∗(r) = 0.\n(iv)asymptotic profile cut-off at ((tε,xε,ωε,xε))ε>0with cut-off time tε:=tε,xε>0,\ntime window ωε:=ωε,xε>0 and profile function G:R→[0,Diam] iftε→ ∞, as\nε→0+, lim\nε→0+ωε\ntε= 0,\n(1.5) lim\nε→0+distε(LawXε\ntε+r·ωε,νε) =:G(r) exists for any r∈R.\nIn addition, it satisfies\nlim\nr→−∞G(r) =Diamand lim\nr→∞G(r) = 0.\nWepointoutthatfindanexplicit profilefunction Gisingeneraldifficult. Althoughit is\nbelieved that many families of processes exhibit asymptotic cut-off p henomenon, showing\nsuch occurrence is often an extremely challenging task even for sim ple Markov chains. It\nrequires the full understanding of the behavior of the underlying d ynamics. It is clear\nthat (iv) implies (iii) and (ii) implies (i).\nIn what follows, we assume that for any fixed ε>0, the map\n(1.6) [0 ,∞)∋t/ma√sto���distε(LawXε\nt,νε) is monotonic decreasing.\nBy (1.6) we have that G∗,G∗are monotonic decreasing. We note that (1.6) is nat-\nural in the context of homogeneous Markov processes and holds t rue for various dis-\ntances/discrepancies of interest, see for instance Lemma B.3 (Mo notonicity) in [26]. Un-\nder (1.6) one can see that (iii) implies (ii).\nInthesequel, wedefinetheso-called η-mixingtime. Forany η∈(0,Diamε)theη-mixing\ntime,τε,distε\nmix(η), is defined by\n(1.7) τε,distε\nmix(η) := inf{t/greaterorequalslant0 :distε(LawXε\nt,νε)/lessorequalslantη}.\nIn other words, τε,distε\nmix(η) is the time required by the process Xεto be close to its limiting\nlaw up to a prescribed error η∈(0,Diamε). For discrete finite state space, the definition\nofη-mixing time additionally has an optimization over the worst possible initia l datum of\nthe random dynamics. However, since our purpose is to describe th e dependence of the\nmixing time with respect to the initial state of the process, we use th e definition (1.7) for\nη-mixing time. We also stress that the cut-off times may have strong d ependence with\nrespect to the initial datum of the random dynamics, see for instan ce [15, 16].\nFor the total variation distance, it is known that the asymptotic cu t-off phenomenon\ncan be also interpreted as a mixing time, see Chapter 18 of [56] or [21 ].\nIn the sequel, we show that the asymptotic cut-off phenomenon in t he preceding frame-\nwork (Definition 1.1) can be also interpreted as a mixing time in the follow ing precise\nsense.\nProposition 1.2 (Asymptotics forthe mixing times) .The following statements are valid.\n(1)Assume that (i) holds true. There exist constants 0<c∗<c∗<∞such that for\nanyη∈(0,Diam)it follows that\nc∗<liminf\nε→0+tε\nτε,distε\nmix(η)/lessorequalslantlimsup\nε→0+tε\nτε,distε\nmix(η)<c∗.8 GERARDO BARRERA AND LILIANA ESQUIVEL\n(2)Assume that (ii) holds true. Then for any η∈(0,Diam)it follows that\nlim\nε→0+tε\nτε,distε\nmix(η)= 1.\n(3)Assume that (iii) holds true. In addition, assume that G∗andG∗are continuous\nand strictly decreasing. Then for any η∈(0,Diam)it follows that\ntε+ωε·(G∗)−1(η)+o(ωε)/lessorequalslantτε,distε\nmix(η)/lessorequalslanttε+ωε·(G∗)−1(η)+o(ωε),\nwhere(G∗)−1(η) := inf{r/greaterorequalslant0 :G∗(r)/lessorequalslantη}and(G∗)−1(η) := inf{r/greaterorequalslant0 :G∗(r)/lessorequalslant\nη}.\n(4)Assume that (iv) holds true. In addition, assume that Gis continuous and strictly\ndecreasing. Then for any η∈(0,Diam)it follows that\nτε,distε\nmix(η) =tε+G−1(η)·ωε+o(ωε),asε→0+,\nwhereG−1(η) := inf{r∈R:G(r)/lessorequalslantη}ando(ωε)/ωε→0+asε→0+.\nThe proof is given in Appendix E.\nWe remark that the previous definition of asymptotic cut-off pheno menon naturally\nappliesfordiscretetimesystems usingthefloorfunctionintherunn ingtimeofthesystem,\nthat is, replacing tby⌊t⌋, accordingly. In the context of ergodic Markov chains with finite\nstate space, the asymptotic cut-off phenomenon is typically measu re in the total variation\ndistance and the complexity parameter is the cardinality of the stat e space. In addition,\nit is shown that it can be expressed at increasingly sharp levels, i.e., (iv ) implies (iii), (iii)\nyields (ii) and (ii) gives (i), but the reciprocal implications are not valid in general.\nBearing all this in mind we provide a complete characterization of when cut-off occurs\nwhich is exactly the content of the following subsection.\n1.2.Main results and their consequences. In this section, we establish the main\nresults of this manuscript. Recall that for any fixed noise intensity ε∈(0,√\n2b) there\nexists a unique invariant probability measure µεfor the random dynamics given in (1.1),\nthat is,Xε\nt(µε)d=µεfor anyt/greaterorequalslant0, whered= denotes equality in distribution sense. In\naddition, for any p>0 and any initial condition x/greaterorequalslant0 of (1.1) it follows that\nlim\nt→∞Wp(Xε\nt(x),µε) = 0 and lim\nt→∞dTV(Xε\nt(x),µε) = 0,\nsee Lemma C.1 in Appendix C. We note that DiamWp=∞andDiamTV= 1.\nFollowing the notation in Subsection 1.1, the complexity parameter is ε∈(0,√\n2b),\nHε:=Requipped with the standard Euclidean distance, xε:=x∈[0,∞),Xε,xε:=Xε,xε\nis given in (1.1) and νε:=µεis the invariant probability measure for (1.1).\nThe first main result of this manuscript is the following profile cut-off p henomenon in\ntotal variation distance, i.e., /tildewideM1(R) :=M1(R) anddistε:= dTV.\nTheorem 1.3 (Asymptotic profile cut-off phenomenon for CIR models I) .Assume that\nx∈[0,∞)\\{b\na}. The family of CIR models (Xε,x)ε>0defined in (1.1)exhibits asymptotic\nprofile cut-off phenomenon as εtends to zero in the total variation distance\n(1.8) at cut-off time tε:=1\naln/parenleftbigg1\nε/parenrightbigg\nand time window ωε:=1\na.\nIn other words, for any r∈Rthe following limit is valid\n(1.9) lim\nε→0+dTV/parenleftbig\nXε\ntε+r·ωε(x),µε/parenrightbig\n= dTV/parenleftbig\n|Cx|·e−r+G,G/parenrightbig\n=:GTV\nx(r),CUT-OFF PHENOMENON IN THE TOTAL VARIATION AND WASSERSTEIN D ISTANCES 9\nwhere the constant Cxis given by\n(1.10) Cx:=√\n2b\nb(ax−b),\nandGdenotes a random variable with standard Gaussian law. Moreo ver,\n(1.11) lim\nr→−∞GTV\nx(r) = 1andlim\nr→∞GTV\nx(r) = 0.\nIn addition, for x=b\naand for any function (sε)ε>0such thatsε→ ∞asε→0+it follows\nthat\nlim\nε→0+dTV/parenleftbig\nXε\nsε(x),µε/parenrightbig\n= 0.\nIn particular, the family of CIR models (Xε,b/a)ε>0defined in (1.1)does not exhibit as-\nymptotic cut-off phenomenon as εtends to zero in the total variation distance.\nFor shorthand and in a conscious abuse of notation, we use indistinc tly the following\nnotations for the exponential function: exp( a) oreafora∈R. In addition, we use |·|for\nthe absolute value function and for the modulus of a complex number .\nWe continue to rely on the notations and assumptions in Theorem 1.3.\nRemark 1.4 (Shape of the asymptotic profile function in the total variation dist ance).\nFor anym∈Rit follows that\ndTV(m+G,G) =2√\n2π|m|/2/integraldisplay\n0exp/parenleftbigg\n−z2\n2/parenrightbigg\ndz,\nfor details we refer to Item (i) of Lemma B.2 in Appendix B of [17]. Hence, for GTV\nx\ndefined in (1.9)it follows that\nGTV\nx(r) =2√\n2π|Cx|·e−r/2/integraldisplay\n0exp/parenleftbigg\n−z2\n2/parenrightbigg\ndzfor anyr∈R,\nwhich with the help of the Monotone Convergence Theorem impl ies\nlim\nr→−∞GTV\nx(r) = 1andlim\nr→∞GTV\nx(r) = 0.\nIn addition, the Fundamental Theorem of Calculus and the Mil l ratio give asymptotics for\nthe profile function GTV\nx. To be more precise, the asymptotically exponential shape o f the\nprofile forr≫1, that is,\nGTV\nx(r)r→∞∼1√\n2π|Cx|·e−r,\nwhereas, the asymptotically doubly exponential shape of th e profile for r≪ −1\n1−GTV\nx(r)r→−∞∼4√\n2π1\n|Cx|·e−rexp/parenleftbigg\n−C2\nx·e−2r\n8/parenrightbigg\n,\nwhere for short we write F1r→∞∼F2andF1r→−∞∼F2in a place of lim\nr→∞F1(r)\nF2(r)= 1and\nlim\nr→−∞F1(r)\nF2(r)= 1, respectively.\nAs a consequence of Theorem 1.3 with the help of Proposition 1.2 and R emark 1.4\nwe obtain the following corollary, which provides the asymptotic beha vior for the mixing\ntimes.10 GERARDO BARRERA AND LILIANA ESQUIVEL\nCorollary 1.5 (η-mixing time in the total variation distance) .For anyη >0it follows\nthat\nτε,dTV\nmix(η) =tε+ωε·/parenleftbig\nGTV\nx/parenrightbig−1(η)+oε→0+(1),\nwhere/parenleftbig\nGTV\nx/parenrightbig−1(η) := inf{r/greaterorequalslant0 :GTV\nx(r)/lessorequalslantη}.\nRemark 1.6 (The mixing time at η= 1/4).Forη=1\n4, we setτε,dTV\nmix:=τε,dTV\nmix(η)and\nthen Corollary 1.5 yields that\nτε,dTV\nmix=tε+ωε·/parenleftbig\nGTV\nx/parenrightbig−1(1/4)+oε→0+(1).\nBy Remark 1.4 we have that\nGTV\nx(r) =2√\n2π|Cx|·e−r/2/integraldisplay\n0exp/parenleftbigg\n−z2\n2/parenrightbigg\ndzfor anyr∈R.\nNow, for simplicity choose the constants a>0,b>0andx/greaterorequalslant0such thatCx= 2and\nthen find the unique r>0such that\ne−r/integraldisplay\n0exp/parenleftbigg\n−z2\n2/parenrightbigg\ndz=√\n2π\n8≈0.3133,that is,r≈1.1435.\nIn summary, we obtain\nτε,dTV\nmix≈1\naln/parenleftbigg1\nε/parenrightbigg\n+1\na×1.1435forε≪1.\nThe second main result of this paper is the following profile cut-off phe nomenon in a\nrenormalized Wasserstein distance of order p >0, i.e.,/tildewideM1(R) :=M1,p(R) anddistε:=\n1\nε1∧pWp.\nTheorem 1.7 (Asymptotic profile cut-off phenomenon for CIR models II) .Assume that\nx∈[0,∞)\\{b\na}. The family of CIR models (Xε,x)ε>0defined in (1.1)exhibits asymptotic\nprofile cut-off phenomenon as εtends to zero in the renormalized Wasserstein distance of\norderp>0,ε−(1∧p)Wp, at cut-off time tεand time window ωεdefined in (1.8). In other\nwords, for any r∈Rthe following limit holds true\n(1.12) lim\nε→0+Wp/parenleftbig\nXε\ntε+r·ωε(x),µε/parenrightbig\nε1∧p=/parenleftBigg√\n2b\n2a/parenrightBigg1∧p\nWp/parenleftbig\n|Cx|·e−r+G,G/parenrightbig\n:=GWp\nx(r),\nwhere the constant Cxis given in (1.10)andGdenotes a random variable with standard\nGaussian law. Moreover,\n(1.13) lim\nr→−∞GWp\nx(r) =∞andlim\nr→∞GWp\nx(r) = 0.\nIn addition, for x=b\naand for any function (sε)ε>0such thatsε→ ∞asε→0+it follows\nthat\nlim\nε→0+Wp/parenleftbig\nXε\nsε(x),µε/parenrightbig\nε1∧p= 0.\nIn particular, the family of CIR models (Xε,b/a)ε>0defined in (1.1)does not exhibit as-\nymptotic cut-off phenomenon as εtends to zero in the renormalized Wasserstein distance.\nWe continue to rely on the notations and assumptions in Theorem 1.7.CUT-OFF PHENOMENON IN THE TOTAL VARIATION AND WASSERSTEIN D ISTANCES 11\nRemark 1.8 (Shape of the asymptotic profile function in the Wasserstein distan ce of\norderp >0).Letp/greaterorequalslant1be fixed. For any random variable Xwith finite p-th absolute\nmoment and any deterministic number m∈Rthe following shift-linearity\nWp(m+X,X) =|m|,\nholds true, for details we refer to (2.6) in Item (d) of Lemma 2 .2 in[9]. Hence, for GWp\nx(r)\ndefined in (1.9)it follows that\nGWp\nx(r) =/parenleftBigg√\n2b\n2a/parenrightBigg\n|Cx|·e−rfor anyr∈R,\nwhich implies lim\nr→−∞GWp\nx(r) =∞forx/\\e}atio\\slash=b\na, andlim\nr→∞GWp\nx(r) = 0. While for p∈(0,1)we\nhave/parenleftBigg√\n2b\n2a/parenrightBiggp\nmax{|Cx|p·e−r·p−2E[|G|p],0}/lessorequalslantGWp\nx(r)/lessorequalslant/parenleftBigg√\n2b\n2a/parenrightBiggp\n|Cx|p·e−r·p\nfor anyr∈R, see (2.7) in Item (d) of Lemma 2.2 in [9]. Then we have\nGWp\nx(r)r→−∞∼/parenleftBigg√\n2b\n2a/parenrightBiggp\n|Cx|p·e−r·p,\nwhereas\nGWp\nx(r)r→∞=O/parenleftBigg/parenleftBigg√\n2b\n2a/parenrightBiggp\n|Cx|p·e−r·p/parenrightBigg\n,\nwhereOdenotes the classical Bachmann–Landau notation.\nAs a consequence of Theorem 1.7 with the help of Proposition 1.2 and R emark 1.8\nwe obtain the following corollary, which provides the asymptotic beha vior for the mixing\ntimes.\nCorollary 1.9 (η-mixing time in the normalized Wasserstein distance) .For anyη >0\nandp/greaterorequalslant1it follows that\nτε,Wp/ε\nmix(η) =tε+ωε·/parenleftbig\nGWp\nx/parenrightbig−1(η)+oε→0+(1),\nwhere\n/parenleftbig\nGWp\nx/parenrightbig−1(η) = ln/parenleftBigg/parenleftBigg√\n2b\n2a/parenrightBigg\n|Cx|\nη/parenrightBigg\nand the constant Cxis given in (1.10).\n2.Proofs of the main results: Theorem 1.3 and Theorem 1.7\nIn this section, we give the proof of Theorem 1.3 and Theorem 1.7. Re call thatx/greaterorequalslant0,\na>0,b>0 andε∈(0,√\n2b). We set\nqε+1 :=2b\nε2, cε(t) :=2a\nε21\n1−e−atfor anyt>0 andcε(∞) :=2a\nε2. (2.1)\nFor any positive numbers randθwe denote the Gamma distribution with parameter r\nandθby Γ(r,θ). In other words, a random variable Zhas distribution Γ( r,θ) if and only\nif its characteristic function is given by\n(2.2) R∋u/ma√sto→E[eiuZ] =/parenleftbig\n1−iθ−1u/parenrightbig−r.12 GERARDO BARRERA AND LILIANA ESQUIVEL\nForconvenience andinaconsciousabuseofnotation,wewrite E[eiuΓ(r,θ)]insteadof E[eiuZ].\n2.1.Proof of Theorem 1.3: The total variation distance. In this subsection, we\nprovide the proof of Theorem 1.3.\n2.1.1.The local limit theorem for the invariant measure .We recall that Xε\n∞d=\nΓ(qε+1,cε(∞)), see Lemma C.1 in Appendix C. In the sequel, we show that the law of\nXε\n∞satisfies a Gaussian local limit theorem.\nLemma 2.1 (Local limit theorem for Xε\n∞).Letmε:=qε+1\ncε(∞)=b\naandσε:=√qε+1\ncε(∞)=√\n2b\n2aε.\nThen it follows that\n(2.3) lim\nε→0+dTV/parenleftbiggXε\n∞−mε\nσε,G/parenrightbigg\n= 0,\nwhereGdenotes the standard Gaussian distribution.\nProof.Recall that Xε\n∞d= Γ(qε+ 1,cε(∞)), whereqε+ 1 =2b\nε2andcε(∞) =2a\nε2. By\nLemma B.1 in Appendix B we have Xε\n∞d=c−1\nε(∞)Γ(qε+1,1). Since\n(2.4)Xε\n∞−mε\nσεd=Γ(qε+1,1)−(qε+1)√qε+1,\nLemma B.2 in Appendix B implies (2.3). The proof is complete. /square\n2.1.2.The decoupling argument in the total variation distance .By Lemma 2.1\nit is natural to consider the normalization\n(2.5) Yε\nt(x) :=Xε\nt(x)−mε\nσεfor allt/greaterorequalslant0,\nwheremε=qε+1\ncε(∞)=b\naandσε=√qε+1\ncε(∞)=√\n2b\n2aε. Straightforward computations yields\n(2.6) Yε\nt(x) =cε(∞)Xε\nt(x)−(qε+1)√qε+1for allt/greaterorequalslant0.\nHence, we define\ndε,x(t) := d TV(Yε\nt(x),G) for all t/greaterorequalslant0, (2.7)\nwhereGdenotes the standard Gaussian distribution.\nIn the sequel, we show that the distance,\n(2.8) dε,x(sε) := d TV/parenleftbig\nXε\nsε(x),µε/parenrightbig\nis asymptotically equivalent to dε,x(sε) for any function ( sε)ε>0such thatsε→ ∞asε→\n0+. Therefore, cut-off/windows cut-off/profile cut-off for the dist ancedε,xis equivalent\nfor the distance dε,x, respectively. In other words, it is enough to show Theorem 1.3 for\nthe distance dε,x.\nLemma 2.2 (Replacement lemma in the total variation distance) .For anyε∈(0,√\n2b),\nx∈[0,∞)andt/greaterorequalslant0it follows that\n(2.9) |dε,x(t)−dε,x(t)|/lessorequalslantdTV(/radicalbig\nqε+1G+(qε+1),Γ(qε+1,1)),\nwheredε,x(t)anddε,x(t)are defined in (2.7)and(2.8), respectively.CUT-OFF PHENOMENON IN THE TOTAL VARIATION AND WASSERSTEIN D ISTANCES 13\nProof.By Item (i) and Item (ii) of Lemma A.1 in Appendix A we have\ndε,x(t) = dTV(cε(∞)Xε\nt(x),/radicalbig\nqε+1G+(qε+1)).\nBy Lemma B.1 in Appendix B and Item (ii) of Lemma A.1 in Appendix A we have\ndε,x(t) = dTV(Xε\nt(x),Xε\n∞) = dTV(Xε\nt(x),Γ(qε+1,cε(∞)))\n= dTV(Xε\nt(x),(cε(∞))−1Γ(qε+1,1)) = d TV(cε(∞)Xε\nt(x),Γ(qε+1,1)).\nThe triangle inequality for the total variation distance yields\ndε,x(t)/lessorequalslantdTV(cε(∞)Xε\nt(x),/radicalbig\nqε+1G+(qε+1))\n+dTV(/radicalbig\nqε+1G+(qε+1),Γ(qε+1,1)).(2.10)\nSimilarly,\ndε,x(t) = dTV(cε(∞)Xε\nt(x),/radicalbig\nqε+1G+(qε+1))\n/lessorequalslantdε,x(t)+dTV(/radicalbig\nqε+1G+(qε+1),Γ(qε+1,1)).(2.11)\nBy (2.10) and (2.11) we obtain (2.9). This completes the proof. /square\nLemma 2.3 (Asymptotically equivalent total variation distance) .Letx∈[0,∞)be fixed.\nFor any function (sε)ε>0such thatsε→ ∞asε→0+it follows that\nlimsup\nε→0+dε,x(sε) = limsup\nε→0+dε,x(sε)andliminf\nε→0+dε,x(sε) = liminf\nε→0+dε,x(sε),\nwheredε,x(t)anddε,x(t)are defined in (2.7)and(2.8), respectively.\nProof.By Item (i) and Item (ii) of Lemma A.1 in Appendix A we obtain\ndTV(Γ(qε+1,1),/radicalbig\nqε+1G+(qε+1)) = d TV/parenleftbiggΓ(qε+1,1)−(qε+1)√qε+1,G/parenrightbigg\n.\nThen Lemma 2.2 with the help of Lemma B.2 in Appendix B implies that the rig ht-hand\nside of (2.9) tends to zero as ε→0+. The proof is complete. /square\nIn the sequel, we show Theorem 1.3 for the distance dε,x. Recall that\ndε,x(t) = dTV(Yε\nt(x),G) for all t/greaterorequalslant0, (2.12)\nwhereGdenotes the standard Gaussian distribution. We stress that dε,x(t) has the ad-\nvantage that its second input Gdoes not depend on ε.\nLemma 2.4 (Limiting profile in the total variation distance) .Letx∈[0,∞)be fixed.\nFortεandωεbeing defined in (1.8), it follows that\nlim\nε→0+dε,x(tε+r·ωε) = dTV(Cx·e−r+G,G)for anyr∈R,\nwhereGdenotes the standard Gaussian distribution and Cx=√\n2√\nb(ax−b).\nProof.By the triangle inequality for the total variationdistance, it is enoug h to show that\n(2.13) d TV(Yε\ntε+r·ωε(x),Cx·e−r+G) = 0 for any r∈R.\nTo show the preceding limit, we apply Proposition A.3 in Appendix A. Then we need to\nshow Item (i) Convergence in distribution, Item (ii) Fourier integrab ility and Item (iii)\nTail behavior.14 GERARDO BARRERA AND LILIANA ESQUIVEL\nNow, we compute the characteristic function of the marginal Yε\nt(x). By Corollary C.3\nin Appendix C we have\n(2.14) E[eizYε\nt(x)] =ψ/parenleftbiggcε(∞)z√qε+1;t,ε,x/parenrightbigg\ne−iz√qε+1, z∈R,\nwhere\nψ(z;t,ε,x) : =E[eizXε\nt(x)]\n=1/parenleftBig\n1−iz\ncε(t)/parenrightBigqε+1exp/parenleftbigg\nizc2\nε(t)xe−at\nc2\nε(t)+z2/parenrightbigg\nexp/parenleftbigg\n−z2cε(t)xe−at\nc2\nε(t)+z2/parenrightbigg\n(2.15)\nfor anyz∈R. Letϕε(z;x) :=E[eizYε\ntε+r·ωε(x)] for anyz∈R, wheretεandωεare defined\nin (1.8), and r∈R. Note that Lemma C.4 in Appendix C implies\nlim\nε→0ϕε(z;x) = exp/parenleftbigg\n−z2\n2/parenrightbigg\nexp/parenleftBigg\nz√\n2√\nb(ax−b)e−r/parenrightBigg\n,\nwhich yields Item (i).\nWe continue with the proof of Item (ii). We note that\nc2\nε(∞)\nc2\nε(t)1\nqε+1=(1−e−at)2ε2\n2bfor anyt>0.\nBy (2.14) and (2.15) for any r∈Rwe have\n(2.16)/integraldisplay\nR|ϕε(z;x)|dz/lessorequalslant/integraldisplay\nR1\n/parenleftBig\n1+(1−εe−ar)2ε2\n2bz2/parenrightBigb\nε2dz <∞whenε∈(0,√\n2b),\nwheretεandωεare defined in (1.8). It is easy to see that/integraltext\nR|ϕ(z;x)|dz <∞. The proof\nof Item (ii) is complete.\nFinally, we verify Item (iii). Observe that\n1\n8/lessorequalslant(1−εe−r)2\n2forε∈/parenleftbigg\n0,1\n2er/parenrightbigg\n.\nBy (2.16) we obtain\n/integraldisplay\n|z|/greaterorequalslantℓ|ϕε(z;x)|dz/lessorequalslant/integraldisplay\n|z|/greaterorequalslantℓ1\n/parenleftbig\n1+z2\n8ε2\nb/parenrightbigb\nε2dz/lessorequalslant/integraldisplay\n|z|/greaterorequalslantℓ1\n/parenleftbigg\n1+(z2/8)\nb\nε2\n0/parenrightbiggb\nε2\n0dz <∞\n(2.17)\nfor anyε∈(0,ε0), whereε0:=1\n2min{1\n2er,√\n2b}. Then the Dominated Convergence\nTheorem yields\nlim\nε→0+/integraldisplay\n|z|/greaterorequalslantℓ|ϕε(z;x)|dz=/integraldisplay\n|z|/greaterorequalslantℓe−z2\n8dz,\nwhich implies\nlim\nℓ→∞limsup\nε→0+/integraldisplay\n|z|/greaterorequalslantℓ|ϕε(z;x)|dz= 0.\nThiscompletetheproofofItem(iii)andthereforetheproofofLem ma 2.4iscomplete. /squareCUT-OFF PHENOMENON IN THE TOTAL VARIATION AND WASSERSTEIN D ISTANCES 15\nNow, we analyze the case x=b/athat yields Cx= 0. Then Lemma 2.4 does not\nimply profile cut-off phenomenon at tε+r·ωε. In fact, in this case, there is no cut-off\nphenomenon as the following lemma states.\nLemma 2.5 (No cut-off phenomenon in the total variation distance for x=b/a).Let\nx=b/abe fixed. For any function (sε)ε>0satisfyingsε→ ∞asε→0+, it follows that\nlim\nε→0+dTV(Xε\nsε(x),Xε\n∞) = 0.\nProof.By Lemma 2.2 it is enough to show lim ε→0+dε,x(sε) = 0, where dε,xis defined\nin (2.12). Similarly to the proof of Lemma 2.4, to show the preceding limit , we apply\nProposition A.3 in Appendix A. Then we need to show Item (i) Converge nce in distribu-\ntion, Item (ii) Fourier integrability and Item (iii) Tail behavior.\nLet/tildewideϕε(z;x) :=E[eizYε\nsε(x)] for anyz∈R. Note that Lemma C.5 in Appendix C implies\n(2.18) lim\nε→0+/tildewideϕε(z;x) =e−z2/2for allz∈R.\nThis completes the proof of Item (i).\nWe continue with the proof of Item (ii). Analogously to (2.16) we obta in\n/integraldisplay\nR|/tildewideϕε(z;x)|dz/lessorequalslant/integraldisplay\nR1\n/parenleftBig\n1+(1−e−asε)2ε2\n2bz2/parenrightBigb\nε2dz <∞whenε∈(0,√\n2b),\nimplying Item (ii).\nFinally, Item (iii) follows similarly to (2.17). Indeed, since sε→ ∞asε→0+, we have\nthe existence of /tildewideε0:=/tildewideε0(a) such that\n1\n8/lessorequalslant(1−e−asε)2\n2forε∈(0,min{/tildewideε0,√\n2b}).\nThen we have\n/integraldisplay\n|z|/greaterorequalslantℓ|/tildewideϕε(z;x)|dz/lessorequalslant/integraldisplay\n|z|/greaterorequalslantℓ1\n/parenleftbig\n1+z2\n8ε2\nb/parenrightbigb\nε2dz/lessorequalslant/integraldisplay\n|z|/greaterorequalslantℓ1\n/parenleftbigg\n1+(z2/8)\nb\nε2\n0/parenrightbiggb\nε2\n0dz <∞,\nwhich implies Item (iii) and therefore the proof of Lemma 2.5 is complete . /square\n2.1.3.Proof of Theorem 1.3 .In this subsection, we stress the fact that Theorem 1.3\nis just a consequence of what we have already proved up to here.\nLetx∈[0,∞) be fixed. For tεandωεbeing defined in (1.8), Lemma 2.4 with the help\nof Lemma 2.2 and Lemma 2.3 implies\nlim\nε→0+dε,x(tε+r·ωε) = dTV(Cx·e−r+G,G) for any r∈R,\nwhereGdenotesthestandardGaussiandistributionand Cx=√\n2√\nb(ax−b). Thisyields(1.9).\nForx/greaterorequalslant0 andx/\\e}atio\\slash=b/awe have that Cx/\\e}atio\\slash= 0, which implies (1.11). and therefore (1.5)\nholds true. Finally, Lemma 2.5 yields no cut-off phenomenon at x=b/ain the sense\nof (1.4). The proof is complete.16 GERARDO BARRERA AND LILIANA ESQUIVEL\n2.2.Proof of Theorem 1.7: The Wasserstein distance. In this section, we show\nTheorem 1.7.\nWe start recalling two basic properties for the Wasserstein distanc e that are useful\nduring this section.\nLemma 2.6 (Basic properties of the Wasserstein distance) .LetXandYbe random\nvectors taking values in Rd. Assume that XandYhave finite p-th moment for some\np>0. Then the following is valid.\n(i)Translation invariance: For any deterministic vectors v1,v2∈Rdit follows that\nWp(v1+X,v2+Y) =Wp(v1−v2+X,Y) =Wp(X,v2−v1+Y).\n(ii)Homogeneity: For any non-zero constant cit follows that\nWp(cX,cY) =|c|1∧pWp(X,Y) =/braceleftBigg\n|c|Wp(X,Y)forp∈[1,∞),\n|c|pWp(X,Y)forp∈(0,1].\nProof.For details, we refer to Lemma 2.2 in [9]. /square\n2.2.1.The limit theorem for the invariant measure .In this subsection, similarly\nas Lemma 2.1, we prove that the marginal Xε\n∞satisfies a Gaussian limit theorem in the\nWasserstein distance.\nLemma 2.7 (Limit theorem for Xε\n∞inWpfor anyp >0).Letmε:=qε+1\ncε(∞)=b\naand\nσε:=√qε+1\ncε(∞)=√\n2b\n2aε. Then for any p>0it follows that\nlim\nε→0+Wp/parenleftbiggXε\n∞−mε\nσε,G/parenrightbigg\n= 0,\nwhereGdenotes the standard Gaussian distribution.\nProof.By (2.4) we have that\n(2.19) Yε\n∞:=Xε\n∞−mε\nσεd=Γ(qε+1,1)−(qε+1)√qε+1\nLemma 2.1 yields thatXε\n∞−mε\nσεconverges in distribution to Gasε→0+. Hence, by\nTheorem 7.12 in [77] it is enough to show that the family ( Yε\n∞)ε∈(0,√\n2b)is uniformly\nintegrable, that is,\n(2.20) lim\nN→∞limsup\nε→0+E[|Yε\n∞|p1{|Yε∞|/greaterorequalslantN}] = 0.\nWe start with the following observation. For any |λ|>0 andp>0 there exists a positive\nconstantCλ,psuch that\n(2.21) |z|p/lessorequalslantCλ,pe|λ||z|/lessorequalslantCλ,peλz+Cλ,pe−λzfor anyz∈R.\nThen the Cauchy–Schwarz inequality with the help of subadditivity of the square root\nmap [0,∞)∋r/ma√sto→√ryields\nE[|Yε\n∞|p1{|Yε∞|/greaterorequalslantN}]/lessorequalslant/radicalbig\nE[|Yε∞|2p]/radicalbig\nP(|Yε∞|/greaterorequalslantN)\n/lessorequalslant/radicalBig\nCλ,2pE[eλYε∞]+Cλ,2pE[e−λYε∞]/radicalbig\nP(|Yε∞|/greaterorequalslantN)\n/lessorequalslant/radicalbig\nCλ,2p/parenleftBig/radicalbig\nE[eλYε∞]+/radicalbig\nE[e−λYε∞]/parenrightBig/radicalbig\nP(|Yε\n∞|/greaterorequalslantN).(2.22)CUT-OFF PHENOMENON IN THE TOTAL VARIATION AND WASSERSTEIN D ISTANCES 17\nWe claim that\n(2.23) lim\nε→0+E[eλYε\n∞] =eλ2/2=:Kλfor anyλ∈R.\nThen for any |λ|>0 we have\nlimsup\nε→0+E[|Yε\n∞|p1{|Yε∞|/greaterorequalslantN}]/lessorequalslant/radicalbig\nCλ,2plimsup\nε→0+(/radicalbig\nE[eλYε∞]+/radicalbig\nE[e−λYε∞])/radicalbig\nP(|Yε∞|/greaterorequalslantN)\n/lessorequalslant2/radicalbig\nCλ,2pKλlimsup\nε→0+/radicalbig\nP(|Yε∞|/greaterorequalslantN)\n= 2/radicalbig\nCλ,2pKλ/radicalbig\nP(|G|/greaterorequalslantN),(2.24)\nwhere in the last equality we have used that Yε\n∞converges in distribution to Gasε→0+.\nNow, sending N→ ∞we obtain (2.20). In the sequel, we show (2.23). By (2.19) we have\nE[eλYε\n∞] =E[eλΓ(qε+1,1)√qε+1]e−λ√qε+1=E[eλ√qε+1Γ(qε+1,1)]e−λ√qε+1\n=/parenleftbigg\n1−λ√qε+1/parenrightbigg−(qε+1)\ne−λ√qε+1=1\n/parenleftBig\n1+−λ√qε+1/parenrightBig(qε+1)1\ne−√qε+1(−λ)\nfor anyλ <√qε+1. By Lemma D.1 in Appendix D we obtain (2.23). The proof is\ncomplete. /square\n2.2.2.The decoupling argument .In the sequel, we estimate\n1\nε1∧pWp(Xε\nt(x),Xε\n∞).\nBy Item (ii) in Lemma 2.6 and the triangle inequality for Wpwe have\n1\nε1∧pWp(Xε\nt(x),Xε\n∞) =σ1∧p\nε\nε1∧pWp/parenleftbiggXε\nt(x)−mε\nσε,Xε\n∞−mε\nσε/parenrightbigg\n/lessorequalslant/parenleftBigg√\n2b\n2a/parenrightBigg1∧p\nWp/parenleftbiggXε\nt(x)−mε\nσε,G/parenrightbigg\n+/parenleftBigg√\n2b\n2a/parenrightBigg1∧p\nWp/parenleftbigg\nG,Xε\n∞−mε\nσε/parenrightbigg\n,\nwhere in the last inequality we have used that σε=√\n2b\n2aεandGdenotes the standard\nGaussian distribution. Similarly, we obtain\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nε1∧pWp(Xε\nt(x),Xε\n∞)−/parenleftBigg√\n2b\n2a/parenrightBigg1∧p\nWp/parenleftbiggXε\nt(x)−mε\nσε,G/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n/lessorequalslant/parenleftBigg√\n2b\n2a/parenrightBigg1∧p\nWp/parenleftbiggXε\n∞−mε\nσε,G/parenrightbigg\n.\nThe preceding inequality yields the following replacement lemma.18 GERARDO BARRERA AND LILIANA ESQUIVEL\nLemma 2.8 (Replacement lemma in the Wasserstein distance of order p>0).Letp>0\nbe fixed. For any x/greaterorequalslant0andt>0it follows that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nε1∧pWp(Xε\nt(x),Xε\n∞)−/parenleftBigg√\n2b\n2a/parenrightBigg1∧p\nWp/parenleftbiggXε\nt(x)−mε\nσε,G/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n/lessorequalslant/parenleftBigg√\n2b\n2a/parenrightBigg1∧p\nWp/parenleftbiggXε\n∞−mε\nσε,G/parenrightbigg\n,\nwhereGdenotes the standard Gaussian distribution.\nBy Lemma 2.7 and Lemma 2.8 we obtain the following lemma.\nLemma 2.9 (Asymptotically equivalent distances in the Wasserstein distance of order\np >0).Letp >0andx/greaterorequalslant0be fixed. For any function (sε)ε>0such thatsε→ ∞as\nε→0+it follows that\nlimsup\nε→0+1\nε1∧pWp(Xε\nsε(x),Xε\n∞) =/parenleftBigg√\n2b\n2a/parenrightBigg1∧p\nlimsup\nε→0+Wp/parenleftbiggXε\nsε(x)−mε\nσε,G/parenrightbigg\nand\nliminf\nε→0+1\nε1∧pWp(Xε\nsε(x),Xε\n∞) =/parenleftBigg√\n2b\n2a/parenrightBigg1∧p\nliminf\nε→0+Wp/parenleftbiggXε\nsε(x)−mε\nσε,G/parenrightbigg\n.\nFinally, the limiting profile is calculated in the following lemma.\nLemma 2.10 (Limiting profile) .Letp>0andx/greaterorequalslant0be fixed. For tεandωεbeing defined\nin(1.8), it follows that\nlim\nε→0+1\nε1∧pWp(Xε\ntε+r·ωε(x),Xε\n∞) =/parenleftBigg√\n2b\n2a/parenrightBigg1∧p\nWp(Cx·e−r+G,G)for anyr∈R,\nwhereGdenotes the standard Gaussian distribution and Cx=√\n2√\nb(ax−b).\nProof.Recall by (2.5) and (2.6) that\nYε\nt(x) =Xε\nt(x)−mε\nσε=cε(∞)Xε\nt(x)−(qε+1)√qε+1for allt/greaterorequalslant0.\nBy Lemma 2.9 it is enough to show that\n(2.25) lim\nε→0+Wp/parenleftbig\nYε\ntε+r·ωε(x),G/parenrightbig\n=Wp/parenleftbig\nCx·e−r+G,G/parenrightbig\nfor anyr∈R.\nBy the triangle inequality for Wpwe note that\nWp/parenleftbig\nYε\ntε+r·ωε(x),G/parenrightbig\n/lessorequalslantWp/parenleftbig\nYε\ntε+r·ωε(x),Cx·e−r+G/parenrightbig\n+Wp/parenleftbig\nCx·e−r+G,G/parenrightbig\nand\nWp/parenleftbig\nG,Cx·e−r+G/parenrightbig\n/lessorequalslantWp/parenleftbig\nG,Yε\ntε+r·ωε(x)/parenrightbig\n+Wp/parenleftbig\nYε\ntε+r·ωε(x),Cx·e−r+G/parenrightbig\n.\nTherefore, to prove (2.25) it is enough to show that\n(2.26) lim\nε→0+Wp/parenleftbig\nYε\ntε+r·ωε(x),Cx·e−r+G/parenrightbig\n= 0 for any r∈R.CUT-OFF PHENOMENON IN THE TOTAL VARIATION AND WASSERSTEIN D ISTANCES 19\nThe proof of (2.26) is analogous to the proof of Lemma 2.7. Indeed, sinceYε\ntε+r·ωε(x)\nconverges in distribution to Cx·e−r+Gasε→0+, see (2.13), Theorem 7.12 in [77] implies\nthatit is enoughto show that thefamily ( Yε\ntε+r·ωε(x))ε∈(0,√\n2bis uniformly integrable. More\nprecisely,\n(2.27) lim\nN→∞limsup\nε→0+E[|Yε\ntε+r·ωε(x)|p1{|Yε\ntε+r·ωε(x)|/greaterorequalslantN}] = 0.\nSimilarly to (2.22) we obtain\nE/bracketleftBig\n|Yε\ntε+r·ωε(x)|p1{|Yε\ntε+r·ωε(x)|/greaterorequalslantN}/bracketrightBig\n/lessorequalslant/radicalbig\nCλ,2p/parenleftbigg/radicalBig\nE[eλYε\ntε+r·ωε(x)]+/radicalBig\nE[e−λYε\ntε+r·ωε(x)]/parenrightbigg\n×/radicalBig\nP/parenleftbig\n|Yε\ntε+r·ωε(x)|/greaterorequalslantN/parenrightbig\n,\nwhere the constant Cλ,2pis given in (2.21). With the help of Lemma C.4 in Appendix C\nwe obtain\nlim\nε→0+E[eλYε\nsε(x)] =eλ2/2eλCxfor anyλ∈R.\nAn analogous reasoning using in (2.24) yields (2.27). The proof is comp lete. /square\nIn the sequel, we analyze the case x=b/athat yields Cx= 0. Hence, Lemma 2.10\ndoes not imply profile cut-off phenomenon at tε+r·ωε. In fact, in this case, there is no\ncut-off phenomenon as the following lemma states.\nLemma 2.11 (No cut-off phenomenon for x=b/ainWpforp>0).Letp>0be fixed.\nForx=b/aand for any function (sε)ε>0satisfyingsε→ ∞asε→0+, it follows that\n(2.28) lim\nε→0+1\nε1∧pWp(Xε\nsε(x),Xε\n∞) = 0.\nProof.By Lemma 2.9, the limit (2.28) is equivalent to the limit\n(2.29) lim\nε→0+Wp/parenleftbig\nYε\nsε(x),G/parenrightbig\n= 0,\nwhereGdenotes the Gaussian distribution and\nYε\nt(x) =Xε\nt(x)−mε\nσε=cε(∞)Xε\nt(x)−(qε+1)√qε+1for allt/greaterorequalslant0.\nBy (2.18) we have that Yε\nsε(x) tends in distribution to Gasε→0+. To conclude (2.29) it\nis enough to show that the family ( Yε\nsε(x))ε∈(0,√\n2b)is uniformly integrable.\nSimilarly to (2.22) we obtain\nE/bracketleftbig\n|Yε\nsε(x)|p1{|Yεsε(x)|/greaterorequalslantN}/bracketrightbig\n/lessorequalslant/radicalbig\nCλ,2p/parenleftbigg/radicalBig\nE[eλYεsε(x)]+/radicalBig\nE[e−λYεsε(x)]/parenrightbigg\n×/radicalBig\nP/parenleftbig\n|Yεsε(x)|/greaterorequalslantN/parenrightbig\nfor|λ|>0, where the constant Cλ,2pis given in (2.21). With the help of Lemma C.5 in\nAppendix C we obtain\nlim\nε→0+E[eλYε\nsε(x)] =eλ2/2for anyλ∈R.\nAn analogous reasoning using in (2.24) yields that the family ( Yε\nsε(x))ε∈(0,√\n2b)is uniformly\nintegrable. The proof is complete. /square20 GERARDO BARRERA AND LILIANA ESQUIVEL\n2.2.3.Proof of Theorem 1.7 .In this subsection, we stress the fact that Theorem 1.7\nis just a consequence of what we have already proved up to here.\nLetp >0 be fixed. For x/greaterorequalslant0 and fortεandωεbeing defined in (1.8), Lemma 2.10\nimplies\nlim\nε→0+1\nε1∧pWp(Xε\ntε+r·ωε(x),Xε\n∞) =/parenleftBigg√\n2b\n2a/parenrightBigg1∧p\nWp(Cx·e−r+G,G) for any r∈R,\nwhereGdenotes the standard Gaussian distribution and Cx=√\n2√\nb(ax−b).\nThis yields (1.12). For x/greaterorequalslant0 andx/\\e}atio\\slash=b/awe have that Cx/\\e}atio\\slash= 0, which implies (1.13).\nand therefore (1.5) holds true.\nFinally, Lemma 2.11 yields no cut-off phenomenon at x=b/ain the sense of (1.4).\nThe proof is complete.\nAppendix A.General properties of the total variation distance and\nParseval–Plancherel–Fourier approach to total variation\nconvergence\nIn this section, we provide general properties of the total variat ion distance. As we\nhave already discussed in Section 1, in general, convergence in distr ibution does not\nimply convergence in the total variation distance. We then impose su itable hypotheses\non the sequence of characteristic functions that ensure conver gence in the total variation\ndistance. Since the latter could be of independent interest, we sta te the results in a\nmultidimensional setting.\nAlong this section, dalways is a positive fixed integer, /⌊ard⌊l·/⌊ard⌊ldenotes the Euclidean norm\ninRd,/a\\}⌊ra⌋ketle{t·,·/a\\}⌊ra⌋ketri}htdenotes the standard inner product in RdandB(Rd) are the Borelian sets of\nRd.\nLemma A.1 (General properties of the total variation distance) .LetXandYbe ran-\ndom vectors defined in the probability space (Ω,F,P)and taking values in Rd. Then the\nfollowing is valid.\n(i)Translation invariance: For any deterministic vectors v1,v2∈Rdit follows that\ndTV(v1+X,v2+Y) = dTV(v1−v2+X,Y) = dTV(X,v2−v1+Y).\n(ii)Scaling invariance: For any non-zero constant cit follows that\ndTV(cX,cY) = dTV(X,Y).\nThe proof is a direct consequence of Theorem 5.2 in [31].\nLemma A.2 (Scheff´ e’s Lemma) .Let(Ω,F,P)be a probability space and consider a se-\nquence(Pn)n∈Nof probability measures on space (Ω,F). For each n∈Nassume that\nfn:Rd→[0,∞)is a density of Pnwith respect to the Lebesgue measure on (Rd,B(Rd)).\nIn addition, assume that f:Rd→[0,∞)is a density of Pwith respect to the Lebesgue\nmeasureon (Rd,B(Rd)). Iflim\nn→∞fn(x) =f(x)forx-almosteverywhere (w.r.t. the Lebesgue\nmeasure), then it follows that\nlim\nn→∞dTV(Pn,P) = 0.\nThe proof is well-known and it can be found in Lemma 3.3.2 of [72].CUT-OFF PHENOMENON IN THE TOTAL VARIATION AND WASSERSTEIN D ISTANCES 21\nProposition A.3 (Fourier approach to total variation convergence) .Let(Xε)ε>0be a\nfamily of random vectors defined in the probability space (Ω,F,P)and taking values\non(Rd,B(Rd)). LetXbe a random vector defined in (Ω,F,P)and taking values on\n(Rd,B(Rd)). Assume that\n(i)Convergence in distribution: Xεd−→Xasε→0+.\n(ii)Fourier integrability:\n(A.1) ϕε,ϕ∈L1(Rd)for eachε>0,\nwhereϕεandϕdenote the characteristic function of XεandX, respectively.\n(iii)Tail behavior (uniform integrability):\n(A.2) limsup\nℓ→∞limsup\nε→0+/integraldisplay\n{|z|/greaterorequalslantℓ}|ϕε(z)|dz= 0.\nThen it follows that\n(A.3) lim\nε→0+dTV(Xε,X) = 0.\nProof.Hypothesis (A.1) with the help of the Fourier Inversion Theorem yield s the exis-\ntence of continuous and bounded densities fεandfsuch that\nfε(x) =1\n(2π)d/integraldisplay\nRde−i/an}bracketle{tz,x/an}bracketri}htϕε(z)dzandf(x) =1\n(2π)d/integraldisplay\nRde−i/an}bracketle{tz,x/an}bracketri}htϕ(z)dz,\nsee Proposition 2.5 Item (xii) in [73]. Then for all x∈Rdwe have\n(2π)d|fε(x)−f(x)|=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nRd/parenleftbig\ne−i/an}bracketle{tz,x/an}bracketri}htϕε(z)−e−i/an}bracketle{tz,x/an}bracketri}htϕ(z)/parenrightbig\ndz/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n/lessorequalslant/integraldisplay\nRd|ϕε(z)−ϕ(z)|dz\n=/integraldisplay\n{|z|/lessorequalslantℓ}|ϕε(z)−ϕ(z)|dz+/integraldisplay\n{|z|>ℓ}|ϕε(z)−ϕ(z)|dz(A.4)\nfor allℓ>0. For each ℓ>0, Item (i) of Theorem 15.24 in [47] yields\n(A.5) lim\nε→0+/integraldisplay\n{|z|/lessorequalslantℓ}|ϕε(z)−ϕ(z)|dz= 0.\nBy (A.4) and (A.5) it follows that\n(2π)dlimsup\nε→0+|fε(x)−f(x)|/lessorequalslantlimsup\nε→0+/integraldisplay\n{|z|>ℓ}|ϕε(z)−ϕ(z)|dz\n/lessorequalslantlimsup\nε→0+/integraldisplay\n{|z|>ℓ}|ϕε(z)|dz+/integraldisplay\n{|z|>ℓ}|ϕ(z)|dz(A.6)\nforx∈Rdand anyℓ>0. Sinceϕ∈L1(Rd), it follows that\n(A.7) lim\nℓ→∞/integraldisplay\n{|z|>ℓ}|ϕ(z)|dz= 0.\nSendingℓto infinity both sides of (A.6), Hypothesis (A.2) and (A.7) imply\nlimsup\nε→0+|fε(x)−f(x)|= 0 for x∈Rd.22 GERARDO BARRERA AND LILIANA ESQUIVEL\nHence, it follows that\nlim\nε→0+fε(x) =f(x) forx∈Rd.\nThe preceding limit with the help of Lemma A.2 in Appendix A implies (A.3). Th e proof\nis complete. /square\nProposition A.4 (Parseval–Plancherel approachtotheconvergence inthetotal variation\ndistance) .Let(Xε)ε>0andXbe as in Proposition A.3. Assume that the Item (i) in\nProposition A.3 holds true. In addition, assume that\n(ii’)L2(Rd)-integrability:\nϕε,ϕ∈L2(Rd)for eachε>0,\nwhereϕεandϕdenote the characteristic function of XεandX, respectively.\n(iii’)Tail behavior ( L1(Rd)-uniform integrability):\nlimsup\nℓ→∞limsup\nε→0+/integraldisplay\n{|z|/greaterorequalslantℓ}|ϕε(z)|2dz= 0.\nThen it follows that\nlim\nε→0+dTV(Xε,X) = 0.\nProof.Item (ii’) with the help of Lemma 9.2.8 in [48] yields that the laws of Xεand\nXare absolutely continuous respect to the Lebesgue measure on Rd, i.e., the existence\nof densities fεandfcorresponding to ϕεandϕ, respectively. By Parseval–Plancherel\nTheorem we have\n(A.8)/integraldisplay\nRd(fε(x)−f(x))2dx=/integraldisplay\nRd|ϕε(z)−ϕ(z)|2dz.\nWe claim that the right-hand side of (A.8) tends to zero as ε→0+. Indeed, let ℓ>0 be\nfixed and note that/integraldisplay\nRd|ϕε(z)−ϕ(z)|2dz=/integraldisplay\n{|z|/lessorequalslantℓ}|ϕε(z)−ϕ(z)|2dz+/integraldisplay\n{|z|>ℓ}|ϕε(z)−ϕ(z)|2dz\nFor eachℓ>0, Item (i) of Theorem 15.24 in [47] with the help of Item (i) gives\nlim\nε→0+/integraldisplay\n{|z|/lessorequalslantℓ}|ϕε(z)−ϕ(z)|2dz= 0.\nSince\nlimsup\nε→0+/integraldisplay\n{|z|>ℓ}|ϕε(z)−ϕ(z)|2dz/lessorequalslant2limsup\nε→0+/integraldisplay\n{|z|>ℓ}|ϕε(z)|2dz\n+2/integraldisplay\n{|z|>ℓ}|ϕ(z)|2dz\nandϕ∈L2(Rd), we have\nlimsup\nℓ→∞limsup\nε→0+/integraldisplay\n{|z|>ℓ}|ϕε(z)−ϕ(z)|2dz/lessorequalslant2limsup\nℓ→∞limsup\nε→0+/integraldisplay\n{|z|>ℓ}|ϕε(z)|2dz.\nHence Item (iii’) gives\nlim\nℓ→∞lim\nε→0+/integraldisplay\n{|z|>ℓ}|ϕε(z)−ϕ(z)|2dz= 0.CUT-OFF PHENOMENON IN THE TOTAL VARIATION AND WASSERSTEIN D ISTANCES 23\nBy (A.8) we obtain\n(A.9) lim\nε→0+/integraldisplay\nRd(fε(x)−f(x))2dx= 0.\nTo conclude we show that the preceding limit yields\n(A.10) lim\nε→0+/integraldisplay\nRd|fε(x)−f(x)|dx= 0.\nIndeed, (A.9) with the help of the Cauchy–Schwarz inequality gives f or anyK >0\n(A.11) lim\nε→0+/integraldisplay\n{|z|/lessorequalslantK}|fε(x)−f(x)|dx= 0.\nRecall that fεandfare the densities of the random vectors XεandX, respectively. Since\nf∈L1(Rd) andXε→Xin distribution as ε→0+, we have\nlimsup\nK→∞limsup\nε→0+/integraldisplay\n{|z|>K}|fε(x)−f(x)|dx/lessorequalslantlimsup\nK→∞limsup\nε→0+/integraldisplay\n{|z|/greaterorequalslantK}fε(x)dx\n/lessorequalslantlimsup\nK→∞limsup\nε→0+P(/⌊ard⌊lXε/⌊ard⌊l/greaterorequalslantK)/lessorequalslantlimsup\nK→∞P(/⌊ard⌊lX/⌊ard⌊l/greaterorequalslantK) = 0.(A.12)\nBy (A.11) and (A.12) we deduce (A.10). /square\nAppendix B.Scaling property and local central limit theorem for Gamma\ndistribution\nIn this section, we show a scaling property for the second paramet er of the Gamma\ndistribution and a Gaussian local central limit theorem. We recall tha t for any positive\nparameters αandθ, Γ(α,θ) denotes a Gamma distribution with characteristic function\ngiven by (2.2).\nLemma B.1 (Scaling property of the Gamma distribution) .For any positive constants\nαandθit follows that Γ(α,θ)d=θ−1Γ(α,1).\nProof.For anyu∈R, (2.2) yields\n(B.1) E/bracketleftBig\neiu(θ−1Γ(α,1))/bracketrightBig\n=E/bracketleftBig\nei(uθ−1)Γ(α,1)/bracketrightBig\n=/parenleftbig\n1−iuθ−1/parenrightbig−r=E/bracketleftbig\neiuΓ(α,θ)/bracketrightbig\n.\nRecall that the distribution of a random variable is characterized by its characteristic\nfunction, see for instance Theorem 15.9 in [47]. Then (B.1) implies the s tatement. /square\nLemma B.2 (Local central limit theorem) .For any positive αsetZα:=Γ(α,1)−α√α. Then\nZαconverges in the total variation distance to a standard Gaus sian distribution as α→\n∞.\nProof.Letα>0 be fixed and let gαbe the density of the random variable Zα. In other\nwords,\n(B.2) gα(z) =√α\nΓ(α)(√αz+α)α−1e−√αz−α1{z/greaterorequalslant−√α},\nwhere Γ denotes the usual Gamma function. Recall that, for shor t, we write Fα→∞∼Gin\na place of lim\nα→∞F(α)\nG(α)= 1. The Stirling formula for the function Γ yields\nΓ(α)α→∞∼/radicalbig\n2π(α−1)/parenleftbiggα−1\ne/parenrightbiggα−1\n.24 GERARDO BARRERA AND LILIANA ESQUIVEL\nThen we have\n(B.3)√α\nΓ(α)α→∞∼1√\n2π√α√α−1/parenleftbigge\nα−1/parenrightbiggα−1\nα→∞∼1√\n2π/parenleftbigge\nα−1/parenrightbiggα−1\n.\nNote that\n(B.4) eα→∞∼1/parenleftbig\n1−1\nα/parenrightbigα−1α→∞∼/parenleftbiggα\nα−1/parenrightbiggα−1\n.\nHence, for any z∈Rfixed, combining (B.3), (B.4) in (B.2) yields\ngα(z)α→∞∼e−1\n√\n2π/parenleftbiggα\nα−1/parenrightbiggα−1/parenleftbigg\n1+√αz\nα/parenrightbigg−1/parenleftbigg\n1+√αz\nα/parenrightbiggα\ne−√αz1{z/greaterorequalslant−√α}\nα→∞∼1√\n2π/parenleftbigg\n1+z√α/parenrightbiggα\ne−√αz.(B.5)\nBy Lemma D.1 in Appendix D we have,\n(B.6)/parenleftbigg\n1+z√α/parenrightbiggα\ne−√αzα→∞∼e−z2/2.\nConsequently, for any z∈R(B.5) and (B.6) imply\nlim\nα→∞gα(z) =1√\n2πe−z2/2,\nwhich with the help of Lemma A.2 in Appendix A gives the statement. /square\nAppendix C.Ergodicity and characteristic function for the SDE (1.1)\nIn the section, we recall the strong ergodicity (total variation co nvergence) and the\nexplicit formulas for the characteristic function and moment gener ating function of (1.1).\nWe recall that\nqε+1 :=2b\nε2, cε(t) :=2a\nε21\n1−e−atfor anyt>0 andcε(∞) :=2a\nε2.\nLemma C.1 (Ergodicity for the CIR model in the Feller regime) .For anyε∈(0,√\n2b)\nandx∈[0,∞)let(Xε\nt(x))t/greaterorequalslant0be the unique strong solution of (1.1). Then there exists a\nrandom variable Xε\n∞such that for any, x∈[0,∞)it follows that\n(C.1) lim\nt→∞dTV(Xε\nt(x),Xε\n∞) = 0\nand\n(C.2) lim\nt→∞Wp(Xε\nt(x),Xε\n∞) = 0.\nIn addition, Xε\n∞d= Γ(qε+1,cε(∞)), whereqε+1andcε(∞)are defined in (2.1).\nThe proof of (C.1) can be found for instance in Theorem 1.2 ] of [44]. U sing the Fourier\nmethod in Proposition A.3 in Appendix A, an alternative proof of (C.1) c an be obtained.\nIn addition, the convergence in (C.2) can be deduced using Theorem 7.12 in [77].CUT-OFF PHENOMENON IN THE TOTAL VARIATION AND WASSERSTEIN D ISTANCES 25\nLemma C.2 (Analytic continuation of the characteristic function for the CIR p rocess).\nFor anyε∈(0,∞)andx∈[0,∞)let(Xε\nt(x))t/greaterorequalslant0be the unique strong solution of (1.1).\nFor eacht>0the complex characteristic function of the time marginal Xε\nt(x)is given by\nE/bracketleftbig\nezXε\nt/bracketrightbig\n=1\n/parenleftBig\n1−z\ncε(t)/parenrightBig2b\nε2exp/parenleftbiggzcε(t)xe−at\ncε(t)−z/parenrightbigg\n1{Re(z)<cε(t)},\nwherecε(t) =2a\nε21\n(1−e−at)andRe(z)denotes the real part of the complex number z. In other\nwords,\nE/bracketleftbig\nezXε\nt/bracketrightbig\n=1\n/parenleftBig\n1−z\ncε(t)/parenrightBig2b\nε2exp/parenleftbiggzc2\nε(t)xe−at\nc2ε(t)−z2/parenrightbigg\nexp/parenleftbiggz2cε(t)xe−at\nc2ε(t)−z2/parenrightbigg\n1{Re(z)<cε(t)}.\nThe proof can be found in Proposition 1.2.4 p. 7 of [2].\nCorollary C.3 (Characteristic function for the CIR process) .For anyε∈(0,∞)and\nx∈[0,∞)let(Xε\nt(x))t/greaterorequalslant0be the unique strong solution of (1.1). For any z∈R, the\ncharacteristic function and the moment generating functio n of the time marginal Xε\nt(x)\nare given respectively by\nψ(z;t,ε,x) :=E/bracketleftbig\neizXε\nt(x)/bracketrightbig\n=1\n/parenleftBig\n1−iz\ncε(t)/parenrightBig2b\nε2exp/parenleftbigg\nizc2\nε(t)xe−at\nc2\nε(t)+z2/parenrightbigg\nexp/parenleftbigg\n−z2cε(t)xe−at\nc2\nε(t)+z2/parenrightbigg\nand\nφ(z;t,ε,x) :=E/bracketleftbig\nezXε\nt/bracketrightbig\n=1\n/parenleftBig\n1−z\ncε(t)/parenrightBig2b\nε2exp/parenleftbiggzc2\nε(t)xe−at\nc2ε(t)−z2/parenrightbigg\nexp/parenleftbiggz2cε(t)xe−at\nc2ε(t)−z2/parenrightbigg\n1{z<cε(t)}.\nMoreover,\n|ψ(z;t,ε,x)|/lessorequalslant1\n/parenleftBig\n1+z2\nc2ε(t)/parenrightBigb\nε2for anyz∈R,\nwhich yields/integraldisplay\nR|ψ(z;t,ε,x)|dz <∞whenε∈(0,√\n2b).\nLemma C.4 (Gaussian central limit theorem at the mixing time) .Letx∈[0,∞)and\nr∈R. Then it follows that\n(C.3) lim\nε→0+E/bracketleftbig\nezYε\ntε+r·ωε(x)/bracketrightbig\n= exp/parenleftbiggz2\n2/parenrightbigg\nexp/parenleftBigg\nz√\n2√\nb(ax−b)e−r/parenrightBigg\nfor anyz∈C,\nwhereYε\nt(x) =cε(∞)Xε\nt(x)−(qε+1)√qε+1for anyt/greaterorequalslant0, andtεandωεare given by\ntε:=1\naln/parenleftbigg1\nε/parenrightbigg\nandωε:=1\na.26 GERARDO BARRERA AND LILIANA ESQUIVEL\nProof.Letz∈Csuch that\n(C.4) Re(z)</radicalbig\nqε+1cε(tε+r·ωε)\ncε(∞)=√\n2b\nε1\n1−εe−r.\nWe denote ϕε(z;x) :=E[ezYε\ntε+rωε(x)]. Straightforward computations with the help of\nLemma C.2 in Appendix C yields\n(C.5) ϕε(z;x) =E1(ε)E2(ε;x)E3(ε;x),\nwhere\n(C.6) E1(ε) :=e−z√qε+1\n/parenleftBig\n1−cε(∞)z\ncε(tε+r·ωε)√qε+1/parenrightBigqε+1,\n(C.7) E2(ε;x) := exp/parenleftBiggcε(∞)z√qε+1c2\nε(tε+r·ωε)xe−a(tε+r·ωε)\nc2ε(tε+r·ωε)−c2ε(∞)z2\nqε+1/parenrightBigg\nand\n(C.8) E3(ε;x) := exp/parenleftBiggc2\nε(∞)z2\nqε+1cε(tε+r·ωε)xe−a(tε+r·ωε)\nc2\nε(tε+r·ωε)−c2ε(∞)z2\nqε+1/parenrightBigg\n.\nIn the sequel, we compute the limiting behavior of (C.6). Let αε:=qε+1,u:=−z,\nuε=−cε(∞)\ncε(tε+r·ωε)z=−(1−e−a(tε+r·ωε))z=u−εe−ru,\nfrom this we observe that√αε(uε−u) =z√\n2be−r. By Lemma D.2 in Appendix D we\nhave\n(C.9) lim\nε→0+E1(ε) =ez2/2e−z√\n2be−r.\nNow, we compute the asymptotic behavior of (C.7). Straightforwa rd computation with\nthe help of (2.1) yields,\nE2(ε;x) = exp/parenleftBigg\nzxcε(∞)√qε+1c2\nε(tε+r·ωε)\nc2ε(∞)e−a(tε+r·ωε)\nc2ε(tε+r·ωε)\nc2ε(∞)−z2\nqε+1/parenrightBigg\n= exp/parenleftBigg\nzxcε(∞)√qε+11\n(1−e−a(tε+r·ωε))2e−a(tε+r·ωε)\n1\n(1−e−a(tε+r·ωε))2−z2\nqε+1/parenrightBigg\n= exp/parenleftBigg\nzx2a√\n2b1\n(1−εe−r)2e−r\n1\n(1−εe−r)2−ε2z2\n2b/parenrightBigg\n,\nwhich implies\n(C.10) lim\nε→0+E2(ε;x) = exp/parenleftbigg\nzx2a√\n2be−r/parenrightbigg\n.CUT-OFF PHENOMENON IN THE TOTAL VARIATION AND WASSERSTEIN D ISTANCES 27\nFinally, we compute the limiting behavior of (C.8). We note that\nE3(ε;x) = exp/parenleftBigg\nz2x1\nqε+1cε(tε+r·ωε)e−a(tε+r·ωε)\nc2ε(tε+r·ωε)\nc2ε(∞)−z2\nqε+1/parenrightBigg\n= exp/parenleftBigg\nz2xε2\n2b2a\nεe−r1\n1−εer\n1\n(1−εe−r)2−z2ε\n2b/parenrightBigg\n,\nwhich yields\n(C.11) lim\nε→0+E3(ε;x) = 1.\nCombining (C.9), (C.10) and (C.11) in (C.5) with the observation that t he right-hand\nside of (C.4) tends to infinity as ε→0+, we obtain (C.3). /square\nLemma C.5 (Gaussian central limit theorem at any time scale) .Letx=b\naandr∈R.\nThen for any function (sε)ε>0such thatsε→ ∞asε→0+it follows that\n(C.12) lim\nε→0+E/bracketleftbig\nezYε\nsε(x)/bracketrightbig\n= exp/parenleftbiggz2\n2/parenrightbigg\nfor anyz∈C,\nwhereYε\nt(x) =cε(∞)Xε\nt(x)−(qε+1)√qε+1for anyt/greaterorequalslant0.\nProof.Letz∈Csuch that\nRe(z)</radicalbig\nqε+1cε(sε)\ncε(∞)=√\n2b\nε1\n1−e−asε,\nand let/tildewideϕε(z;x) :=E[ezYε\nsε(x)]. Similarly to (C.5) we obtain\n(C.13) /tildewideϕε(z;x) =/tildewideE1(ε)/tildewideE2(ε;x)/tildewideE3(ε;x),\nwhere\n(C.14)/tildewideE1(ε) :=e−z√qε+1\n/parenleftBig\n1−cε(∞)z\ncε(sε)√qε+1/parenrightBigqε+1=1/parenleftBig\n1−(1−e−asε)z√qε+1/parenrightBigqε+1e−z√qε+1,\n/tildewideE2(ε;x) : = exp/parenleftBiggcε(∞)z√qε+1c2\nε(sε)xe−asε\nc2ε(sε)−c2ε(∞)z2\nqε+1/parenrightBigg\n= exp/parenleftBigg\nz/radicalbig\nqε+1e−asε1\n1−(1−e−asε)2z2\nqε+1/parenrightBigg\n(C.15)\nand\n(C.16) /tildewideE3(ε;x) := exp/parenleftBiggc2\nε(∞)z2\nqε+1cε(sε)xe−asε\nc2ε(sε)−c2ε(∞)z2\nqε+1/parenrightBigg\n= exp/parenleftBigg\nz2e−asε1\n1−e−asε\n1\n(1−e−asε)2−z2\nqε+1/parenrightBigg\n.\nIn the sequel, we show that\nlim\nε→0+/tildewideϕε(z;x) =ez2/2for allz∈C.\nBy (C.16) we observe that\n(C.17) lim\nε→0+/tildewideE3(ε;x) = 1.28 GERARDO BARRERA AND LILIANA ESQUIVEL\nNow, by (C.14) we note that\n/tildewideE1(ε) =1/parenleftBig\n1−(1−e−asε)z√qε+1/parenrightBigqε+1ez√qε+1(1−e−asε)\nez√qε+1(1−e−asε)e−z√qε+1\n=1/parenleftBig\n1−(1−e−asε)z√qε+1/parenrightBigqε+11\nez√qε+1(1−e−asε)e−z√qε+1e−asε.(C.18)\nBy (C.15) and (C.18) we obtain\n/tildewideE1(ε)/tildewideE2(ε;x) =1/parenleftBig\n1−(1−e−asε)z√qε+1/parenrightBigqε+11\nez√qε+1(1−e−asε)\n×exp/parenleftBigg\n−z/radicalbig\nqε+1e−asε1\nqε+1/parenleftBigg\nz2(1−e−asε)2\n1+(1−e−asε)2z2\nqε+1/parenrightBigg/parenrightBigg\n.\nThe preceding equality with the help of Lemma D.3 in Appendix D yields\n(C.19) lim\nε→0+/tildewideE1(ε)/tildewideE2(ε;x) =ez2/2for allz∈C.\nCombining (C.17) and (C.19) in (C.13) implies (C.12). /square\nAppendix D.Second order Taylor’s expansions\nIn this section, we show refined asymptotics that were used along t he manuscript.\nLemma D.1 (Second order asymptotics I) .For anyz∈Rit follows that\nlim\nα→∞/parenleftbigg\n1+z√α/parenrightbiggα\ne−√αz=e−z2/2.\nProof.Letz∈Rbe fixed. It is enough to show that\nlim\nα→∞/bracketleftbigg\nαln/parenleftbigg\n1+z√α/parenrightbigg\n−√αz/bracketrightbigg\n=−z2\n2.\nBy Taylor’s expansion we have for α≫1\nln/parenleftbigg\n1+z√α/parenrightbigg\n=z√α−z2\n2α+H(z,α),\nwhere\n(D.1) limsup\nr→∞(α3/2|H(z,α)|)<∞.\nThen we have\nαln/parenleftbigg\n1+z√α/parenrightbigg\n−√αz=−z2\n2+αH(z,α).\nBy (D.1) we deduce α|G(z,α)| →0 asα→ ∞and conclude the statement. /square\nLemma D.2 (Second order asymptotics II) .Let(uα)α>0⊂Cbe a function such that\n(D.2) lim\nα→∞√α(uα−u) =cfor some u∈Candc∈C.\nThen it follows that\nlim\nα→∞/parenleftbigg\n1+uα√α/parenrightbiggα\ne−√αu=e−u2/2+c.CUT-OFF PHENOMENON IN THE TOTAL VARIATION AND WASSERSTEIN D ISTANCES 29\nProof.It is enough to show that,\nlim\nα→∞/bracketleftbigg\nαln/parenleftbigg\n1+uα√α/parenrightbigg\n−√αu/bracketrightbigg\n=−u2\n2+c.\nHere, we are considering the principal branch of the complex logarit hmic function log :=\nln. By Taylor’s expansion we have for α≫1\nln/parenleftbigg\n1+uα√α/parenrightbigg\n=uα√α−u2\nα\n2α+H(uα,α),\nwhere\n(D.3) limsup\nα→∞(α3/2|H(uα,α)|)<∞.\nThen we have\nαln/parenleftbigg\n1+uα√α/parenrightbigg\n−√αuα=−u2\nα\n2+αH(uα,α),\nwhich can be written as\nαln/parenleftbigg\n1+uα√α/parenrightbigg\n−√αu=√α(uα−u)−u2\nα\n2+αH(uα,α).\nBy (D.3) we deduce that α|H(uα,α)| →0 asα→ ∞and with the help of (D.2) we\nconclude the statement. /square\nLemma D.3 (Second order asymptotics III) .Let(uα)α>0⊂Cbe a function such that\n(D.4) lim\nα→∞uα=ufor some u∈C.\nThen it follows that\nlim\nα→∞/parenleftbigg\n1+uα√α/parenrightbiggα\ne−√αuα=e−u2/2.\nProof.It is enough to prove that,\nlim\nα→∞/bracketleftbigg\nαln/parenleftbigg\n1+uα√α/parenrightbigg\n−√αuα/bracketrightbigg\n=−u2\n2.\nHere, we are considering the principal branch of the complex logarit hmic function log :=\nln.By Taylor’s expansion we have for α≫1\nln/parenleftbigg\n1+uα√α/parenrightbigg\n=uα√α−u2\nα\n2α+H(uα,α),\nwhere\n(D.5) limsup\nα→∞(α3/2|H(uα,α)|)<∞.\nThen we have\nαln/parenleftbigg\n1+uα√α/parenrightbigg\n−√αuα=−u2\nα\n2+αH(uα,α).\nBy (D.5) we deduce that α|H(uα,α)| →0 asα→ ∞and with the help of (D.4) we\nconclude the statement. /square\nRemark D.4. We point out that (D.2)implies(D.4). However, in general (D.4)does\nnot imply (D.2).30 GERARDO BARRERA AND LILIANA ESQUIVEL\nAppendix E.Mixing times equivalence of asymptotic profile cut-off\nIn this section, we show Item (4) of Proposition 1.2, which gives a rela tion between\nthe mixing time with the cut-off time and the time window. The proofs of the remainder\nitems are analogous.\nLemma E.1 (Asymptotics of mixing times and profile cut-off phenomenon) .Assume\nthat the function G:R→(0,Diam)is strictly decreasing and continuous. In addition,\nassume that lim\nr→−∞G(r) =Diamandlim\nr→∞G(r) = 0, and the existence of a function\n((tε,ωε))ε>0satisfying\nlim\nε→0+tε=∞andlim\nε→0+ωε\ntε= 0.\nThen the following statements are equivalent.\n(a)The following limit\n(E.1) lim\nε→0+distε(LawXε\ntε+r·ωε,νε) =G(r)for anyr∈R\nholds true.\n(b)For anyη∈(0,Diam)it follows that\n(E.2) τε,distε\nmix(η) =tε+G−1(η)·ωε+o(ωε),asε→0+,\nis valid, where G−1(η)is the unique r:=rη∈Rsuch thatG(r) =ηand the\nfunction o(ωε)satisfieso(ωε)\nωε→0asε→0+.\nProof.We start assuming that the statement of Item (a) is valid.\nLetη∈(0,Diam) be fixed. We first show that\n(E.3) limsup\nε→0+τε,x\nmix(η)−(tε+G−1(η)·ωε)\nωε/lessorequalslant0.\nWe takeδ∗such that 0 < δ∗< η. SinceGis a bijection, there exists a unique r∗∈R\nsuch thatG(r∗) =η−δ∗, i.e,r∗=G−1(η−δ∗). For the preceding choice ηandδ∗, the\nlimit (E.1) with the help of (1.3) gives the existence of ε∗=ε∗(η,δ∗)>0 such that\n0<η<Diamεand\ndistε(tε+r∗·ωε)<δ∗+G(r∗) =η(E.4)\nfor all 0<ε<ε∗. Thus (E.4) with the help of the definition of η-mixing time yields\nτε,x\nmix(η)/lessorequalslanttε+r∗·ωε=tε+G−1(η−δ∗)·ωε\n=tε+G−1(η)·ωε+/parenleftbig\nG−1(η−δ∗)−G−1(η)/parenrightbig\n·ωε\nfor all 0<ε<ε∗. Hence, we obtain\n(E.5) limsup\nε→0+τε,x\nmix(η)−(tε+G−1(η)·ωε)\nωε/lessorequalslantG−1(η−δ∗)−G−1(η)\nfor anyη >0 andδ∗∈(0,η). SinceGis a strictly decreasing continuous function, then\nG−1is also a continuous function, see Proposition 3.6.6. in [53]. Moreover , the left-hand\nof (E.5) does not depend on δ∗, then tending δ∗→0+we deduce (E.3).\nWe now prove that\n(E.6) liminf\nε→0+τε,x\nmix(η)−(tε+G−1(η)·ωε)\nωε/greaterorequalslant0.CUT-OFF PHENOMENON IN THE TOTAL VARIATION AND WASSERSTEIN D ISTANCES 31\nWe takeδ∗>0 such that η+δ∗<Diam. SinceGis a bijection, there exists a unique\nr∗∈Rsuch thatG(r∗) =η+δ∗, i.e,r∗=G−1(η+δ∗). For the preceding choice ηandδ∗,\nthe limit (E.1) with the help of (1.3) implies the existence of ε∗=ε∗(η,δ∗)>0 such that\n0<η<Diamεand\nη=−δ∗+G(r∗)<distε(tε+r∗·ωε)(E.7)\nfor all 0<ε<ε ∗. Thus (E.7) with the help of the definition of η-mixing time yields\nτε,x\nmix(η)/greaterorequalslanttε+r∗·ωε=tε+G−1(η+δ∗)·ωε\n=tε+G−1(η)·ωε+/parenleftbig\nG−1(η+δ∗)−G−1(η)/parenrightbig\n·ωε\nfor all 0<ε<ε∗, and therefore\n(E.8) liminf\nε→0+τε,x\nmix(η)−(tε+G−1(η)·ωε)\nωε/greaterorequalslantG−1(η+δ∗)−G−1(η)\nfor anyη>0 andδ∗such thatη+δ∗<Diam. SinceG−1is a continuous function and the\nright-hand of (E.8) does not depend on δ∗, tendingδ∗→0+we deduce (E.6). By (E.3)\nand (E.6) we obtain the statement of Item (b).\nIn the sequel, we assume that the statement of Item (b) is valid. Le tr∈Rbe fixed.\nWe start with the proof of\n(E.9) limsup\nε→0+distε(LawXε\ntε+r·ωε,νε)/lessorequalslantG(r).\nAssume that δ∗>0 and take η∗:=G(r−δ∗)∈(0,Diam). By hypothesis Gis a\nbijection and then we have that r=G−1(η∗) +δ∗. For the preceding choice of δ∗>0\nandη∗∈(0,Diam), the asymptotic (E.2) with the help of (1.3) yields the existence of\nε∗:=ε∗(r,δ∗)>0 such that\n0<η∗<Diamεand\nτε,distε\nmix(η∗)<tε+(G−1(η∗)+δ∗)·ωε=tε+r·ωε\nfor allε∈(0,ε∗). The definition of η∗-mixing time with the help of (1.6) yields\ndistε(LawXε\ntε+r·ωε,νε)/lessorequalslantη∗=G(r−δ∗) for all ε∈(0,ε∗).\nTherefore,\n(E.10) limsup\nε→0+distε(LawXε\ntε+r·ωε,νε)/lessorequalslantG(r−δ∗).\nSince theleft right-handof (E.10) doesnot depend on δ∗andthefunction Giscontinuous,\ntendingδ∗→0+we deduce (E.9).\nWe continue with the proof of\n(E.11) liminf\nε→0+distε(LawXε\ntε+r·ωε,νε)/greaterorequalslantG(r).\nAssume that δ∗>0 and take η∗:=G(r+δ∗)∈(0,Diam), i.e.,r=G−1(η∗)−δ∗. For the\npreceding choice of δ∗>0 andη∗∈(0,Diam), the asymptotic (E.2) with the help of (1.3)\nyields the existence of ε∗:=ε∗(r,δ∗)>0 such that\n0<η∗<Diamεand\n0<tε+r·ωε=tε+(G−1(η∗)−δ∗)·ωε<τε,distε\nmix(η∗)\nfor allε∈(0,ε∗). The definition of η∗-mixing time with the help of (1.6) yields\ndistε(LawXε\ntε+r·ωε,νε)/greaterorequalslantη∗=G(r+δ∗) for all ε∈(0,ε∗).32 GERARDO BARRERA AND LILIANA ESQUIVEL\nTherefore,\nliminf\nε→0+distε(LawXε\ntε+r·ωε,νε)/greaterorequalslantG(r+δ∗).\nSince the left right-handof (E.10) does not depend on δ∗and the function Gis continuous,\ntendingδ∗→0+we deduce (E.11).\nBy (E.9) and (E.11) we obtain the statement of Item (a). /square\nAcknowledgments\nG. Barrera would like to express his gratitude to the Department of Mathematics and\nStatistics at University of Helsinki, Helsinki, Finland, for all the facilitie s used along the\nrealization of this work. G. Barrera would also like to thank IMPA for s upport and\nhospitality during the 2023 Post-Doctoral Summer Program, wher e partial work on this\npaper was undertaken. L. Esquivel would like to express her gratit ude to the Department\nof Mathematics at Universidad del Valle, Cali, Colombia, for all the fac ilities used along\nthe realization of this work.\nEthical approval. Not applicable.\nCompeting interests. The authors declare that they have no conflict of interest.\nAuthors’ contributions. Both authors have contributed equally to the paper.\nAvailability of data and materials. Data sharing not applicable to this article, as no\ndata-sets were generated or analyzed during the current study .\nFunding The research of G. Barrera has been supported by the Academy o f Finland,\nvia an Academy project (project No. 339228) and the Finnish Cent re of Excellence in\nRandomness and Structures (project No. 346306).\nReferences\n[1]Aldous, D. and Diaconis, P. (1986). Shuffling cards and stopping times. Am. Math. Monthly\n93(5) 333–348.\n[2]Alfonsi, A. 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Exac-\ntum in Kumpula Campus. PL 68, Pietari Kalmin katu 5. Postal Co de: 00560. Helsinki,\nFinland.\nEmail address :gerardo.barreravargas@helsinki.fi\nUniversity of Valle, Department of mathematics. Cali, Colo mbia. Postal Code:\n760034.\nEmail address :liliana.esquivel@correounivalle.edu.co"    },    {        "title": "2402.15459v2.Magnetoelectric_effect_in_superconductors_with_d_wave_magnetization.pdf",        "content": "Magnetoelectric effect in superconductors with d-wave magnetization\nAlexander A. Zyuzin\nDepartment of Applied Physics, Aalto University, P. O. Box 15100, FI-00076 AALTO, Finland\nWe report on a study of the interplay between the supercurrent and spin-polarization in a two-\ndimensional superconducting system in the presence of a d-wave symmetric antiferromagnetic ex-\nchange interaction (altermagnetism). It is demonstrated that the supercurrent exhibits a transverse\ncontribution in the presence of both constant and momentum-dependent exchange interactions. We\nalso discuss the analog of the Edelstein effect for such material, showing that the induced spin\npolarization is quadratic in the supercurrent and d-wave symmetric.\nI. INTRODUCTION\nThe magnetoelectrics in non-centrosymmetric super-\nconductors [1] has recently gathered substantial attention\nparticularly in its experimental application in the non-\nreciprocal superconducting response [2], as discussed, for\nexample, in a recent review [3]. Specifically, the Edelstein\nmagnetoelectric effect is the generation of spin polariza-\ntion induced by an applied supercurrent, while its inverse\nscenario is the diode-like effect, i.e. the critical current\nis different for two opposite directions, generated in the\npresence of an external magnetic field. One of the un-\nderlying reasons for these phenomena is the violation of\nspatial inversion symmetry caused by the spin-orbit in-\nteraction or the inhomogeneous magnetic exchange field\nacting on a momentum-dependent spin splitting of the\nenergy bands [4–6], all giving rise to the coupling be-\ntween electron spin polarization and charge current [7].\nIn this paper, we consider a case of centrosymmetric\nmetals hosting a collinear antiferromagnetic (AFM) or-\nder parameter with a d-wave symmetry, [8–11]. Such\nAFM order induces a specific d-wave momentum-\ndependent spin-splitting of the Fermi surface of con-\nducting fermions [4–6]. The extended symmetry clas-\nsification of anisotropic magnetic order has been recently\nreviewed in [12–14]. The representative materials dis-\nplaying this feature include, for example, collinear-type\nAFMs: metallic RuO 2, Mn 5Si3, VNb 3S6, semiconduct-\ning MnTe, and many more [12–17]. Furthermore, strain-\nstabilized superconductivity has been recently observed\nin thin films RuO 2with Tc≈1.8Kdepending on the film\nthickness, [18–20]. Motivated by the recent experimen-\ntal progress, the theoretical investigation of the d-wave\nAFM exchange coupling on superconductivity became an\nintensive area of research, including the study of Andreev\nreflection and Josephson current [21–25], inhomogeneous\nstates in a d-wave superconductor with d-wave AFM [26],\nor even exotic Majorana modes [27]. For a recent high-\nlight article, see [28].\nIn this context, the question of superconductivity and\nmagnetoelectrics comes up naturally. Clearly, in the cen-\ntrosymmetric superconductors with d-wave magnetiza-\ntion, in contrast to the Edelstein effect in polar super-\nconductors, the induced spin polarization of carriers is\nproportional to the even power of supercurrent and ex-\nhibits a d-wave symmetry. Application of a constant ex-change field leads to a transverse supercurrent response.\nII. MODEL\nThe Bogoliubov - deGennes Hamiltonian, describing\nclean two-dimensional superconducting material subject\nto an isotropic magnetic exchange (or the Zeeman effect\nof a magnetic field) and d-wave AFM exchange interac-\ntions, is given by\nHBdG(k) =\u0012\nH0(k)−µ ∆\n∆∗−σyH∗\n0(−k)σy+µ\u0013\n,(1a)\nH0(k) =k2\n2m+βkxkyσz+γ(k2\nx−k2\ny)σz+h·σ,(1b)\nwhere k= (kx, ky) and mare the momentum and mass\nof electrons, µis the chemical potential, ∆ is the s-wave\nsuperconducting gap, and σ= (σx, σy, σz) is a vector of\nPauli matrices in spin space. We will use ℏ=kB= 1\nunits henceforth. The second and third terms in (1b)\ndescribe the d-wave exchange interaction (dubbed ”al-\ntermagnetism” in some literature [29]) characterized by\na parameters βandγ, while the forth term describes\nthe Zeeman (or constant exchange) field spin-splitting\nh= (hx, hy, hz), which can have both in-plane and out-\nof-plane components. We assume that hdoes not affect\nthe AFM order.\nAlthough we consider a specific dxy−dx2−y2wave sym-\nmetry field for metals with two-dimensional Fermi sur-\nface, the results can be readily extended to situations\nincluding more general combinations of the ( d, g, i )-wave\nsymmetry terms [13] and to higher dimension as well.\nHere will consider a regime of weak exchange, m|β|<\n1,m|γ|<1 and|h|< µ. The system tuned to a quadratic\nband-touching point van-Hove singularity m|β|>1,\nm|γ|>1 will be studied separately. It is useful to explore\nthe basic properties of superconducting systems with d-\nwave exchange interaction before going into the studies\nof magnetoelectric effect.\nInitially, one must distinguish between various mani-\nfestations of superconducting correlations in the system.\nNamely, one can realize a scenario in which the supercon-\nducting gap in the material is induced by the proximity\neffect from other superconductor. In this case, ∆ is a\nproximity induced minigap which might be treated asarXiv:2402.15459v2  [cond-mat.supr-con]  15 Mar 20242\nky\nkxky\nkx\nFIG. 1. Dispersion relation E±,s(k) = 0 for the two-\ndimensional d-wave exchange field with proximity induced su-\nperconducting minigap ∆ at h= (0,0, hz) and γ= 0. (a).\nAthz= 0, β̸= 0 and ∆ = 0. (b). At hz= 0, β̸= 0, turning\non ∆≪m|β|µ, gaps out certain regions of the Fermi surface\nuntil at ∆ = 2 m|β|µit becomes all gapped. (c) and (d). Now\natβ̸= 0 and ∆ ≪m|β|µ, the increase of the hz-term closes\nthe gap and tend to spin-split the Fermi surface isotropically.\na model parameter depending on the delicate interplay\nbetween the material properties and the contact trans-\nparency. On the other hand, one can consider a situation\nin which the superconducting gap is intrinsic. Let us first\nrevisit the former case.\nDiagonalizing the Hamiltonian (8), one obtains\nthe following quasiparticle dispersion: E±,s(k) =\n±p\nξ2\nk+|∆|2+s{h2\nx+h2\ny+[hz+βkxky+γ(k2\nx−k2\ny)]2}1/2,\nwhere s=±1 denotes the exchange field induced band\nsplitting and ξk=k2/(2m)−µ. Keeping ∆ as a param-\neter, one observes that the application of the exchange\nfield results in the emergence of nodes within the gap\nfunction. Setting h= 0, γ= 0, one finds the exis-\ntence of nodes provided the following inequality holds\n|sin(2ϕ)|⩾1\nm|β||∆|√\n|∆|2+µ2, where ϕis the momentum\nangular coordinate. With the increase of parameter |β|,\nthe gap closes at four points on the kx−kyplane with co-\nordinates determined by |sin(2ϕ)|=1\nm|β||∆|√\n|∆|2+µ2and\nk={2mµ(1 +|∆|2/µ2)}1/2. With the further increase\nof|β|, each nodal points transform into two Fermi arcs.\nThe application of hzeliminates one pair of Fermi arcs\nstretching the other pair. The effect of hx-term is to close\nthe gap isotropically, while the effect of γis to rotate the\nplot around the centre of origin. The evolution of the gap\nnodes as a function of the βandhzis illustrated in Fig.\n(1). All in all, the β, γandhterms break time-reversal\nsymmetry, therefore, contributing to the vanishing of the\n0.0 0.2 0.4 0.6 0.8 1.00.00.51.01.52.0\nT/Tc0hz\nTc0\n0.0 0.2 0.4 0.6 0.8 1.00.00.51.01.52.0\nT/Tc0mβ μ\nTc0FIG. 2. The phase transition curves between the BCS and\nthe normal metal states are shown in two scenarios ( γ= 0).\n(a) Fixed parameter mβµ/T c0= (0,0.7), where Tc0is the su-\nperconducting transition temperature at β= 0 and hz= 0.\nThe d-wave term tends to increase the transition temperature\nat large hz. (b) Fixed parameter hz/Tc0= (0,0.9,1.1,1.3).\nThe increase of the d-wave term gives rise to a superconduct-\ning pocket at large hz.\ngap function ∆.\nTo get more insight into superconducting correlations,\nit is helpful to examine the spatial dependence of the\nCooper pair wave function. It can be analyzed by ex-\nploring the kernel in particle - particle ladder, Π( r) =\nTP\nntr2G(ωn,r)[G(−ωn,r)|h,β,γ→−h,−β,−γ], where tr 2\nis the trace over the spin Pauli matrices. At γ= 0 and\nhx=hy= 0, without loosing the generality, the elec-\ntron Green function in spatial coordinate representation,\nG(ωn,r) =Rd2k\n(2π)2[iωn+µ−H0(k)]−1e−ik·r, is given by\nG(ωn,r) =−m\n2πX\ns=±11 +sσzp\n1−β2m2\n×K0 \n−isgn(ωn)kFrfs(ϕr)s\n1 +iωn−shz\nµ!\n,(2)\nwhere K0(z) is the modified Bessel function, kF=√2mµ\nis the Fermi momentum, ωn= (2n+ 1)πTwith n∈Zis\nthe Matsubara frequency at temperature T, and fs(ϕr) =q\n1−sβmsin(2ϕr)\n1−β2m2 is introduced for brevity with ϕrbeing\nthe spatial coordinate azimuth angle. Using asymptotic\nexpansion K0(z)≈pπ\n2ze−zat|z| ≫1, one obtains\nΠ(r) =Tm2\n2π2rkFcsch{πT\nvFr[f+(ϕr) +f−(ϕr)]}\n(1−β2m2)p\nf+(ϕr)f−(ϕr)(3)\n×cos\u001ahzr\nvF[f+(ϕr) +f−(ϕr)] +kFr[f−(ϕr)−f+(ϕr)]\u001b\n,\nwhere vF=kF/mis the Fermi velocity, at m|β|<1,\nexpression on the second line might be estimated as\n∝cosn\n2r\nvF[hz+βmµ sin(2ϕr)]o\n. At finite exchange in-\nteractions, the paring correlations decay and oscillate in\nspace, as expected for systems with magnetic Cooper\npair-breaking source. Thus, it can emerge via 0- πtran-\nsitions in the Josephson junctions through the d-wave\nAFM, as demonstrated in Ref. [23–25].3\nNext, it is instructive to comment on the supercon-\nducting transition temperature of the intrinsic supercon-\nductivity. At ∆ →0, assuming superconductivity in\nthe spatially homogeneous regime, the BCS transition\ntemperature can be found from the solution of the self-\nconsistently equation\nlnT\nTc0= Ψ\u00121\n2\u0013\n−Re\u001c\nΨ0\u00121\n2−ihz+mβµ sin(ϕ)\n2πT\u0013\u001d\n,\n(4)\nwhere, Ψ 0(x) is the polygamma function, ⟨f(ϕ)⟩ ≡R2π\n0dϕ\n2πf(ϕ) denotes integration over the directions of mo-\nmentum and Tc0is the transition temperature at hz= 0\nandβ= 0.\nIt is evident that both hzandβterms suppress the\ntransition temperature, in accord with the spatial oscilla-\ntory dependence of the correlator (3). However, they may\npartially compensate each other giving rise to a residual\nsuperconducting state at larger hzat small Tregime, as\nillustrated in Fig. (2).\nOne might argue that the strength of d-wave AFM or-\nder can depend on the position of the chemical potential,\nβ(µ). In this situation, the superconducting transition\ntemperature can exhibit unusual behaviour as a function\nof electron density in thin films.\nFurthermore, it can be shown that the momentum de-\npendence of the β-term suppresses the realization of in-\nhomogeneous bulk Larkin - Ovchinnikov - Fulde - Ferrell\nstate compared in contrast to the effect of hz-term con-\ntribution. However, in finite-size systems with lengths on\nthe order of several vF/|βmµ| ∼1/|βmk F|, one might ex-\npect the stabilization of such inhomogeneous state. Ad-\nditionally, the inhomogeneous state might be stabilized\nin a d-wave superconductor when brought in contact with\nthe d-wave AFM under certain symmetry-matching con-\nditions, [26]. This prediction can be tested, for example,\nvia the anomalous Little-Parks oscillations [30, 31]. Af-\nter this general introduction, let us now investigate the\nmagnetoelectric effect in the system.\nIII. MAGNETOELECTRIC EFFECT\nConsider a simple model to demonstrate the magne-\ntoelectric effect in the superconductor with d-wave ex-\nchange interaction, by setting h= 0 and focusing on the\nsmall gap regime |∆| ≪T. It allows addressing the spin\npolarization in the lowest order in ∆.\nBy construction of the d-wave exchange field in Hamil-\ntonian (1b), the superconducting current can only in-\nduce an out-of-plane component of the spin polariza-\ntion density, which in momentum representation Sz(q) =\n-1 -0.8 -0.6 -0.4 -0.2 0.2 0.4 0.6 0.8 1x\n-0.3-0.2-0.10.10.20.3A(x,0)FIG. 3. The function A(x, y) aty= 0 in (9) of the main text.\nAt a given x, the finite value of ysuppresses the amplitude of\nfunction A(x, y).\nR\ndrSz(r)e−iq·rcan be expressed as [1]\nSz(q) =−TX\nnZdpdk\n(2π)4tr2\u001a\nσzG\u0010\nωn,p+k+q\n2\u0011\n×∆\u0010\nk+q\n2\u0011\nσyG(−ωn,p)σy∆∗\u0010\nk−q\n2\u0011\n×G\u0010\nωn,p+k−q\n2\u0011\u001b\n, (5)\nwhere the electron Green function in momentum repre-\nsentation is given by\nG(ωn,p) =1\n2X\ns=±11 +sσz\niωn−ξp−s[βpxpy+γ(p2x−p2y)].(6)\nTo proceed, we consider the long-wave limit not-\ningp≈pF≫k, qand expand the Green functions\nG(ωn,p+k±q/2) in (5) over the powers of momentum\nk±q/2. Performing integration over pand summing\nover n, we obtain the leading contribution to the spin\npolarization in the lowest non-vanishing order ( m|β|<\n1, m|γ|<1) in the following form\nSz(q) =−2µ\n(2πT)3Zdk\n(2π)2∆\u0010\nk+q\n2\u0011\n∆∗\u0010\nk−q\n2\u0011\n×\u001ah\n3kxky+qxqy\n4i\nA\u0012βmµ\n2πT,γmµ\n2πT\u0013\n+1\n2\"\n3(k2\nx−k2\ny) +q2\nx−q2\ny\n4#\nA\u0012γmµ\n2πT,βmµ\n2πT\u0013\u001b\n,(7)\nwhere\nA(x, y) =−Im\n48π\u001c\nΨ3\u00141\n2+i(xsinϕ+ycosϕ)\u0015\nsinϕ\u001d\n(8)\nis a function of xandyas plotted in Fig. (3). It is noted\nthat the spin polarization vanishes at large |β|mµ/T . It\ncan be attributed to the suppression of the supercurrent\namplitude induced by the time-reversal-breaking pertur-\nbation. This stands in contrast to the Edelstein effect4\nin superconductors with Rashba spin-orbit interaction,\nwherein the spin polarization increases with the increase\nof spin-orbit interaction strength [1].\nWe also note that intrinsic superconductivity requires\n|β|mµ/T ≪1, while the region of parameters |β|mµ/T ≳\n1 might be achieved in systems with proximity induced\nsuperconducting correlations.\nTransforming the remaining integrals over momentum\nnoting thatRdk\n(2π)2[3kxky+qxqy\n4]∆\u0000\nk+q\n2\u0001\n∆∗\u0000\nk−q\n2\u0001\n=\n1\n2R\ndre−iq·r[(∂x∆)∂y∆∗−2∆∂2\nxy∆∗+h.c.], we obtain the\nspin polarization density in spatial coordinate represen-\ntation in the following form\nSz(r) =−µ\n(2πT)3A\u0012βmµ\n2πT,γmµ\n2πT\u0013\n(9)\n×[(∂x∆)∂y∆∗−2∆∂2\nxy∆∗+ h.c.]\n−µ\n(2πT)3A\u0012γmµ\n2πT,βmµ\n2πT\u0013\n×[|∂x∆|2− |∂y∆|2−2∆(∂2\nx∆∗−∂2\ny∆∗) + h.c.].\nFinally, expanding in the lowest order in βandγin\n(9), one obtains Sz(r) =−31\n32ζ(5)µ2\nπ4T4ν{β[(∂x∆)∂y∆∗−\n2∆∂2\nxy∆∗+ h.c.] + γ[(∂x∆)∂x∆∗−(∂y∆)∂y∆∗−\n2∆(∂2\nx∆∗−∂2\ny∆∗) + h .c.]}, where ν=m/2πis the elec-\ntron density of states per spin and 31 ζ(5)/(32π4)≈0.01.\nIt is observed that spin polarization is quadratic in the\ngradients of the order parameter and d-wave symmet-\nric, namely comprises terms proportional to the product\nof supercurrents applied in transverse directions and to\nthe current acceleration. Specifically, in the case when\n∆(r) =|∆|eiϕ(r), we find\nSz(r)∝ −β\u0002\n(∂xϕ)∂yϕ+ 2∂2\nxyϕ\u0003\n(10)\n−γ\u0002\n(∂xϕ)2−(∂yϕ)2+ 2∂2\nxϕ−2∂2\nyϕ\u0003\n.\nTo further explore magnetoelectrics, one might recall\nthe inverse Edelstein effect in non-centrosymmetric sys-\ntems: a supercurrent diode-like response induced by the\nconstant magnetization in combination with the spin-\norbit coupling.\nIn our system, however, the hz-term gives rise to\nthe transverse supercurrent component. To demonstrate\nthis, it is convenient to examine the Ginzburg-Landau\nfunctional density for the order parameter. Keeping\nh= (0,0, hz), in the lowest order in powers of |β|m < 1,\n|γ|m < 1 and hz< T, we obtain a quadratic part in the\nform\nFGL=a|∆|2+ (b−b1)|∂x∆|2+ (b+b1)|∂y∆|2\n−b2[(∂x∆)∂y∆∗+ h.c.], (11)\nwhere a=ν(T−Tc)/Tcwith Tcdetermined by Eq.\n(4),b=7ζ(3)\n32π2v2\nF\nT2νandb1,2=93ζ(5)\n32π4v2\nFµhz\nT4νm(γ, β) with\n|b1,2|/b < 1. Here we dropped unnecessary isotropic\ncorrections to the b-term, which are ∼O(h2\nz/T2) and\n∼O((mβµ/T )2).Therefore, the supercurrent exhibits a transverse con-\ntribution in the presence of an exchange field hz:\nJ=−2ie\u001a\n[ˆey(b+b1)−ˆexb2](∆∗∂y∆−∆∂y∆∗)\n+ [ˆex(b−b1)−ˆeyb2](∆∗∂x∆−∆∂x∆∗)\u001b\n.(12)\nwhere e <0 is the electron charge. The proposed effect\nis the superconducting analog of the anisotropic linear\nmagnetoconductivity in metallic collinear AFM investi-\ngated recently in [32].\nIV. DISCUSSION AND CONCLUSIONS\nLet us briefly comment on the experimental observabil-\nity of the proposed effect. To this end, we estimate the\nenergy scale associated with the superconducting phase\ntransition in the presence of the d-wave field. Taking\nµ∼1eV and Tc∼1K, one estimates µ/Tc∼104, so\nthat for m|β|µ/Tc∼1, one has to require m|β| ∼10−4.\nThe typical value of AFM parameter in normal metal is\nm|β| ∼0.1 [14], however it is not known for the recently\nobserved superconducting RuO 2films. Thus, exploring\nthe magnetoelectric effect might require utilizing Joseph-\nson junctions through the d-wave AFM [25].\nIt is also instructive to compare our result for the spin\npolarization density with the one in Ref. [1]. In the limit\nαkF/πT≪1 (where αis the parameter of Rashba spin-\norbit interaction), taking β∼γ, we estimate the ratio of\nspin densities as |Sz|/|SSOI| ∝(vF\nα)3|β|m\nLkF, where Lis the\ntypical length scale of the superconducting phase varia-\ntion. For example taking L∼1/|βmk F|to be the short-\nest length, [25], we find |Sz|/|SSOI| ∝(βm)2(vF/α)3.\nTo summarize, we investigated the magnetoelectric re-\nsponse in a centrosymmetric superconducting system in\nthe presence of d-wave exchange interaction. We dis-\ncussed an analog of the Edelstein effect, demonstrating\nthat the spin-charge coupling leads to the spin polariza-\ntion quadratic in the supercurrent and d-wave symmet-\nric. Moreover, we showed that the supercurrent exhibits\na transverse contribution in the presence of an isotropic\nmagnetic exchange field.\nOur results may be used as a starting point for further\ninvestigation of spin-charge coupling in superconduct-\ning structures with ( d, g, i )-wave magnetizations. Here\nwe have focused on the ballistic two-dimensional case.\nHowever, experimentally realistic superconducting sys-\ntems are three-dimensional and inevitably involve disor-\nder scattering processes [19]. The investigation of trans-\nport phenomena in such structures is of interest [33].\nV. ACKNOWLEDGEMENTS\nWe thank Jakub Tworzyd lo and Vladimir Zyuzin for\nhelpful discussions. This work was performed as part5\nof the Academy of Finland Centre of Excellence pro-\ngram (project 352925). We acknowledge support from\nthe QuantERA II Programme that has received fund-\ning from the European Union’s Horizon 2020 researchand innovation programme under Grant Agreement No\n101017733. We thank the Pirinem School of Theoretical\nPhysics for warm hospitality.\n[1] V. M. Edelstein, “Magnetoelectric effect in polar super-\nconductors,” Phys. Rev. Lett. 75, 2004 (1995).\n[2] V. M. Edelstein, “The Ginzburg - Landau equation for\nsuperconductors of polar symmetry,” Journal of Physics:\nCondensed Matter 8, 339 (1996).\n[3] M. Nadeem, M. S. Fuhrer, and X. Wang, “ The super-\nconducting diode effect,” Nat. Rev. Phys. 5, 558 (2023).\n[4] S. I. Pekar and E. I. Rashba, “Combined resonance in\ncrystals in inhomogeneous magnetic fields,” Sov. Phys.\nJETP 20, 1295 (1965).\n[5] N. R. Ogg, “Conduction-band g factor anisotropy in in-\ndium antimonide,” Proc. Phys. Soc. 89, 431 (1966).\n[6] E. L. Ivchenko and A. A. Kiselev, “Electron g-\nfactor in quantum wells and superlattices,” Fiz. 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Ralph, “Tilted spin current generated\nby the collinear antiferromagnet ruthenium dioxide,” Na-\nture Electronics 5, 267 (2022).\n[17] R. D. Gonzalez Betancourt, J. Zub´ aˇ c, R. Gonzalez-\nHernandez, K. Geishendorf, Z. ˇSob´ aˇ n, G. Springholz,\nK. Olejn´ ık, L. ˇSmejkal, J. Sinova, T. Jungwirth, S. T. B.Goennenwein, A. Thomas, H. Reichlov´ a, J. ˇZelezn´ y, and\nD. Kriegner, “Spontaneous anomalous Hall effect arising\nfrom an unconventional compensated magnetic phase in\na semiconductor,” Phys. Rev. Lett. 130, 036702 (2023).\n[18] M. Uchida, T. Nomoto, M. Musashi, R. Arita, and\nM. Kawasaki, “Superconductivity in uniquely strained\nRuO 2Films,” Phys. Rev. Lett. 125, 147001 (2020).\n[19] J. P. Ruf, H. Paik, N. J. Schreiber, H. P. Nair, L. Miao,\nJ. K. Kawasaki, J. N. Nelson, B. D. Faeth, Y. Lee, B. H.\nGoodge, B. Pamuk, C. J. Fennie, L. F. Kourkoutis, D. G.\nSchlom, and K. M. Shen, “Strain-stabilized supercon-\nductivity,” Nature Communications 12, 59 (2021).\n[20] C. A. 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Black-Schaffer, “Zero-field\nfinite-momentum and field-induced superconductivity in\naltermagnets,” arxiv:2309.14427 .\n[27] S. A. A. Ghorashi, T. L. Hughes, and J. Cano, “Alter-\nmagnetic routes to Majorana modes in zero net magne-\ntization,” arxiv:2306.09413 .\n[28] I. Mazin, “Notes on altermagnetism and superconductiv-\nity,” arxiv:2203.05000 .\n[29] We will refrain from using altermagnetism and adopt con-\nventional notations of collinear antiferromagnets.\n[30] A. A. Zyuzin and A. Yu. Zyuzin, “Aharonov-Bohm effect\nin the superconducting LOFF state,” JETP Letters 88,\n136 (2008).\n[31] A. A. Zyuzin and A. Yu. Zyuzin, “Anomalous transi-\ntion temperature oscillations in the Larkin-Ovchinnikov-\nFulde-Ferrell state,” Phys. Rev. B 79, 174514 (2009).\n[32] D. L. Vorobiev and V. A. Zyuzin, “d-Wave Hall effect\nand linear magnetoconductivity in metallic collinear an-\ntiferromagnets,” arxiv:2311.04890 .\n[33] G. A. Bobkov, I. V. Bobkova, and A. M. Bobkov, “Prox-\nimity effect in superconductor/antiferromagnet hybrids:\nN´ eel triplets and impurity suppression of superconduc-\ntivity,” Phys. Rev. B 108, 054510 (2023)."    },    {        "title": "2402.15478v1.Transformers_are_Expressive__But_Are_They_Expressive_Enough_for_Regression_.pdf",        "content": "TRANSFORMERS ARE EXPRESSIVE , BUTARETHEY EXPRESSIVE ENOUGH\nFOR REGRESSION ?\nSwaroop Nath1Harshad Khadilkar1 2Pushpak Bhattacharyya1\nAbstract\nTransformers have become pivotal in Natural Lan-\nguage Processing, demonstrating remarkable suc-\ncess in applications like Machine Translation and\nSummarization. Given their widespread adop-\ntion, several works have attempted to analyze the\nexpressivity of Transformers. Expressivity of a\nneural network is the class of functions it can ap-\nproximate. A neural network is fully expressive\nif it can act as a universal function approximator.\nWe attempt to analyze the same for Transformers.\nContrary to existing claims, our findings reveal\nthat Transformers struggle to reliably approximate\ncontinuous functions, relying on piecewise con-\nstant approximations with sizable intervals. The\ncentral question emerges as: “ Are Transformers\ntruly Universal Function Approximators ?” To ad-\ndress this, we conduct a thorough investigation,\nproviding theoretical insights and supporting ev-\nidence through experiments. Our contributions\ninclude a theoretical analysis pinpointing the root\nof Transformers’ limitation in function approxi-\nmation and extensive experiments to verify the\nlimitation. By shedding light on these challenges,\nwe advocate a refined understanding of Trans-\nformers’ capabilities.\n1. Introduction\nTransformers (Vaswani et al., 2017) have become the de-\nfacto backbone models in several NLP applications: Ma-\nchine Translation, Summarization, Question Answering,\netc. By modeling recurrence relations through self-attention\nonly, they have enabled large-scale pre-training (Radford\net al., 2019; Brown et al., 2020; Zhang et al., 2022; Touvron\net al., 2023). This has led to drastic advancements in Lan-\nguage Technologies over competing architectures, such as\n1{swaroopnath ,harshadk ,pb}@cse.iitb.ac.in\nIndian Institute of Technology, Bombay2Franklin\nTempleton. Correspondence to: Swaroop Nath\n<swaroopnath@cse.iitb.ac.in >.\nPREPRINT , W ORK UNDER PROGRESSLSTM (Hochreiter & Schmidhuber, 1997) and GRU (Cho\net al., 2014).\nIn light of the success of Transformers, several works (De-\nhghani et al., 2018; Yun et al., 2020a; Perez et al., 2021;\nMerrill & Sabharwal, 2023) have studied the expressivity1of\nTransformers. Specifically, two lenses have been employed\nto study them: ( a) Lens of Universal Function Approxima-\ntion, and ( b) Lens of Formal Languages and Complexity\nClasses. In the latter theme, several works have contributed\nto prove that expressivity of Transformers is upper bounded\nby the TC 0complexity class. In the former theme, Yun et al.\n(2020a;b); Zaheer et al. (2020) indicated that Transformers\n(and their variants) are likely to be Universal Function Ap-\nproximators. However, our experimental results (Section\n5) contradicted this: they are unable to reliably approxi-\nmate continuous functions. We witnessed that Transformers\ncould approximate the function only after a piecewise con-\nstant approximation, with large sized pieces . Perplexed by\nsuch experimental results, we ask a simple question: Are\nTransformers truly Universal Function Approximators?\nWhile we acknowledge the theoretical results by Yun et al.\n(2020a), we find that there are further implications and anal-\nyses that can provide tighter bounds.\nIn this work, we conduct both theoretical and experimental\nanalysis on the approximation capabilities of Transformers.\nWe find that – Transformers are bad at approximating\ncontinuous functions . We provide a relevant theoretical\nanalysis in Section 4, to see where the difficulty stems from,\nand the magnitude of this difficulty. To further verify our\nclaim, we conduct experiments2to extensively test the func-\ntion approximation capabilities of the Transformer. Specifi-\ncally, we conduct experiments for two axes: ( a) verifying\nfunction approximation capabilities for continuous func-\ntions, and ( b) verifying function approximation capabilities\nfor piecewise constant functions. The results along these\ntwo axes provide a boundary of the approximation capabili-\nties of the Transformers. We provide details on the datasets,\nevaluation measures, and report all our results in Section 5.\nOur contributions are:\n1Expressivity of a neural network characterizes its capability\nin complexity in data.\n2We would be releasing our code in the camera-ready paper.\n1arXiv:2402.15478v1  [cs.LG]  23 Feb 2024TRANSFORMERS ARE EXPRESSIVE , BUTARETHEY EXPRESSIVE ENOUGH FOR REGRESSION ?\n1.Theoretical analysis on expressivity of Transformers,\nwhich leads to the finding that Transformers are bad at\napproximating continuous functions.\n2.Extensive experimentation evaluating the continuous\nfunction approximation capabilities of Transformers,\nverifying our claim and supporting our theoretical anal-\nysis.\n2. Related Works\nWe categorize the previous works in this line into two cat-\negories: Analyzing the expressivity of Transformers from\nthe lens of ( a) Universal Function Approximation, and ( b)\nFormal Languages and Complexity Classes. We first sum-\nmarize the works in these two lines, and finally provide a\nrecap of works that question the Transformers’ expressivity.\nLens of Universal Function Approximation . Yun et al.\n(2020a;b) provided some of the first works in analyzing\nTransformers as Universal Function Approximators. No-\ntably, Yun et al. (2020a) contributed a step-by-step strategy\nfor analyzing Transformers under such a lens, which has\nbeen utilized in many further works. In our work too, we\nbuild up on the strategy to show the limitations in expressiv-\nity of Transformers. Zaheer et al. (2020) used the strategy to\ndeduce the expressivity of their new sparse-attention based\nTransformer. More recently, Luo et al. (2022) used the\nstrategy to prove that Transformers with Relative Positional\nEmbeddings are notUniversal Function Approximators.\nLens of Formal Languages . Perez et al. (2021); Dehghani\net al. (2018) provided two foundational works in this line,\nwith opposing conclusions. Perez et al. (2021) concluded,\nunder assumptions like arbitrary precision, that Transform-\ners are Turing Complete. While, Dehghani et al. (2018)\nprovided intuition-based arguments to claim that Transform-\ners are not Turing Complete. Bhattamishra et al. (2020)\nprovide a lower limit on the expressivity of Transformers,\nconcluding that Transformers are at least as powerful as Sim-\nplified Stateless Counter Machines (SSCM). On the other\nhand, Merrill et al. (2022); Hao et al. (2022); Merrill & Sab-\nharwal (2023) show that the expressivity of Transformers is\nupper-bounded by the TC 0complexity class. More recently,\nChiang et al. (2023) provide a tighter bound than TC 0for\nthe expressivity of Transformers.\nWe see that many works have probed the expressivity of\nTransformers from the lens of formal languages and com-\nplexity classes. Additionally, Yun et al. (2020a) have at-\ntempted to show that Transformers are Universal Function\nApproximators. However, perplexed by our initial exper-\nimental results on the same, we start an investigation on\nthe Universal Function Approximation capabilities of the\nTransformer. Like previous works, we provide a theoretical\nanalysis to examine the capability. And, unlike previousworks, we extensively test our claim through experiments.\n3. Notations, Definitions and Preliminaries\n3.1. Transformer Architecture\nA Transformer (Vaswani et al., 2017) is a Sequence-to-\nSequence mapper, composed of two main components: En-\ncoder and Decoder. In turn, encoders and decoders are\ncomposed of several stacked blocks. Each block has two\nkey components: Self-Attention andToken-wise Feed For-\nward Network , The decoder has an additional component:\nCross-Attention . Using these blocks, Transformers essen-\ntially perform a mapping from X(∈Rm×d) toY(∈Rn×d),\nwhere mandnare sequence lengths of the input and output,\nanddis the embedding dimension.\nThe Self-Attention component operates according to Equa-\ntion 3.0.1. Specifically, it is a dot-product attention (Luong\net al., 2015), conducted across several heads (the equation\ndepicts hheads). Token-wise Feed Forward Network op-\nerates on each token according to Equation 3.0.3. Cross-\nAttention is a variant of Self-Attention where a prefix of Y\nattends to X, according to Equation 3.0.2.\nSA(X) =X+\nWOhM\ni=1Wi\nV·X·σ\u0002\n(Wi\nK·X)TWi\nQ·X\u0003\n(3.0.1)\nCA(X,Y) =Y:j+\nW′\nOhM\ni=1W′i\nV·X·σ\u0002\n(W′i\nK·X)TW′i\nQ·Y:j\u0003\n(3.0.2)\nWhereLh\ni=1denotes the concatenation operator along the\nembedding dimension axis, W′\nO,WO(∈Rd×hd),W′\nV,\nWV(∈Rd×d),W′\nK,WK(∈Rd×d), and W′\nQ,WQ(∈\nRd×d)denote learnable matrices. Y:jdenotes the j-length\nprefix of the n-length sequence Y.\nFFN(X) =X+W2·relu(W1·X+b1) +b2(3.0.3)\nWhere W1(∈Rr×d),W2(∈Rd×r),b1(∈Rr) and b2\n(∈Rd) are learnable matrices and vectors, respectively.\nWith these notations for the operations in a Transformer, we\ndenote a Transformer as Th,d,rin further sections.\n3.2. Necessary Definitions\nIn this section, we provide definitions necessary for the rest\nof the paper.\n2TRANSFORMERS ARE EXPRESSIVE , BUTARETHEY EXPRESSIVE ENOUGH FOR REGRESSION ?\nx y \n𝛿 \n(a)\nx y \n𝛿 (b)\nFigure 1: Effect of changing the size of resolution factor. In\n(a) we have a smaller resolution factor, leading to a smaller\nerror in approximation than (b).\nDefinition 3.1 (Resolution Factor ).Letfbe a function,\nandfbe a piecewise constant approximation to f. Then,\nResolution Factor ,δ, is the minimum size of the pieces in\nf.\nA large δindicates that the interval for which fis constant\nis large, Figure 1. This leads to a bad approximation if f\nhas a high rate of change (derivative).\nDefinition 3.2 (Adequacy of Approximation ).Letgbe\nan approximation to a function f. We say that gadequately\napproximates fif the following holds, for some ϵ >0and\n1≤p <∞.\ndp(f, g) =\u0012Z\r\rg(x)−f(x)\r\rp\npdx\u00131\np\n≤ϵ\ndp(f, g)defines a distance between fandgfor different\nvalues of x. It is normalized by the norm ( p). Prior works\n(Yun et al., 2020a; Zaheer et al., 2020) have frequently used\nthis definition for adequacy of approximation.\nDefinition 3.3 (Function with Compact Support ).A func-\ntion,f, is said to have a compact support iff it is non-zero\nfor an input from a compact set . A set is compact iff it is\nclosed andbounded .\n3.3.Transformers as Universal Function Approximators\nIn this section, we outline the 3-step process followed by\nYun et al. (2020a). The work attempts to prove that Trans-\nformers, Th,d,r, approximate continuous permutation equiv-\nariant functions with compact support. We denote the set of\nsuch functions by FPE.\nStep 1 .Approximate FPEwith piecewise constant func-\ntions . In this step, Yun et al. (2020a) map the class of\nfunctions FPEtoFPE, such that f(∈FPE) is piecewise\nconstant. We have seen in Definition 3.1 that the goodness\nof this approximation is governed by δ.\nStep 2 .Approximate FPEwith modified Transformers .\nYun et al. (2020a) show that FPEcan be approximated bya Transformer with some simplifying modifications, such as\nreplacing the softmax operator with the hardmax operator.\nStep 3 .Approximate modified Transformers with (orig-\ninal) Transformers . In this step, Yun et al. (2020a) show\nthat the original Transformers can approximate the modified\nTransformers.\nUsing this 3-step process, Yun et al. (2020a;b); Zaheer\net al. (2020) have analyzed the expressivity of Transform-\ners for their respective architectures. Additionally, Yun\net al. (2020a) also note that with a resolution factor δ, in\nStep 1, the number of layers in the Transformer, needed\nto approximate a piecewise constant function, grows as:\nO(n(1/δ)dn).\n4. Expressivity of Transformers in the\nContinuous Space\nYun et al. (2020a) show that Transformers can approximate\ncontinuous functions, with a condition on the number of\nTransformer layers. We briefly describe their arguments\nin Section 3.3, which show that: Transformers ( Th,d,r)\napproximate a piecewise constant function ( f), which is,\nin turn, an approximation of the target function ( f). The\nconstruction of the piecewise constant approximation is\ngoverned by the resolution factor, δ(Definition 3.1). Again,\nwe see that this resolution factor also affects (Section 3.3)\nthe number of layers needed for an adequate approximation\n(Definition 3.2) by the Transformer. Thus, we can deduce\nthat: there is one factor, δ, which governs the adequacy of\nboth approximations: ( a) between fandf, and ( b) between\nTh,d,randf. We are interested in finding how δaffects\nthese approximations: Does a change in δaffect both the\napproximations similarly3, or dissimilarly4?We answer\nthis in steps, by answering the following questions:\n1.What governs the choice of the resolution factor,\nδ?We provide an intuition (Section 4.1) and a mathe-\nmatical expression to answer this question (Theorem\n4.1).\n2.How does a well-chosen resolution factor affect the\nconclusion of Yun et al. (2020a) , about the expressiv-\nity of Transformers?\nTheorem 4.1. Letf(∈F) be an approximation to f ( ∈ F),\nwith goodness of approximation defined by dp(f,f)≤ϵ\n(for some ϵ >0). Then an upper bound on the resolution\nfactor, δ, can be expressed by the equation:\n3By similar effect we mean that adequacy of both approxima-\ntions change in the same direction (worse or better).\n4By dissimilar effect we mean that adequacy of both approxi-\nmations change in the opposite directions.\n3TRANSFORMERS ARE EXPRESSIVE , BUTARETHEY EXPRESSIVE ENOUGH FOR REGRESSION ?\nδ≤ \n2p·(p+ 1)·ϵp\nP\nX0∈X0Pd\ni=1\f\f\f∂f(X)\n∂xi\f\f\nX0\f\f\fp! 1\n(p+d)\n(4.1.1)\nwhere X0is a covering over the compact support ( S) for\nthe function f,pis the norm in dp, and dis the embedding\ndimension.\nFrom Theorem 4.1, we understand that for a decently chang-\ning function, a small resolution factor is necessary, for an\nadequate approximation between fandf. And, from Sec-\ntion 3.3, we understand that Th,d,rneedsO(n(1/δ)dn)lay-\ners to approximate a permutation invariant f. Thus, for a\ndecently changing continuous function ( f), with a small\nδ, large (exponential in sequence length) number of layers\nare required for Th,d,r, to adequately approximate f. We\nprovide a small example quantification below, to put this\nrequirement into perspective.\nLetϵ= 0.1,p= 1,d= 1, and|X0|=K; for illustration,\nletK∼10. Assuming fto be a 1-Lipshitz continuous\nfunction in its compact support, we can have the bound\nd f(x)\ndx≤1. With such a setup, we have a bound on resolution\nfactor as: δ≤0.2. Therefore, the required number of layers\nforTh,d,rgrows as O(n·5n). We can see how this goes\nvery high for even an input of sequence length 10.\nFrom the discussion above, we understand the following:\n1.The choice of δis governed by the derivative of f, as\nhighlighted by Theorem 4.1.\n2.A well-chosen δ, for even a decently changing function\n(say, Lipshitz constant = 1), leads to a requirement of\nlarge number of layers for Th,d,r.\nWhat if a function is already piecewise constant? In such\na case, the resolution factor is no longer constrained by the\nderivative (Equation 4.1.1). Rather it depends on the size\nof the pieces (step-sizes). As an example, let us consider\nthe following piecewise constant function, defined over the\ninterval [0,1):\nf(x) =(\n0.5 0≤x <0.75\n1.0 0.75≤x <1.0\nHere, the step-sizes are 0.75and0.25. According to Defi-\nnition 3.1, we can deduce that the resolution factor, in this\ncase, would be 0.25. A piecewise constant function, with\nsmall step-sizes, would still pose a challenge for Transform-\ners.\nFrom the discussion, we can deduce the following things:\n(a) For a continuous function, a constraint on δis imposed asper Equation 4.1.1, and ( b) For a piecwise constant function,\nstep-sizes constrain δ, specifically δ= min( step sizes ).\n4.1. Proof Sketch for Theorem 4.1\nIn this section, we provide an intuition and a short proof\nto motivate Theorem 4.1. From Figure 1, we can see how\nvarying the resolution factor can change the goodness of ap-\nproximation. We can understand from the figure that having\na smaller step size becomes necessary for functions with a\nlarge rate of change, to have a certain degree of goodness\nin approximation. With this intuition, we start deriving a\nbound on the resolution factor, based on the derivative of\nthe function. We provide proof for the 1-dimensional case,\nusing the ℓ1norm. We include a proof for the general case\nofd-dimensional function, using ℓpnorm in Appendix A.\nx y \nx0 𝛿 \n2 𝛿 \n2 f(x0) \n(a)\nx0 𝜽\n𝜽\n𝛿 \n2 𝛿 \n2 (b)\nFigure 2: (a) Behaviour of f(x)in the δ/2neighborhood\nofx0. In this neighborhood, we can approximate f(x)by\nf(x0), hence f(x) =f(x0). (b) The shaded area in δ/2\nneighborhood of x0forf(x)can be approximated by two\ntriangles.\nProof. We have: dp(f,f)≤ϵ;ϵ >0\n=⇒\u0012Z\nS\r\rf(x)−f(x)\r\rp\npdx\u00131\np\n≤ϵ\nNote that for x̸∈ S,f(x) =f(x) = 0 . With p= 1, the\nexpression becomes:\nZ\nS\f\ff(x)−f(x)\f\fdx≤ϵ\nNote that the expression on the left represents the area\nbounded within the curves y=f(x)andy=f(x).\nConsider the δ/2neighborhood of the point x0in Fig-\nure 2a. A good piecewise constant approximation of f(x)\nin this neighborhood is f(x0), that is f(x) =f(x0)for\u0000\nx0−δ/2\u0001\n≤x <\u0000\nx0+δ/2\u0001\n. With this approximation,\nthe expression on the left represents the area of the shaded\nregion (Figure 2a), for\u0000\nx0−δ/2\u0001\n≤x <\u0000\nx0+δ/2\u0001\n. The\n4TRANSFORMERS ARE EXPRESSIVE , BUTARETHEY EXPRESSIVE ENOUGH FOR REGRESSION ?\nshaded regions can be approximated by two triangles, as\nshown in Figure 2b. Thus, we have:\nZx0+δ\n2\nx0−δ\n2\f\ff(x)−f(x)\f\fdx= 2·1\n2·δ\n2·(δ\n2tan(θ))\n=δ2\n4\f\f\f\f\fd f(x)\ndx\f\f\f\f\nx0\f\f\f\f\f\nWhered f(x)\ndx\f\f\f\f\nx0represents the derivative at x0. Now, if we\nconsidering a covering X0ofS, such that x0∈ X0, we\nhave:\nZ\nS\f\ff(x)−f(x)\f\fdx=X\nx0∈X0Zx0+δ\n2\nx0−δ\n2\f\ff(x)−f(x)\f\fdx\n=δ2\n4X\nx0∈X0\f\f\f\f\fd f(x)\ndx\f\f\f\f\nx0\f\f\f\f\f\nFinally,\nδ2\n4X\nx0∈X0\f\f\f\f\fd f(x)\ndx\f\f\f\f\nx0\f\f\f\f\f≤ϵ\n=⇒δ≤vuuuut4ϵ\nP\nx0∈X0\f\f\f\f\fd f(x)\ndx\f\f\f\f\nx0\f\f\f\f\f\nThis matches with our expression in Equation 4.1.1.\nAs per our intuition, we see that for functions with high rate\nof change, the resolution factor has to be small.\n5. Experiments\nWe use the Transformer Architecture proposed by Vaswani\net al. (2017) for our experiments. We are aware of several\nmodifications to the vanilla architecture (Xiong et al., 2020;\nZaheer et al., 2020; Child et al., 2019; Kitaev et al., 2020;\nBeltagy et al., 2020). Tay et al. (2022) provide a compre-\nhensive survey of all variants of Transformers. We urge\nthe interested reader to refer to that for more details. These\nworks attempt to improve stability in training and reducing\ncomputational costs, and do not focus on improving the ex-\npressivity of Transformers. Hence, it suffices to experiment\non the vanilla Transformer architecture.\nWe perform experiments on sythetic datasets, to ensure that\ndata quality does not affect our conclusions. Specifically,\nwe perform two experiments:\n•EXPT-I(Regression ): In this experiment, we train the\nTransformer to generate vectors to directly approxi-\nmate a continuous function.•EXPT-II(Quantized Classification ): In this experi-\nment, we quantize the outputs to certain classes, and\ntrain the Transformer to predict the class.\nEXPT-Iaims to show how Transformers fare in approximat-\ning continuous functions. In EXPT-II, the output space is\nquantized to piecewise constants, thereby essentially testing\nthe Transformer’s ability to approximate piecewise constant\nfunctions. From Section 4, we understand that Transformers\nwould suffer for continuous functions and small step-size\npiecewise constant functions. However, for piecewise con-\nstant functions with high step-sizes ( ≥1), Transformers can\nperform well. Keeping this in mind, in EXPT-II, we train the\nTransformer to predict into kclasses, where kis reasonably\nsmall. Transformer failing in EXPT-Iand succeeding in\nEXPT-II would validate our hypothesis.\nNote that, in all of our experiments, we respect the assump-\ntion of compact support (Definition 3.3) for the functions.\n5.1. Evaluation Measures\nComparing models for EXPT-IandEXPT-IIis non-trivial,\nas one is a regression task and the other is a classification\ntask. In order to compare them, we propose a unified metric:\nfailure-rate . We define this metric as follows:\nfailure-rate : It is the fraction of times the ground\ntruth output is not the nearest one to the generated output.\nForEXPT-II,failure-rate is essentially inaccuracy\n(1−accuracy). For EXPT-I, it is computed as the fraction of\ntimes the ground truth vector is not the nearest-neighbor of\nthe generated vector. We realize that this metric is very strin-\ngent for EXPT-I, as there can be a lot of points in a close-by\nregion, and also additionally propose failure-rate@ k.\nForEXPT-II,failure-rate@ kdefines the fraction of\ntimes the correct class is not in the top- kprobable outputs.\nForEXPT-I, it is computed as the fraction of times the\nground truth vector is not in the k-nearest-neighbor set of\nthe generated vector.\nThe design of the metric is guided by our initial criteria of\njudging an approximation (Definition 3.2). Mathematically,\nfailure-rate is linked to the Adequacy of Approxima-\ntion as follows:\nfailure-rate =P\nx∼DF(Th,d,r, x)\n|D|\nwhere,\nF(Th,d,r, x) = 1\u0000\n∃y∈ D:d1(Th,d,r(x), f(y))\n< d1(Th,d,r(x), f(x))\u0001\nwhere, 1(z)if1if and only if zevaluates to true, andDis\nthe test set.\n5TRANSFORMERS ARE EXPRESSIVE , BUTARETHEY EXPRESSIVE ENOUGH FOR REGRESSION ?\n3 4 5 6 7 8 9 10\nlog(tw-ffn-dim)0.00.20.40.60.81.0failure-rate\nExpt-I\nExpt-II\n(a)Trend of failure-rate vs. token-wise feed-forward\nnetwork of the Transformer. We vary the latter from 8to1024 ,\nwhile keeping the embedding dimension ( d) constant to 64.\n3 4 5 6 7 8 9 10\nlog(emb-dim)0.00.20.40.60.81.0failure-rate\nExpt-I\nExpt-II(b)Trend of failure-rate vs. embedding dimension of\nthe Transformer. We vary the latter from 4to512, while\nkeeping the token-wise feed-forward network dimension ( r)\nequal to the embedding dimension.\nFigure 3: Trend of failure-rate with respect to embedding dimension and the dimension of token-wise feed-forward\nnetwork. We perform each experiment ( training -validation -testing pipeline) 5times. The line in the graph corresponds to\nthe mean across runs, and the bands around the line indicate the standard deviation.\n5.2. Synthetic Datasets & Design Considerations\nWe generate a synthetic dataset using the equations in Table\n1. Notice that we cover various functional forms ( loga-\nrithmic ,exponential , and polynomial ), various forms of\ninteractions ( multiplicative , and additive ) among variables.\nSuch coverage adequately checks the expressivity in one go,\nrather than designing separate datasets for various functional\nforms and/or interactions. We generate 200000 ,10000 and\n20000 samples for training ,validation andtesting , respec-\ntively.\nTable 1: Equations governing the synthetic data generation.\nType: Input signifies that these variables are fed to the Trans-\nformer Encoder. Type: Output signifies that the Transformer\nis trained to generate these variables through the Decoder.\nBy design, we have functions covering the whole spectrum\nof first-order derivatives ( >1,= 1and<1).\nType Generator Function\nInputX1∼Uniform (−1,1)\nX2=3√X1\nX3=2·log(2 + X1) +X2/10\nX4=eX2+X3\nOutputY1=1\n5(X1+X2+X3+X4)\nY2=X1·Y1+eX2+X3+ log( X4)\nY3=1\n5(X1+Y2+Y1+√X2+X3·X4)\nWith this dataset, we ask the following empirical questions:•How does the failure-rate vary for change in the\ndimension of the Token-Wise Feed Forward Network\nin the Transformer?\n•How does the failure-rate vary for change in the\nembedding dimension of the Transformer?\n•How does the failure-rate vary with the chosen\nPositional Embedding scheme?\n•How does the failure-rate vary with the number\nof inputs and outputs?5\n5.3. Implementation, Results & Analysis\nWe use PyTorch (Paszke et al., 2017) to implement Trans-\nformers. We perform hyperparameter tuning on batch\nsize ,gradient accumulation ,epochs , etc. We\nrepeat each training -validation -testing pipeline 5times, to\nsee the variation in performance across runs. Refer to Ap-\npendix B for training details.\nWe use mean squared error loss for EXPT-Iandcross-\nentropy loss for EXPT-II. For EXPT-II, we use 5classes\nfor testing the Transformer across all settings. Addition-\nally, we also report how the Transformer behaves for higher\nnumber of classes, later on.\nWe observe that for all of the questions asked in Section 5.2,\nthe Transformer architecture fails significantly for EXPT-I.\nIn Figure 3a, we analyze the trend of performance when\nthe token-wise feed-forward network dimension is var-\nied. We observe that the performance for EXPT-IIdoes\nnot have a discernible trend. However, for EXPT-I, the\n5We include the data generation equations in Appendix C .\n6TRANSFORMERS ARE EXPRESSIVE , BUTARETHEY EXPRESSIVE ENOUGH FOR REGRESSION ?\n1 2 3\n#outputs0.00.20.40.60.81.0failure-rateExpt-I\nExpt-II\n(a)Trend of failure-rate with respect to the number of\noutputs. We run experiments for 1,2and3outputs.\n2 3 4\n#outputs0.00.20.40.60.81.0failure-rateExpt-I\nExpt-II(b)Trend of failure-rate with respect to the number of\ninputs. We run experiments for 2,3and4inputs.\nFigure 4: Trend of failure-rate with respect to the number of inputs and the number of outputs. We perform each\nexperiment ( training -validation -testing pipeline) 5times. The colored bars correspond to the mean across runs, while the\nblack lines indicate the standard deviation. For EXPT-I, we keep r=d= 32 , and for EXPT-II, we keep r=d= 128 .\nThese configurations were found to be best performing from Figure 3.\nfailure-rate oscillates a lot. Additionally, the stan-\ndard deviation in performance across runs is significantly\nmore for EXPT-I. In further experiments, we make a sim-\nplifying assumption – we keep the token-wise feed-forward\ndimension equal to the embedding dimension. This helps re-\nduce the possible space of experimental settings, and Figure\n3a shows that it is a reasonable assumption.\nIn Figure 3b, we analyze the trend of performance when\nthe embedding dimension of the Transformer is varied. We\nobserve that for EXPT-II, there is a downward trend in\nfailure-rate till embedding dimension = 128 . And,\nforEXPT-I, there is a similar oscillatory trend. Also, as\nbefore, the standard deviation in performance remains high,\nthus further proving our hypothesis – the inability of Trans-\nformers to reliably generate vectors to directly approximate\na continuous function. From Figure 3b, we understand that\nTransformers perform best in EXPT-Ifor embedding dimen-\nsion= 32 , and in EXPT-IIfor embedding dimension = 128 .\nIn our further experiments, we follow this configuration.\nWe test the Transformer by varying the number of inputs\n(Figure 4b) and outputs (Figure 4a). We see that there is a\nsignificant difference between performance in EXPT-Iand\nEXPT-IIirrespective of the number of inputs and outputs.\nWe also examine the performance across two popular po-\nsitional embedding schemes: sinusoidal (Vaswani et al.,\n2017), learned (Devlin et al., 2019), Figure 6. We observe\na similar thing – there is a significant difference between\nperformance in EXPT-IandEXPT-IIirrespective of the\npositional embedding scheme. These experiments answer\nall the questions we set in Section 5.2. We test for a few\nmore things too: ( a) checking performance in EXPT-IIfor ahigher number of classes, and ( b) checking performance as\nkincreases in failure-rate@ k.\nTo check the effect of higher number of classes in EXPT-\nII, we repeat the trend analysis study (similar to Fig-\nure 3b) with 20classes, Figure 7. We posited earlier\nthat for piecewise constant functions, small-sized pieces\nwould adversely affect performance. The number of pieces\nwould increase as the number of classes increases. With\nthe range constant, this implies that a higher number of\nclasses would lead to smaller pieces. Hence, we expect\na higher failure-rate as the number of classes in-\ncreases. However, as the embedding dimension increases,\nthe positioning of the constant pieces becomes sparse, if\nthe number of classes is kept constant. Hence, we expect\nfailure-rate to reduce as the number of embedding\ndimensions is increased. That is exactly what we observe in\nthe figure.\nWe had stated earlier, failure-rate can be a stringent\nmetric to judge performance in EXPT-II. To alleviate the\nsituation, we proposed failure-rate@ k. We examine\nhow the performance varies across various values of kfor\nEXPT-IandEXPT-II, Figure 8. We see that even if kis\nincreased, the Transformer fails significantly for E XPT-I.\nWe also qualitatively analyze the generations in EXPT-Ito\nvisualize the failure of the Transformer in generating Y1,Y2,\nandY3. We include T-SNEplots of the generated and ground\ntruth values for these three variables, Figures 5a, 5b and 5c.\nFor all of the variables, we see that the Transformer fails to\ngenerate representative vectors – the densities of generated\nand ground truth points differ significantly.\n7TRANSFORMERS ARE EXPRESSIVE , BUTARETHEY EXPRESSIVE ENOUGH FOR REGRESSION ?\n80\n 60\n 40\n 20\n 0 20 40 6080\n60\n40\n20\n0204060T-SNE plot for the output variable Y1\nGround Truth\nGenerated\n(a)\n8\n 6\n 4\n 2\n 0 2 4 6 88\n6\n4\n2\n02468T-SNE plot for the output variable Y2\nGround Truth\nGenerated (b)\n60\n 40\n 20\n 0 20 40 6060\n40\n20\n020406080T-SNE plot for the output variable Y3\nGround Truth\nGenerated (c)\nFigure 5: T-SNE plot for the generated and ground truth Y1(a),Y2(b) and Y3(c). Deeper colors indicate higher density.\nExpt-II Expt-I\n#outputs0.00.20.40.60.81.0failure-rateSinusoidal\nLearned\nFigure 6: Trend of failure-rate with respect to\ntwo popular Positional Embedding schemes: sinusoidal\n(Vaswani et al., 2017) and learned (Devlin et al., 2019). We\nsee that changing the scheme does not affect the perfor-\nmance significantly. For EXPT-I, we keep r=d= 32 , and\nforEXPT-II, we keep r=d= 128 . These configurations\nwere found to be best performing from Figure 3.\n6. Summary, Conclusion and Future Work\nIn this work, we analyze the function approximation capabil-\nities of the Transformer. We provide a theoretical treatment,\nanalyzing the capability, in Section 4. Following that, we\nprovide experimental results on several settings to exper-\nimentally evaluate the capability, in Section 5. We find\nthatTransformers are bad at approximating continuous\nfunctions . However, they are quite adept at approximating\npiecewise constant functions with moderately large-sized\npieces . We provide an analysis of how the approximation\ncapability is affected by piece size in Section 5.3. In future\nworks, we would analyze the individual components of the\nTransformer to pinpoint the source of reduced expressivity\nand would attempt to expand its expressivity.\n2 3 4 5 6 7 8 9\nlog(emb-dim)0.20.30.40.50.6failure-rate\nExpt-II w/ 20 classesFigure 7: Trend of failure-rate with respect to em-\nbedding dimension for 20classes in EXPT-II. We keep\nr=d= 128 for this experiment, found to be the best con-\nfiguration for E XPT-II from Figure 3.\n1 2 3 4\nk0.00.20.40.60.81.0failure-rate\nExpt-I\nExpt-II\nFigure 8: Trend of performance of Transformer for EXPT-\nIandEXPT-IIfor various kinfailure-rate@ k. For\nEXPT-I, we keep r=d= 32 , and for EXPT-II, we keep\nr=d= 128 . 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A., Ainslie, J.,\nAlberti, C., Ontanon, S., Pham, P., Ravula, A., Wang,\nQ., Yang, L., and Ahmed, A. Big bird: Transformers\nfor longer sequences. In Larochelle, H., Ranzato,\nM., Hadsell, R., Balcan, M., and Lin, H. (eds.),\nAdvances in Neural Information Processing Systems ,\nvolume 33, pp. 17283–17297. Curran Associates, Inc.,\n2020. URL https://proceedings.neurips.\ncc/paper_files/paper/2020/file/\nc8512d142a2d849725f31a9a7a361ab9-Paper.\npdf.\nZhang, S., Roller, S., Goyal, N., Artetxe, M., Chen, M.,\nChen, S., Dewan, C., Diab, M., Li, X., Lin, X. V ., Mi-\nhaylov, T., Ott, M., Shleifer, S., Shuster, K., Simig, D.,\nKoura, P. S., Sridhar, A., Wang, T., and Zettlemoyer,\nL. Opt: Open pre-trained transformer language models,\n2022.\nA. Proof for Theorem 4.1\nProof. We have: dp(f, f)≤ϵ\n=⇒\u0012Z\nS\r\rf(X)−f(X)\r\rp\npdx\u00131\np\n≤ϵ\nNote that for X̸∈ S,f(X) =f(X) = 0 . Now,\n\r\rf(X)−f(X)\r\rp\np=\u001adX\ni=1\f\ff(X)i−f(X)i\f\fp\u001bp×1\np\n=dX\ni=1\f\ff(X)i−f(X)i\f\fp\nSo, we have,\n\u0012Z\nS\r\rf(X)−f(X)\r\rp\npdx\u00131\np\n≤ϵ\n=⇒\u0012Z\nSdX\ni=1\f\ff(X)i−f(X)i\f\fpdX\u00131\np\n≤ϵ\nSimilar to the proof in Section 4.1, we look at the δ/2neigh-\nborhood of X0. Note that the neighborhood is a hypercube.\nAs before, we have f(X) =f(X0)in this neighborhood.\nAgain, in this neighborhood, f(X)can be rewritten as:\nf(X)i=f(X0)i+∂f(X)\n∂xi\f\f\f\f\nX0(xi−x0\ni)∀i\nwhere xidenotes the i-th component of X. With f(X) =\nf(X0), we have,\n10TRANSFORMERS ARE EXPRESSIVE , BUTARETHEY EXPRESSIVE ENOUGH FOR REGRESSION ?\nf(X)i−f(X)i=−∂f(X)\n∂xi\f\f\f\f\nX0(xi−x0\ni)∀i\nThus, in the δ/2neighborhood of X0, we can compute\ntheℓperror of approximation as ( [δ/2]ddenotes a d-\ndimensional vector, with each element as δ/2):\nZX0+[δ/2]d\nX0−[δ/2]ddX\ni=1\f\ff(X)i−f(X)i\f\fpdX\n=ZX0+[δ/2]d\nX0−[δ/2]ddX\ni=1\f\f\f\f\f∂f(X)\n∂xi\f\f\f\f\nX0(xi−x0\ni)\f\f\f\f\fp\ndX\n=ZX0+[δ/2]d\nX0−[δ/2]ddX\ni=1\f\f\f\f\f∂f(X)\n∂xi\f\f\f\f\nX0\f\f\f\f\fp\f\fxi−x0\ni\f\fpdX\n=dX\ni=1\f\f\f\f\f∂f(X)\n∂xi\f\f\f\f\nX0\f\f\f\f\fpZX0+[δ/2]d\nX0−[δ/2]d\f\fxi−x0\ni\f\fpdX\nThe integral can be computed as follows:\nZX0+[δ/2]d\nX0−[δ/2]d\f\fxi−x0\ni\f\fpdX\n=Zx0\ni+δ/2\nx0\ni−δ/2\f\fxi−x0\ni\f\fpdxiZX0\n−i+[δ/2]d−1\nX0\n−i−[δ/2]d−1dX−i\nX−idenotes the components of Xother than the i-th com-\nponent. Note that the integral is truly a multiple integral\nover all the dimensions. This lets us compute the integral for\nthei-th and other dimensions separately. Now, the second\nintegral is essentially the volume of a d−1dimensional\nhypercube of side δ. It evaluates to δd−1. We can compute\nthe first integral as:\nZx0\ni+δ/2\nx0\ni−δ/2\f\fxi−x0\ni\f\fpdxi\n=Zδ/2\n−δ/2|u|pdxi[Change of variable; u=xi−x0\ni]\n= 2·Zδ/2\n0updxi[|u|pis symmetric about 0]\n= 2·up+1\np+ 1\f\f\f\fδ/2\n0=1\np+ 1·2−p·δp+1Thus, we have,\nZX0+[δ/2]d\nX0−[δ/2]ddX\ni=1\f\ff(X)i−f(X)i\f\fpdX\n=dX\ni=1\f\f\f\f\f∂f(X)\n∂xi\f\f\f\f\nX0\f\f\f\f\fp\n1\np+ 1·2−p·δp+1·δd−1\n=dX\ni=1\f\f\f\f\f∂f(X)\n∂xi\f\f\f\f\nX0\f\f\f\f\fp\n1\np+ 1·2−p·δp+d\nAs in Section 4.1, if we consider a covering X0ofS, such\nthatX0∈ X0, we have:\n\u0012Z\nSdX\ni=1\f\ff(X)i−f(X)i\f\fpdX\u00131\np\n≤ϵ\n⇒X\nX0∈X0ZX0+[δ/2]d\nX0−[δ/2]ddX\ni=1\f\ff(X)i−f(X)i\f\fpdX≤ϵp\n⇒X\nX0∈X0dX\ni=1\f\f\f\f\f∂f(X)\n∂xi\f\f\f\f\nX0\f\f\f\f\fp\nδp+d\n2p·(p+ 1)≤ϵp\n⇒δp+d\n2p·(p+ 1) X\nX0∈X0dX\ni=1\f\f\f\f\f∂f(X)\n∂xi\f\f\f\f\nX0\f\f\f\f\fp!\n≤ϵp\n⇒δ≤ \n2p·(p+ 1)·ϵp\nP\nX0∈X0Pd\ni=1\f\f\f∂f(X)\n∂xi\f\f\nX0\f\f\fp! 1\n(p+d)\nB. Training Details\nWe perform an extensive hyperparameter search for each ex-\nperimental setting using wandb.ai . We search for the best\ncombination for the following hyperparameters: batch size ,\nmaximum number of training steps ,learning rate ,number\nof attention heads ,number of transformer layers ,dropout\nandwarmup steps for learning rate . We list the range of\neach in Table 2. We found that the best combination for\neach setup is not the same.\nWe train all our models on a single A 100GPU, with each\ntraining consuming ∼2.5GB GPU memory and ∼1−3\nhours of training.\n11TRANSFORMERS ARE EXPRESSIVE , BUTARETHEY EXPRESSIVE ENOUGH FOR REGRESSION ?\nTable 2: Search Space for Hyperparamters. {·}denotes\na set,∼ U(l, h)denotes sampling from a Uniform distri-\nbution within landh.num-layers corresponds to the\nnumber of transformer layers ,max-steps corresponds\nto the maximum training steps ,num-attn-heads corre-\nsponds to the number of attention heads ,warmup-steps\ncorresponds to the fraction of training steps used for learn-\ning rate warmup , and emb-dim denotes the embedding\ndimension of the transformer.\nHyperparameter Values\nbatch-size {128,256,512}\nmax-steps {1200 ,1400 ,1600}\nnum-layers {2,4,6}\nnum-attn-heads {2,4,···,min(16,emb-dim )}\nlearning-rate ∼ U(10−2,5×10−6)\nwarmup-steps ∼ U(0.2,0.4)\ndropout ∼ U(0.1,0.2)\nC. Details on Auxiliary Datasets\nWe had provided ablation studies on how the performance\nvaries for different number of inputs and outputs. Here,\nwe provide the equations governing the generation of those\ndatasets. These generation equations are minor variations of\nequations in Table 1, which generates data for 4inputs and\n3outputs. Tables 3, 4, 5, 6 show the equations for ablations\non2inputs, 3inputs, 1output and 2outputs.\nTable 3: Equations governing the synthetic data generation\nfor2inputs and 3outputs.\nType Generator Function\nInputX1∼Uniform (−1,1)\nX2=3√X1\nOutputY1=1\n5(X1+X2)\nY2=X1·Y1+eX2\nY3=1\n5(X1+Y2+Y1+√X2)Table 4: Equations governing the synthetic data generation\nfor3inputs and 3outputs.\nType Generator Function\nInputX1∼Uniform (−1,1)\nX2=3√X1\nX3=2·log(2 + X1) +X2/10\nOutputY1=1\n5(X1+X2+X3)\nY2=X1·Y1+eX2+X3\nY3=1\n5(X1+Y2+Y1+√X2)\nTable 5: Equations governing the synthetic data generation\nfor4inputs and 1output.\nType Generator Function\nInputX1∼Uniform (−1,1)\nX2=3√X1\nX3=2·log(2 + X1) +X2/10\nX4=eX2+X3\nOutput Y1=1\n5(X1+X2+X3+X4)\nTable 6: Equations governing the synthetic data generation\nfor4inputs and 2outputs.\nType Generator Function\nInputX1∼Uniform (−1,1)\nX2=3√X1\nX3=2·log(2 + X1) +X2/10\nX4=eX2+X3\nOutputY1=1\n5(X1+X2+X3+X4)\nY2=X1·Y1+eX2+X3+ log( X4)\n12"    },    {        "title": "2402.15485v1.Graph_Partitioning_With_Limited_Moves.pdf",        "content": "Graph Partitioning With Limited Moves\nMajid Behbahani* Mina Dalirrooyfard†Elaheh Fata‡Yuriy Nevmyvaka§\nAbstract\nIn many real world networks, there already exists a (not necessarily optimal) k-partitioning of the network.\nOftentimes, one aims to find a k-partitioning with a smaller cut value for such networks by moving only a few nodes\nacross partitions. The number of nodes that can be moved across partitions is often a constraint forced by budgetary\nlimitations. Motivated by such real-world applications, we introduce and study the r-move k-partitioning problem,\na natural variant of the Multiway cut problem. Given a graph, a set of kterminals and an initial partitioning of the\ngraph, the r-move k-partitioning problem aims to find a k-partitioning with the minimum-weighted cut among all the\nk-partitionings that can be obtained by moving at most rnon-terminal nodes to partitions different from their initial\nones. Our main result is a polynomial time 3(r+ 1) approximation algorithm for this problem. We further show that\nthis problem is W[1]-hard, and give an FPTAS for when ris a small constant.\n1 Introduction\nGraph partitioning problems are among the most fundamental and widely used graph problems in the fields of artificial\nintelligence, theoretical computer science, operations research and operations management. A k-partitioning of a graph\nGrefers to partitioning the graph’s nodes into k-disjoint sets, clusters, or partitions. Oftentimes, the goal in partitioing\nproblems is to find a k-partitioning with the least cut value (also referred to as cut size in the literature), which is the\nsum of the weights of the edges that are between different partitions. Graph partitioning is widely used in machine\nlearning, parallel computing, computer vision, VLSI design, political districting, epidemiology and many more areas\n[1, 2, 3, 4, 5, 6].\nGiven a graph*G, an integer k≥2andkterminals, the well-known Multiway cut problem asks to find a partitioning\nof the nodes of Gintokclusters with the smallest cut value among all k-partitionings that have exactly one terminal in\neach partition [ 7]. In this paper, we use the terms partitions andclusters , interchangeably. It is known that for k >2,\nthe Multiway cut problem is APX-hard [8].\nWe introduce and study a natural variant of the Multiway cut problem called the r-move k-partitioning problem.\nSuppose that we are given a graph Gwith an initial k-partitioning, an integer k≥2,kterminals and a move parameter\nr∈N. The problem only considers k-partitionings for which at most rnon-terminal nodes have been moved from their\ninitial partitions to new clusters. The r-move k-partitioning problem asks to find a k-partitioning that minimizes the cut\nvalue over all the k-partitionings considered.\nNumerous examples of the r-move k-partitioning problem can be seen in real-world networks. In fact, in many\nnetworks, the underlying graph already has a k-partitioning and this k-partitioning is not optimal, i.e., the cut value\nis not minimized. For instance, the edge weights in a network may change over time, resulting in the sub-optimality\nof the initial partitioning. However, in most applications, we cannot afford moving many nodes from their initial\nk-partitioning, and so the goal is to improve the cut value by moving only “a few” nodes.\nFor example, consider a company with many offices across the country. The company can be modeled as a graph,\nwhere the nodes are the employees and the edge weights indicate how beneficial it would be if these two employees\n*Machine Learning Research, Morgan Stanley. Email: majid.behbahani@morganstanley.com.\n†Machine Learning Research, Morgan Stanley. Email: mina.dalirrooyfard@morganstanley.com.\n‡Smith School of Business, Queen’s University. Email: elaheh.fata@queensu.ca.\n§Machine Learning Research, Morgan Stanley. Email: yuriy.nevmyvaka@morganstanley.com.\n*Authors are ordered alphabetically.\n*We use the terms “graphs” and “networks”, interchangeably.\n1arXiv:2402.15485v1  [cs.DS]  23 Feb 2024are in the same office, for example it could be proportional to the amount they communicate, or any other metric. The\ncompany’s goal is to find a k-partitioning with a low cut value, where kis the number of available offices. Employees\nhave initial assignments to these koffices, which might not be optimal. However, due to budget constraints, companies\nare often unable to conduct large-scale reorganizations and can only move a limited number of employees between\noffices. Moreover, not all employees can or are willing to move. For each office, we can model these employees as a\nterminal node by “contracting” all the employees of that office (partition) that cannot relocate. If the company’s budget\ndictates that at most remployees can be relocated, then we seek an r-move k-partitioning where the initial partitioning\nis the original distribution of people across offices. Modifications of the office reorganization example are applicable\nto many other applications such as public housing relocation to improve the housing assignments and enhance the\nresidents’ community access, or disaster management (see [9]).\nAnother application of the r-move k-partitioning problem is its use as a sub-problem in a local search algorithm\nfor the Multiway cut problem. Consider an initial feasible multiway cut Cfrom which we aim to achieve another\nfeasible multiway cut C′with a lower cost, allowing only rmoves for the transformation from CtoC′. This problem\ncan be exactly formulated as the r-move k-partitioning problem. For more information on local search approximation\nalgorithms for the Multiway cut problem, the reader is referred to [ 10]. Moreover, the r-move k-partitioning problem\ncan be explored in the context of learning-augmented algorithms. In this case, the initial multiway cut is obtained from\na learning-based algorithm and the goal is to use this initial multiway cut to reach multiway cuts with lower costs, while\nmoving at most rnodes to partitions different from their initial ones. The question here is whether there exists an\nalgorithm for the r-move k-partitioning problem that can improve the initial multiway cut whose guarantees depend on\nthe quality of the initial multiway cut. While this direction is interesting, it falls outside the focus of this paper.\nThere has been a long line of works providing approximation algorithms for the Multiway cut problem [ 8,11,12,\n13,14], most of which use a simple linear programming (LP) relaxation of the problem called the CKR LP , in honor of\nthe authors who introduced it, C ˘alinescu, Karloff and Rabani [ 11]. The best approximation factor for the Multiway\ncut problem is 1.2965 due to [ 15], who round the CKR LP. Interestingly, [ 14] showed that assuming the unique games\nconjecture [ 16], if the integrality gap of the CKR LP is α, then the Multiway cut problem cannot be approximated better\nthanα. [17] showed that the integrality gap of the CKR LP is at least 1.20016 , hence the best approximation factor for\nthe Multiway cut problem lies between 1.20016 and1.2965 . Closing this gap is an interesting open problem.\nPerhaps the closest known variant of the Multiway cut problem to the r-move k-partitioning problem is the mins-t\ncut with at most rnodes , also known as the Minr-sizes-tcutproblem [ 18]. This problem asks to find a minimum value\ns-tcut, where there are at most rnodes in the partition that contains terminal node s(i.e., the s-side). This problem is a\nspecial case of the (r−1)-move 2-partitioning problem, when in the initial partitioning all non-terminal nodes are in\nthe partition that contains node t. By moving at most r−1nodes to the s-side, the s-side will have size at most r.\nEven though the Min r-sizes-tcut problem is a variant of the s-tcut problem, which is polynomially solvable, the\nMinr-sizes-tcut problem is NP-hard [ 19]. Thus, the r-move 2-partitioning and the r-move k-partitioning problems\nare NP-hard. It is also known that the Min r-sizes-tcut problem admits a Fixed-Parameter Tractable (FPT) solution\nwith parameter rwhen the graph is unweighted [ 20] (see Section 2 for a definition of FPT problems). Intuitively, one\nmight wonder if the r-move 2-partitioning problem and the Min r-sizes-tcut problem are equivalent. We show that the\nr-move 2-partitioning problem is in fact W[1]-hard; thus, not equivalent to the Min r-sizes-tcut problem. We resort to\ndesigning approximation algorithms for the r-move k-partitioning problem for all k≥2. Hence, the main theme of this\npaper is around the following question: How well can we approximate r-move k-partitioning?\n1.1 Our Results\nWe provide a comprehensive study of the r-move k-partitioning problem. Recall that the r-move k-partitioning problem\nis NP-hard (for any constant k≥2), hence, there exists no exact polynomial time algorithm for it, unless P = NP.\nThroughout the paper we think of kas a constant.\nFirst, note that when ris also a constant, there exist a simple O((nk)r)algorithm for the problem. This algorithm\ntries every combination of r′nodes, for all 1≤r′≤r, to be moved to any of the kpartitions. It is natural to ask\nwhether we can devise an FPT algorithm with parameter rfor the r-move k-partitioning problem. Our first result shows\nthat the answer to this question is negative.\nTheorem 1.1 Ther-move k-partitioning problem with parameter ris W[1]-hard.\n2In Theorem 1.1, we use a polynomial time reduction from the Densest r-Subgraph problem, where the complete\nproof can be found in Section 7. It is well-known that the Densest r-Subgraph problem with parameter ris W[1]-hard\n[21].\nNext, we give a (1 + ϵ)-approximation algorithm for the r-move k-partitioning problem with running time\nf(r, ϵ)O(n2)for any ϵ >0, where f(·,·)is a function of randϵ. When ris a constant, this gives us an FPTAS\n(with parameter ϵ) with running time O(n2/ϵr), faster than the O(nr)brute-force algorithm, in the expense of an\napproximation factor.\nTheorem 1.2 Given any r-move k-partitioning instance graph Gwithnnodes and any constant ϵ >0, there exists a\n(1 +ϵ)-approximation algorithm for the r-move k-partitioning problem on Gwith running time (2(k−1)(1+ ϵ)\nϵ)rr!cn2,\nwhere cis a universal constant independent of randϵ.\nThe proof of Theorem 1.2 can be found in Section 3. The running time of the algorithm in Theorem 1.2 grows\nfast as rgrows, which leads us to the following question: Can we design an approximation algorithm with a running\ntime polynomial in both nandr? Our main result focuses on answering this question. We give a polynomial time\napproximation algorithm with approximation factor at most 3(r+1). This is done by extending the CKR linear program\nto an LP for the r-move k-partitioning problem by adding a single linear constraint reflecting the move constraint of\nthe problem. Our approximation algorithm is a simple rounding scheme for this LP, which makes our approach very\npractical. The main challenge is in proving the approximation guarantees of our rounding, which is given in Section 4.\nTheorem 1.3 Given a positive integer r, there exists a randomized algorithm for the r-move k-partitioning problem\non any n-node m-edge graph Gthat with approximation factor2k\nk−1(r+ 1) and running time TLP(n, m) +O(mk),\nwhere TLP(n, m)is the running time of solving a linear program with O(n)variables and O(m)constraints.\nFirstly, note that we can de-randomize this algorithm in the expense of an additional factor of nin the running\ntime (see Section 4.1). Secondly, note that unlike Theorem 1.2, the running time of our algorithm in Theorem 1.3 is\nindependent of r. Fork= 2, we give a rounding scheme for our LP with approximation factor at most r+ 1, resulting\nin an (r+ 1) -approximation algorithm for the r-move 2-partitioning problem (see Section 6).We further show that the\nintegrality gap of the LP used in Theorem 1.3 is at least r+ 1, demonstrating that our rounding algorithm is tight within\na constant factor (see Section 7.1).\nFinally, as our approximation factor is dependent on r, we resort to bicriteria algorithms to make it constant.\nTheorem 1.4 For any 1/2< γ < 1, there exists a polynomial time (1\n1−γ,5\n2γ−1)-approximation randomized algorithm\nforr-move k-partitioning problem, where the first criterion is the number of nodes moved and the second criterion is\nthe cut value.\nThe proof of Theorem 1.4 can be found in Section 5. We provide two remarks to better understand the applicability\nof our results. Firstly, we show in Section 7 that r-move k-partitioning without terminals is also W[1] hard and so\neven though for a lot of partitioning problems the “without terminals” version is easier, this is not the case for r-move\nk-partitioning. Our second remark is stated below. We include simple complementary numerical evaluations of our\nalgorithm in Section 8.\nMeaningful range for r:The range of rin which Theorems 1.3 and 1.2 provide meaningful results can be understood\nas follows. For the case of k= 2, the Multiway cut problem is tractable and so it is possible to compute an optimal\nmultiway cut Cfor the input graph in polynomial time. Suppose that cut Cis obtained by moving ˆrnodes from their\ninitial partitions. Then, Theorems 1.3 and 1.2 are meaningful only when r <ˆr. This is because if r≥ˆr, then Cis an\noptimal output for the r-move 2-partitioning problem. On the other hand, when k >2, there is no polynomial time\nalgorithm for the Multiway cut problem. Therefore, in this case, one can obtain a cut Cwhose cut value is within a\nconstant approximation factor of an optimal multiway cut, using an approximation algorithm for the Multiway cut\nproblem (for example the3\n2-approximation algorithm of [ 11]). Let ˆrbe the number of nodes that Cmoves from their\ninitial partitions. Once again, Theorems 1.3 and 1.2 are useful only when r <ˆr.\n3FPT Apprx. Fctr Apprx. Fctr LB\nMinr-sizes-tcut 1 +ϵ* min(r+ 1,log(n))[18, 27] W[1]-hard*\nMinr-size cut 1[20] 1[20]\nr-move 2-partitioning 1 +ϵ* r+ 1* W[1]-hard*\nr-move 2-partitioning without terminals 1 +ϵ* r+ 1* W[1]-hard*\nTable 1: Known and new results on different variants of the 2-partitioning problem. Asterisks are used to show the results\nprovided by this paper. The second column indicates the best known approximation factors for fixed-parameterized\napproximation algorithms for the problems with parameter r. The third column is dedicated to the approximation\nfactors of general approximation algorithms with running times polynomial in both nandr, and the fourth column\nindicates lower bounds. Note that the NP-hardness of the Min r-sizes-tcut problem was already known [ 19] and this\npaper shows the W[1]-hardness of this problem in its weighted version.\n1.2 Related Works\nThe Multiway cut problem without terminals is called the k-partitioning problem. While the k-partitioning problem is\npolynomially solvable for fixed k[22], the Multiway cut problem is NP-hard for k≥3[7]. There exist many “budgeted”\nvariants of the Multiway cut and k-partitioning problems in the literature, we name a few of these variants here.\nThe Min r-sizes-tcut problem: First, recall that the Min r-sizes-tcut problem is NP-hard, even though the “without\nterminals” version of this problem, that is, the Min r-size cut problem, is polynomially solvable [ 23,24]. The Min\nr-size cut problem asks to find a cut with the minimum value among all the cuts with one side having at most rnodes.\nThe result of [ 23] is based on the result of [ 25] which states that the number of cuts with value at most αtimes the value\nof the min cut is at most n2α. Such result does not exist for s-tcuts and, in fact, the number of min s-tcuts can be\nexponentially many*. The Min r-sizes-tcut problem has been studied by other names such as the problem of “cutting\nat most rnodes by edge-cut with terminal” [ 26,20]. Moreover, [ 27] gives an O(log(n))-approximation algorithm for\nthe Min r-size s-tcut problem using R ¨acke’s tree decomposition method [ 28]. See Table 1 for a comparison of the\nrelated variants of the r-move 2-partitioning problem and their associated algorithms.\nTable 1 compares the variants of r-move k-partitioning and Min r-sizes-tcut. Next we include omitted proofs.\nOther variants: Most of the variants of the Multiway k-cut problem that we are aware of are for k= 2. Basically\nhaving a budget on the number of nodes in one side of the cut, and having terminals or no terminals in the graph makes\nup of most of these variants.\nThe exact version of the Min r-sizes-tcut problem is called the Min E r-Size s-tcutproblem [ 29], which asks to\nfind an s-tcut with minimum cut value among all the cuts that have exactly rnodes in the s-side. The (r, n−r)-cut\nproblem [ 30] is the above problem without any terminals. The (r, n−r)-cut problem is known to be W[1]-hard [ 30],\nand as a result the Min E r-Size s-tcut problem is also W[1]-hard. For both these problems, there is a randomized\nO(r/log(n))-approximation algorithm due to [ 29]. These problems have been studied when parameterized by cut\nvalue as well [26].\nFork >2, the most relevant variant of the Multiway k-cut problem is the k-balanced partitioning problem, which is\nproved to be APX-hard [ 31]. Other problems that seem to be indirectly relevant are the k-route cut problem [ 32] and\nthe MinSBCC problem [33].\n[9] study a variant of the min s-tcut problem with fairness considerations. They introduce the Demographic Fair\nCut problem, which is informally defined as follows: Given a terminal node sand a labeling of nodes into various\ndemographics, the goal is to find a minimum cut that disconnects at least a certain given fraction of each demographic\nfrom s. If all nodes of the graph belong to a single demographic, then this problem is equivalent to the Min r-sizes-t\ncut problem. Thus, their log(n)-approximation algorithm can also be applied to the Min r-sizes-tcut problem, resulting\nin amin(r+ 1,log(n))-approximation algorithm for the Min r-sizes-tcut problem. Whether a log(n)-approximation\n*Consider a graph which consists of n/2paths of length 2from terminal sto terminal t. Ans-tcut has at least one edge from each path, and so\nthe number of min s-tcuts is 2n/2.\n4algorithm exists for the r-move k-partitioning problem (either for k= 2ork >2) is an interesting open question that\nis outside the scope of this paper.\n2 Preliminaries\nLetG= (V, E)with weight function c:E→R, that is, an edge (u, v)∈Ehas weight c(u, v)orcuv. We refer to\nan edge with endpoints uandvby(u, v)oruv. We only consider undirected graphs throughout this paper. Let E(G)\nandV(G)denote the edge set and node set of a graph G. A problem of size nwith parameter ris said to be FPT if\nthere is an algorithm that solves it in f(r)O(nc)time, where f(·)is an arbitrary function depending only on randcis\na constant. If any W[1]-complete problem is FTP, then FPT = W[1]and every problem in W[1]is FPT.\nWe formally define the r-move k-partitioning problem using the definition of a k-cut (also known as k-partitioning).\nGiven a weighted graph G= (V, E)with weight function c:E→R, ak-cutis a subset of edges E′⊆Esuch that\nremoving the set of edges E′from the graph results in another graph G′= (V, E\\E′)that has kconnected components.\nWe refer to these connected components that appear after removing edge set E′asclusters orpartitions . The weight of\nak-cutE′is the sum of the weight of the edges in E′and it is called the cut value . Any set of edges that introduce a cut\nin a graph, such as E′, can be referred to as a cut-set .\nDefinition 2.1 (The r-move k-partitioning problem) LetG= (V, E)be a weighted graph with a weight function\nc:E→R, a set of terminals S={s1, . . . , s k} ⊆Vand|V|=n. Suppose that Ghas a given initial partitioning,\nwhere ℓv∈ {1, . . . , k }denotes the initial partition that node vbelongs to and for a terminal node si, we have ℓsi=i, for\neachi∈ {1, . . . , k }. The r-move k-partitioning problem asks to find a minimum-weighted k-cutℓ∗:V→ {1, . . . , k }\nsuch that ℓ∗\nv̸=ℓvfor at most rnon-terminal nodes v.\nThe parameter ris referred to as the move parameter . The linear program we use throughout this paper is represented\nin Table 2 and will be referred to as LP 2. Suppose that kis the number of partitions of interest. For each v∈V, let\nXv= (X1\nv, . . . , Xk\nv)be a vector of size kof positive real decision variables in [0,1]. To understand the role of decision\nvariable Xi\nvbetter, observe that if we were only considering integer solutions, then Xi\nv= 1would represent node v\nbeing in partition i, and constraint (C4) would ensure that each node vis assigned to exactly one partition. We define\nthe distance between pairs of vertices u, vasd(u, v) =1\n2Pk\ni=1|Xi\nu−Xi\nv|. In case of integer solutions, if uandvare\nin the same partition, then d(u, v) = 0 and if they are in different partitions, d(u, v) = 1 . This further motivates the\nobjective function in LP 2. To represent |Xi\nu−Xi\nv|in the LP, we define variables Yi\nuv’s for each (u, v)∈E, and add\nconstraints (C2) and (C3) to make sure Yi\nuv=|Xi\nu−Xi\nv|. Finally, we call constraint (C7) the move constraints , which\nensures that at most rnodes are moved from their initial partitions, in case of integer solutions. Note that LP 2 without\nconstraint (C7) is the CKR LP.\nMinimize:P\n(u,v)∈Ecuvd(u, v)\nSubject to:\nd(u, v) =1\n2Pk\ni=1Yi\nuv∀(u, v)∈E (C1)\nYi\nuv≥Xi\nu−Xi\nv ∀(u, v)∈E,∀i∈ {1···k}(C2)\nYi\nuv≥Xi\nv−Xi\nu ∀(u, v)∈E,∀i∈ {1···k}(C3)Pk\ni=1Xi\nu= 1 ∀u∈V (C4)\nXsi=ei ∀si∈S (C5)\nXi\nv≥0 ∀v∈V,∀i∈ {1, . . . , k } (C6)P\nv∈V(1−Xℓvv)≤r (C7)\nTable 2: r-move k-partitioning LP (LP 2)\nSuppose Xis a feasible, but not necessarily optimal, solution to LP 2. Notation dX(u, v) =1\n2Pk\ni=1|Xi\nu−Xi\nv|is\nused to better clarify that the distance function is calculated specifically for solution X. We use the notation ¯Xto denote\nthe optimal solution of LP 2. We conclude this section with the following lemmas that will be later used in our proofs.\n5Lemma 2.1 Consider a feasible solution Xto LP 2. Given a number 0≤λ <1, ifLis the set of nodes vfor which\nXℓvv< λ, then|L|<r\n1−λ.\nProof. For each v∈L, we have 1−Xℓvv>1−λ. Since Xis a feasible solution of LP 2, r≥P\nv∈L(1−Xℓvv). So\nr >(1−λ)|L|, and|L|<r\n1−λ. □\nLemma 2.2 LetXbe any feasible solution to LP 2. For any i∈ {1, . . . , k }andu, v∈Vwe have dX(u, v)≥\n|Xi\nu−Xi\nv|.\nProof. By the triangle inequality we haveP\nj̸=i|Xj\nu−Xj\nv| ≥ |P\nj̸=iXj\nu−P\nj̸=iXj\nv|=|(1−Xi\nu)−(1−Xi\nv)|, where\nthe last equality usedP\njXj\nu=P\njXj\nv= 1. Therefore, dX(u, v) =1\n2(|Xi\nu−Xi\nv|+P\nj̸=i|Xj\nu−Xj\nv|)≥ |Xi\nu−Xi\nv|.\n□\n3 FPTAS\nHere, we first provide high level intuition on the algorithm of Theorem 1.2 and prove it formally in section 3.1. In\nthis result, we use this simple observation that by moving a node vfrom its initial partition to another partition, the\ncut value reduces by at most d−(v) :=P\nu,ℓu̸=ℓvcuv. LetC0denote the initial cut-set. Let COPTdenote the optimal\ncut-set for the r-move k-partitioning problem and SOPTbe the set of nodes that have to be moved to partitions different\nfrom their initial ones in order to obtain COPT. By the problem definition, |SOPT| ≤r. Let the value of the optimal\ncut be c(COPT)and the value of the initial cut be c(C0). Ifc(C0)is already a (1 +ϵ)approximation of c(COPT), then\nwe can simply output the initial cut as our solution. Otherwise, we argue that there exists a node v∈SOPTfor which\nd−(v)≥ϵ\nr(1+ϵ)c(C0). Let Abe the set of nodes uwithd−(u)≥ϵ\nr(1+ϵ)c(C0). We show that |A| ≤2r(1+ϵ)\nϵ. This\nallows us to achieve a recursive algorithm as follows: For each node v∈Aand each partition i̸=ℓv, move vto\npartition i, contract vwith the terminal node si, then reduce the move parameter by 1and recurse on this instance.\nContraction of node vwith the terminal node siis to forbid the algorithm from moving node vagain. Since we make\n(k−1)|A|instances of the (r−1)-move k-partitioning problem, we can show that the running time of our algorithm\nis polynomial in nand1/ϵ. Note that this algorithm is polynomial only if ris constant. Theorem 1.3 provides an\napproximation algorithm that is polynomial in nand is independent of r.\n3.1 Proof of Theorem 1.2\nLetα= 1 + ϵ. We solve an instance of the r-move k-partitioning problem with at most (k−1)2rα\nα−1instances of the\n(r−1)-move k-partitioning problem.\nRecall that the weight of any edge (u, v)∈E(G)is denoted by cuv. Let the initial cut-set in graph GbeC0, i.e.,\nC0={(u, v)∈E(G)|ℓu̸=ℓv}, and let c(C0)denote its cut value. For each node v∈V(G), letd−(v)denote the\nsum of the weight of all edges incident to vthat lie in the initial cut-set C0i.e.,d−(v) =P\n(u,v)∈C0cuv. Observe that\nc(C0) =1\n2P\nv∈V(G)d−(v), since the weight of each edge (u, v)∈C0appears in both d−(u)andd−(v). LetAbe the\nset of nodes vwithd−(v)≥α−1\nrαc(C0). Our FPTAS makes k−1instances of (r−1)-move k-partitioning for each\nv∈Aas follows: For any j̸=ℓv, letGv,jbe the graph obtained from Gby contracting node vwith terminal sj. After\nthis contraction, for any node u, the weight of edge (sj, u)is increased by cvu. Let set C={C0}. The algorithm then\nrecurses on graph Gv,jwith move parameter r−1for all v′∈Aandj̸=ℓv′, and adds the output of this recursion to\nthe set of cut-sets C. Then the algorithm outputs the cut-set with the least value among all those in C. Note that a small\npost processing is needed for the output as we need to undo the contractions which are at most rmany, see Algorithm 1.\nWe begin the analysis of this algorithm by showing that the running time of the algorithm is at most (2(k−1)α\nα−1)rr!O(n).\nSince c(C0) =1\n2P\nv∈V(G)d−(v)we have c(C0)≥1\n2P\nv∈Ad−(v)≥ |A|α−1\n2rαc(C0); therefor, |A| ≤2rα\nα−1. Let\nF(r, n)be the running time of the algorithm on any n-node graph. We have F(r, n)≤(k−1)2rα\nα−1F(r−1, n−1) +\nO(n2), where the second term is the time required to compute d−(v)for all v∈V(G). Note that the constant behind\nn2is independent of r, kandα. Since F(0, n) =O(1)for any n, we have F(r, n)≤(2(k−1)α\nα−1)rr!O(n2).\nNow we prove that the algorithm outputs a cut-set C′such that c(C′)≤αc(COPT), where COPTis an optimal\ncut-set. Recall that C′= arg max ¯C∈Cc(¯C). LetSOPTbe the set of nodes that were moved from their initial partition\n6Algorithm 1 FPTAS for constant move parameter\nRequire: graph Gwith terminals s1, . . . , s k, move parameter randα= 1 + ϵ.\n1:C0={(u, v)∈E(G)|ℓu̸=ℓv} ← initial cut-set in graph G.\n2:c(C0)←P\nu,vsuch that (u,v)∈C0cuv. ▷Cut value of C0.\n3:ifr= 0then\n4: return C0. ▷Return the initial cut, without moving anyone.\n5:else\n6: C={C0}.\n7: forv∈V(G)do\n8: d−(v) =P\nusuch that (u,v)∈C0cuv.\n9: end for\n10: A← {v∈V(G)|d−(v)≥α−1\nrαc(C0)}.\n11: forv∈A∩V(G)do\n12: forj∈ {1, . . . , k } \\ {lv}do ▷All partitions except ℓv.\n13: Gv,j←Graph obtained from Gby contracting vwithsjand increasing the weight of each edge (sj, u)\nbycvu.\n14: Cv,j←The cut returned by running Algorithm 1 on Gv,jwith move parameter r−1.\n15: C←C∪Cv,j.\n16: end for\n17: end for\n18: return arg min C∈Cc(C). ▷Return the cut with the smallest value in C.\n19:end if\nto obtain COPT. IfSOPT=∅, then COPT=C0. Since C0∈C, we have that c(COPT)≤c(C′)≤c(C0), so\nc(C′) =c(COPT)and the problem is trivial. Therefore, suppose that SOPT̸=∅. Here, two cases are possible: (i)\nc(C0)≤αc(COPT), and (ii) c(C0)> αc (COPT). In case (i), since c(C′)≤c(C0), we have c(C′)≤αc(COPT); thus,\nC′is a valid αapproximation of COPT. For case (ii), we show that SOPT∩A̸=∅. This can be achieved by first showing\nthatc(COPT)≥c(C0)−P\nv∈SOPTd−(v).\nLet graph G′be obtained by removing all nodes in SOPTand edges incident to such nodes from graph G. LetC′\n0\nbe the equivalent of initial cut C0onG′, that is, C′\n0=C0\\ {(u, v)∈E(G)|v∈SOPT}. Similarly, we define C′\nOPT\nas the portion of the optimal cut-set COPTthat exists in G′, i.e., C′\nOPT=COPT\\ {(u, v)∈E(G)|v∈SOPT}. Since\nC0andCOPTonly differed in some edges incident to nodes in SOPT, we have C′\n0=C′\nOPT. Asc(C′\nOPT)≤c(COPT),\nwe have c(C′\n0)≤c(COPT). On the other hand, since by removing each vinSOPTfrom graph Gthe cut value\ndrops by at most d−(v), we have that c(C′\n0)is at least c(C0)−P\nv∈SOPTd−(v). Putting these together provides\nc(COPT)≥c(C0)−P\nv∈SOPTd−(v). Recall that in case (ii), c(C0)> αc (COPT); therefore,P\nv∈SOPTd−(v)≥\nc(C0)−c(COPT)>α−1\nαc(C0). So there must be a node v∗inSOPTsuch that d−(v∗)>α−1\nαrc(C0), concluding that\nSOPT∩A̸=∅.\nSuppose that in the optimal solution, node v∗∈Ais moved to partition jfor some j̸=ℓv∗. We argue that the\noptimal cut value for Gv∗,jwhen at most r−1nodes can be moved to partitions different from their initial ones is\nat most c(COPT). This is because contracting v∗andsjinGv∗,jis equivalent to moving v∗to partition jand not\nmoving it to another partition again. By moving the nodes in SOPT\\ {v∗}in order to obtain the cut COPT , the resulting\ncut value is c(COPT). Let C′\nv∗,jbe the output of the recursion on Gv∗,j. By induction, c(C′\nv∗,j)≤αc(COPT). Since\nC′\nv∗,j∈C, the algorithm outputs an α-approximation for G.\n43(r+ 1)-Approximation Algorithm for Parametric r\nIn this section, we show a2k\nk−1(r+ 1)-approximation algorithm for the r-move k-partitioning problem when k >2.\nFork >2, we have that2k\nk−1≤3; therefore, our algorithm provides a 3(r+ 1)-approximation guarantee for such k.\nNote that, this algorithm works for k= 2as well; however, we already have an (r+ 1) -approximation algorithm for\n7this case.\nAlgorithm 22k\nk−1(r+ 1) -approximation algorithm for the r-move k-partitioning problem\n1:¯X←solution to LP 2.\n2:g=k−1\nk(r+1).\n3:ρchosen uniformly at random from (0, g).\n4:A← {z= (z1, . . . ,zk)|1\ngzi∈Z≥0for all 1≤i≤kandPk\ni=1zi<1 +kg}.\n5:forifrom 1tokdo\n6: forv∈V(G)do\n7: ˜Xi\nv=g⌊¯Xi\nv+ρ\ng⌋.\n8: end for\n9:end for\n10:forz∈Ado\n11: Hz← {v∈V(G)|˜Xv=z}.\n12: ifHzcontains a terminal sithen ▷ Hzcan contain at most one terminal\n13: i∗\nz←i.\n14: else\n15: i∗\nz←arg maxk\nj=1|{v∈Hz|ℓv=j}|.\n16: end if\n17: Assign all the nodes in Hztoi∗\nz.\n18:end for\nThe algorithm operates as follows: The optimal solution to LP 2 gets rounded in two phases. Let g=k−1\nk(r+1)\nandρbe a real number chosen uniformly at random from the interval (0, g). In the first phase, for each node v, we\nround the entries of ¯Xvto obtain ˜Xvsuch that for each 1≤i≤k,˜Xi\nvis an integer multiple of g. More precisely, let\n˜Xi\nv=⌊¯Xi\nv+ρ\ng⌋g, i.e., ˜Xi\nvis the largest multiple of gno larger than ¯Xi\nv+ρ.\nIn the second phase of rounding, the algorithm puts all nodes that are rounded to the same vector in the same\npartition, see Algorithm 2 for a description of how this partition is chosen. To clarify the role of ρ, for two nodes\nu, v∈V(G)ifd¯X(u, v)is small, then we want ¯Xuand¯Xvbe rounded to the same vector with high probability,\ni.e., ˜Xu=˜Xvwith high probability. If we let ρ= 0 at all times, then uandvnever get rounded to the same\nvector in the following case: For some small ϵ >0, let ¯X1\nv=g+ϵ=¯X2\nu,¯X2\nv=g−ϵ=¯X1\nuand¯Xi\nv=¯Xi\nu\nfori >2. Thus, d¯X(u, v) = 2 ϵis small. For this example and ρ= 0 we have ˜X1\nv=⌊¯X1\nv\ng⌋g=⌊g+ϵ\ng⌋g=gand\n˜X1\nu=⌊¯X1\nu\ng⌋g=⌊g−ϵ\ng⌋g= 0. In other words, even though d¯X(u, v)is small ¯Xuand¯Xvare rounded to different\nvectors with certainty. This is in fact true for any constant value of ρand a random ρis key to be able to round two\nnodes uandvthat are very close to each other to the same vector, with high probability.\nLemma 4.1 For each v∈V(G)and each i∈ {1, . . . , k }we have |˜Xi\nv−¯Xi\nv|< g. Moreover, if for two nodes\nu, v∈V(G)and an i∈ {1, . . . , k }we have ˜Xi\nv=˜Xi\nu, then|¯Xi\nv−¯Xi\nu|< g.\nProof. If˜Xi\nv≥¯Xi\nv, then since ˜Xi\nv≤¯Xi\nv+ρandρ < g , we have that |˜Xi\nv−¯Xi\nv|< g. If˜Xi\nv<¯Xi\nv, then since\n˜Xi\nv=⌊¯Xi\nv+ρ\ng⌋g, we have (¯Xi\nv+ρ\ng−1)g < ˜Xi\nv, thus ¯Xi\nv+ρ < ˜Xi\nv+g. Therefore, |˜Xi\nv−¯Xi\nv|< g−ρ < g .\nFor the second part of the lemma, since both ¯Xi\nv+ρand¯Xi\nu+ρare rounded down to the same value α=˜Xi\nv, they\nare both in the [α, α+g)interval. Therefore, ¯Xi\nvand¯Xi\nuare both in the interval [α−ρ, α+g−ρ), concluding that\n|¯Xi\nv−¯Xi\nu|< g. □\nBy Lemma 4.1, we have |˜Xi\nv−¯Xi\nv|< gand by constraint (C4) in LP 2 we havePk\ni=1¯Xi\nv= 1, thus,Pk\ni=1˜Xi\nv<\n1 +kg. LetAbe the set of all k-sized vectors with entries being non-negative integer multiples of gsuch that the sum\nof the entries of each vector is at most 1 +kg. By definition, each entry of ˜Xvis non-negative integer multiple of gand,\nas shown, the sum of entries of ˜Xvis at most 1 +kg. Therefore, Algorithm 2 rounds any ¯Xvto a vector in A.\nNow we show that if d¯X(u, v)is very small, then with high probability ˜Xu=˜Xv. Note that for two nodes uandv\nwith ˜Xu=˜Xv, Algorithm 2 puts uandvin the same partition.\n8Lemma 4.2 For any two distinct nodes u, v∈V(G)such that (u, v)∈E(G), the probability that Algorithm 2 puts\n(u, v)in the cut-set is at most2\ngd¯X(u, v).\nProof. Ifd¯X(u, v)>g\n2, then 1<2\ngd¯X(u, v)and the lemma holds trivially. Assume that d¯X(u, v)≤g\n2. Observe that\nuandvare assigned to different partitions only if they are rounded to different vectors, that is, ˜Xu̸=˜Xv. This happens\nif there exists an i∈ {1, . . . , k }and an integer 1≤t≤1\ngsuch that tgis between ¯Xi\nv+ρand¯Xi\nu+ρ. To measure this\nprobability, let i∈ {1, . . . , k }be fixed. Without loss of generality, suppose that ¯Xi\nv<¯Xi\nu. Then, we need to compute\nthe probability that\n¯Xi\nv+ρ < tg ≤¯Xi\nu+ρ. (1)\nTo see feasible values of ρandt, we consider the following two cases: (i) ⌊¯Xi\nv\ng⌋=⌊¯Xi\nu\ng⌋, and (ii) ⌊¯Xi\nv\ng⌋<⌊¯Xi\nu\ng⌋. For\nthe first case, suppose that ⌊¯Xi\nv\ng⌋=⌊¯Xi\nu\ng⌋=ˆt. This means that ˆtg≤¯Xi\nv<¯Xi\nu<(ˆt+ 1)g. Since ρ < g , the only\nfeasible value for tisˆt+ 1and inequality (1) is equivalent to\n0<(ˆt+ 1)g−¯Xi\nu≤ρ <(ˆt+ 1)g−¯Xi\nv< g. (2)\nThe length of the [(ˆt+ 1)g−¯Xi\nu,(ˆt+ 1)g−¯Xi\nv)]interval is |¯Xi\nu−¯Xi\nv|. Therefore, in case (i), with probability at\nmost1\ng|¯Xi\nu−¯Xi\nv|there exists a tsatisfying inequality (1).\nFor the second case, suppose that ˆt=⌊¯Xi\nv\ng⌋<⌊¯Xi\nu\ng⌋. Since g >g\n2> d ¯X(u, v)≥¯Xi\nu−¯Xi\nv, we must have\n⌊¯Xi\nu\ng⌋=ˆt+ 1. The only accepted values for tisˆt+ 1andˆt+ 2, and inequality (1)is satisfied when ρis in one of the\nfollowing two intervals:\n0≤ρ <(ˆt+ 1)g−¯Xi\nvor(ˆt+ 2)g−¯Xi\nu≤ρ≤g.\nThe sum of the lengths of these two intervals is |¯Xi\nu−¯Xi\nv|, so in case (ii) edge (u, v)is in the cut-set with probability\nat most1\ng|¯Xi\nu−¯Xi\nv|. Putting cases (i) and (ii) together, any edge (u, v)∈E(G)is in the cut-set with probability at\nmost1\ng|¯Xi\nu−¯Xi\nv|. Finally, since d(u, v) =1\n2Pk\ni=1|¯Xi\nu−¯Xi\nv|, the probability that edge (u, v)is in the cut-set is at\nmost2\ngd¯X(u, v). □\nLetzbe a vector in Aandzidenote its i−th entry. Let Hzbe the set of all nodes v∈V(G)for which ˜Xv=z.\nThe following lemma discusses the number of terminal nodes that Hzmay contain.\nLemma 4.3 For any vector z∈A, there can be at most one terminal node in Hz.\nProof. Suppose there exists two distinct terminal nodes siandsjinHz. By constraint (C5) of LP 2, we have ¯Xi\nsi= 1\nand¯Xi\nsj= 0, providing |¯Xi\nsi−¯Xi\nsj|= 1. On the other hand, by Lemma 4.1 we have |¯Xi\nsi−¯Xi\nsj|< g < 1. This is a\ncontradiction and there can be at most one terminal in Hz. □\nIf there exists a terminal node si∈Hz, then Algorithm 2 puts all the nodes in Hzin partition i. Otherwise, the\nalgorithm puts all the nodes in Hzin the partition where most nodes of Hzcome from. Let rz=P\nv∈Hz(1−¯Xℓvv).\nLemma 4.4 Ifr≥1, then for each vector z∈A, there exists a partition izfor which the number of nodes v∈Hz\nwithℓv̸=izis less than rz+rz\nr. If there is a terminal node siinHz, then iz=i.\nProof. Consider a vector z∈Aand its associated set of nodes Hz. LetBibe the set of nodes v∈Hzwithℓv=i, i.e.,\nthe set of nodes in Hzwhose initial partition is i. Similarly, we define B−ito be the set of nodes v∈Hzwithℓv̸=i,\nthat is, the set of nodes in Hzwhose initial partition is not i. To prove this lemma, we consider the following two cases:\n(i) there exists a terminal node si∈Hz, and (ii) Hzcontains no terminal node.\nIn case (i), for a node v∈Hz, it holds that ˜Xi\nv=˜Xi\nsi. So, by Lemma 4.1, we have ¯Xi\nv>¯Xi\nsi−g= 1−g. Using the\nfact that for any v∈V(G)it holds thatP\ni∈{1,...,k}¯Xi\nv= 1, we have rz=P\nv∈Hz(1−¯Xℓvv)≥P\nv∈B−i(1−¯Xℓvv)≥P\nv∈B−i¯Xi\nv>|B−i|(1−g),providing that |B−i|<1\n1−grz. Combining this with g=k−1\nk(r+1)<1\n1+r, we have\n|B−i|< rz+rz\nr.\nIn case (ii), Hzcontains no terminal node. Without loss of generality, suppose that for all 2< i≤kwe have\n|B1| ≤ |B2| ≤ |Bi|. Defining B′=Hz\\(B1∪B2), we have |B1| ≤|B′|\nk−2.For the sake of contradiction, suppose for all\n9i∈ {1, . . . , k }we have |B−i|=|Hz\\Bi| ≥rz+rz\nr. In particular, rz+rz\nr≤ |Hz\\B2|=|B′|+|B1| ≤ |B′|+|B′|\nk−2.\nSo|B′| ≥k−2\nk−1(rz+rz\nr). By Lemma 4.1, for any two nodes v, u∈Hz, we have |¯Xℓvv−¯Xℓvu|< g; thus, 1−¯Xℓvv>1−\n¯Xℓvu−g. Ifℓv/∈ {1,2}, that is, v∈B′, thenP\ni∈{1,...,k},i̸=ℓv¯Xi\nv> g+P\ni∈{1,2},i̸=ℓv(¯Xi\nu−g)+P\ni∈{3,...,k},i̸=ℓv¯Xi\nu.\nIfℓv∈ {1,2}, then we haveP\ni∈{1,...,k},i̸=ℓv¯Xi\nv>P\ni∈{1,2},i̸=ℓv(¯Xi\nu−g) +P\ni∈{3,...,k},i̸=ℓv¯Xi\nu. Therefore,\nrz=P\nv∈Hz(1−¯Xℓvv) =P\nv∈HzP\ni∈{1,...,k},i̸=ℓv¯Xi\nvand so rz>|B′|g+P\nv∈Hz\u0014P\ni∈{1,2},i̸=ℓv(¯Xi\nu−g) +\nP\ni∈{3,...,k},i̸=ℓv¯Xi\nu\u0015\nrz>|B′|g+X\ni∈{1,2}X\nv∈Hz,ℓv̸=i(¯Xi\nu−g)\n+X\ni∈{3,...,k}X\nv∈Hz,ℓv̸=i¯Xi\nu\n≥k−2\nk−1(rz+rz\nr)g+X\ni∈{1,2}(rz+rz\nr)(¯Xi\nu−g)\n+X\ni∈{3,...,k}(rz+rz\nr)¯Xi\nu (3)\n= (rz+rz\nr)(1−k\nk−1g) =rz,\nwhere inequality (3)uses the assumption that |B−i| ≥rz+rz\nrfor all i∈ {1, . . . , k }. The above calculations reaches a\ncontradiction, rz> rz. Therefore, there exists an iz∈ {1, . . . , k }such that the number of nodes v∈Hzwithℓv̸=iz\nis at most rz+rz\nr, concluding the proof of the lemma. □\nProof of Theorem 1.3. Algorithm 2 sets g=k−1\nk(r+1). Firstly, note that by Lemma 4.2 the probability that an edge\n(u, v)is in the cut-set is at most2k\nk−1(r+ 1)d¯X(u, v), where ¯Xis an optimal solution to LP 2. Therefore, Algorithm 2\nreturns a cut-set with cut value at most2k\nk−1(r+ 1) OPT r, where OPT rdenotes the optimal value of LP 2 with a move\nparameter of r.\nSecondly, we prove that Algorithm 2 moves at most rnodes. To do this, we show that for each vector z∈A, the\nnumber of nodes in Hzthat are moved from their initial partitions is less than rz+rz\nr. There are two possibilities to\nconsider here. Case (i): There exists a terminal in si∈Hz(recall that by Lemma 4.3, there can be at most one terminal\ninHz). In this case, Algorithm 2 assigns all the nodes in Hzto partition i. Therefore, the number of nodes that are\nmoved to partitions different from their original partitions is equal to the number of nodes v∈Hzwithℓv̸=i. By\nLemma 4.4, the number of such nodes is less than rz+rz\nr. Case (ii): Hzhas no terminal nodes. In this case, Algorithm\n2 assigns all the nodes to partition i∗\nz←arg maxk\nj=1|{v∈Hz|ℓv=j}|. Defining Bias the set of nodes v∈Hzwith\nℓv=i, the number of nodes that are moved to partitions different from their original ones is equal to |Hz| − |Bi∗z|.\nBy Lemma 4.4, there exists a partition ifor which |Hz| − |Bi|< rz+rz\nr. By definition of i∗\nz, we have |Bi| ≤ |Bi∗z|;\ntherefore, |Hz| − |Bi∗z|< rz+rz\nr.\nThe total number of nodes that are moved by Algorithm 2 in cases (i) and (ii) is strictly less thanP\nz��Arz+rz\nr. Using\nthe definition of rzand constraint (C7) of LP 2, we haveP\nz∈Arz+rz\nr= (r+1\nr)P\nz∈Arz= (r+1\nr)P\nz∈AP\nv∈Hz(1−\n¯Xℓvv) = (r+1\nr)P\nv∈V(G)(1−¯Xℓvv)≤r+ 1. This shows that the total number of nodes that Algorithm 2 moves is\nstrictly less than r+ 1, in other words, at most rnodes are moved to partitions different from their original ones by the\nalgorithm. □\n4.1 Run Time and De-randomization\nFor computing the run time, note that finding the solution to LP2 takes TLP(n, m)time. The rest of the algorithm\nconsists of simple loops and takes O(mk)time.\nOne can de-randomize the algorithm using the technique of Calinescu, Karloff and Rabani [ 11]. They de-randomize\ntheir rounding for their 1.5−1/kapproximation algorithm for the multiway-cut problem (at the expense of polynomial\n10increase in running time), and they argue that there are O(n)“interesting” values of ρand hence one only needs to run\nthe rounding algorithm for those values. To see this in our algorithm at a high level, note that for each i, there is a value\nρiwhere for all ρ < ρ i,˜Xi\nvisg⌊¯Xi\nv\ng⌋and for ρ≥ρi, we have ˜Xi\nv=g⌊¯Xi\nv\ng+ 1⌋. The value of ρican be computed\nfrom ¯Xi\nv. So one can only run the algorithm for these values of ρi. For the running time, it increases by a factor of at\nmost n, however one could decrease this factor by sorting the values of ρ.\n5 Constant-Factor Bicriteria Approximation Algorithm\nTo develop our constant-factor bicriteria approximation algorithm, we use some of the techniques used in [ 11]. We first\nprovide a detailed review on the rounding LP 3 technique of [11].\n5.1 Review of the Rounding Technique of LP 3\nIn this section, we explain the 1.5-approximation rounding of [ 11]. Let an n-node graph Gbe the instance of the\nMultiway cut problem and Xbe a solution to the Multiway cut CKR LP (LP 3). We first add some nodes to graph G\nvia edge subdivision and then extend solution Xto the added nodes such that for each edge (u, v)in the new graph, Xu\nandXvdiffer in at most two entries. This edge subdivision mechanism is discussed below and its steps are shown in\nAlgorithm 3.\nMinimize:P\n(u,v)∈Ecuvd(u, v)\nSubject to:\nd(u, v) =1\n2Pk\ni=1Yi\nuv∀(u, v)∈E,∀i∈ {1···k} (C1)\nYi\nuv≥Xi\nu−Xi\nv ∀(u, v)∈E,∀i∈ {1, . . . , k }(C2)\nYi\nuv≥Xi\nv−Xi\nu ∀(u, v)∈E,∀i∈ {1, . . . , k }(C3)Pk\ni=1Xi\nu= 1 ∀u∈V,∀i∈ {1, . . . , k } (C4)\nXsi=ei ∀si∈S (C5)\nXi\nv≥0 ∀v∈V,∀i∈ {1, . . . , k } (C6)\nTable 3: Multiway cut CKR LP (LP 3)\nConsider two nodes uandv, where XuandXvdiffer in more than two entries. Subdivide the edge (u, v)using a\nnew node wand let the weight of edges (u, w)and(w, v)be the same as the weight of the deleted edge (u, v). Define\nXwas follows: Let ibe the entry that holds the minimum non-zero value among |Xt\nu−Xt\nv|for all entries t∈ {1, . . . , k }.\nWithout loss of generality, assume that Xi\nu< Xi\nvand let α=Xi\nv−Xi\nu. SincePk\nt=1Xt\nu=Pk\nt=1Xt\nv= 1, there\nexists an entry j̸=ifor which Xj\nu−Xj\nv≥α. We define Xi\nw=Xi\nu+α=Xi\nv,Xj\nw=Xj\nu−α. Moreover, for\nany entry t̸=i, j,Xt\nwis defined as Xt\nw=Xt\nu; therefore, nodes uandwdiffer in only two entries. Observe that\nd(u, v) =d(u, w) +d(w, v). Suppose uandvare different in anumber of entries, then vandware different in a−1\nnumber of entries. Graph Ghasnnodes and by going through the explained procedure the number of different entries\nbetween any two nodes decreases by one. Each two nodes in graph Gcan have at most kdifferent entries. Therefore,\nafterO(nk)steps this procedure ends. The resulting graph by the end of this procedure is denoted by G∗= (V∗, E∗)\nand the vectors of endpoints of each edge of it are different in at most two entries, that is, for any edge (u, v)∈E∗,Xu\nandXvdiffer in at most two entries. Note that since d(u, v) =d(u, w) +d(w, v)andcuv=cwu=cwv, the values of\noptimal solutions of LP 3 on graphs GandG∗are the the same.\nUsing Algorithm 3, without loss of generality, it can be assumed that for any edge (u, v)of an instance of the\nMultiway cut problem G,¯Xuand¯Xvdiffer in at most two entries, where ¯Xis an optimal solution to LP 3 on graph\nG. Letρbe a parameter chosen uniformly at random from [0,1]. Let B(ρ, i)be the set of nodes v∈V(G)for which\n¯Xi\nv> ρ. Let σ= (σ(1), . . . , σ (k))be either (1,2, . . . , k −1, k)or(k−1, k−2, . . . , 1, k)with equal probabilities\n1/2. The partitions are processed in the order σ(1), σ(2), . . . , σ (k). In the first step, all nodes in B(ρ, σ(1)) are\nplaced in partition σ(1). Note that for any i̸=j, there might be some nodes that are in both B(ρ, i)andB(ρ, j).\nFor each jfrom 2tok−1, the remaining nodes in B(ρ, σ(j))are placed in partition σ(j), that is, all the nodes in\nB(ρ, σ(j))\\ ∪j−1\ni=1B(ρ, σ(i)). All the remaining nodes are then placed in partition k. LetCbe the cut-set made by the\n11partitions produced by this rounding process. For an edge (u, v)∈E(G), we have Pr((u, v)∈C)is the probability\nthat edge (u, v)is in cut-set C.\nLemma 5.1 ([11]) For any edge (u, v)∈E(G), we have Pr((u, v)∈C)≤1.5×d(u, v).\nProof. Suppose that ¯Xuand¯Xvdiffer in entries iandj. Without loss of generality, assume that ¯Xi\nu≥¯Xi\nv,¯Xj\nu,¯Xj\nv.\nSincePk\nt=1¯Xt\nu=Pk\nt=1¯Xt\nv= 1, we must have that either ¯Xi\nu≥¯Xj\nv≥¯Xi\nv≥¯Xj\nuor¯Xi\nu≥¯Xi\nv≥¯Xj\nv≥¯Xj\nu, thus,\n¯Xj\nu≤¯Xi\nu,¯Xi\nv,¯Xj\nv.\nIn order for edge (u, v)to be in cut-set C, one of its endpoints must belong in partition iand the other in partition\nj, not any other partitions. This is because for any ℓ̸=i, j,¯Xℓ\nv=¯Xℓ\nuand for the chosen ρ, either both uandvare\ninB(ρ, ℓ)or neither are in it. Let Li(and similarly Lj) be defined as the interval between values ¯Xi\nvand¯Xi\nu, that\nis,Li= (¯Xi\nv,¯Xi\nu)andLj= (¯Xj\nu,¯Xj\nv). Edge (u, v)is in cut-set Conly if either ρ∈Liorρ∈Lj. By constraint\n(C4) of LP 3, both intervals LiandLjare of the same length. However, LiandLjmight be overlapping; thus,\n|Li∪Lj| ≤2|Li|= 2d¯X(u, v), where d¯X(u, v) =1\n2\u0000\n|¯Xi\nu−¯Xi\nv|+|¯Xj\nu−¯Xj\nv|\u0001\n.\nWith equal probabilities, the aforementioned partitioning method processes partition jbefore i. We claim that (u, v)\nis going to be in Cifρ∈Li∪Lj. To see that, consider the following three scenarios: (i) If partition jis processed\nbefore partition iandρ∈Lj, then vgoes to partition jbutudoes not go to partition j. (ii) If partition jis processed\nbefore partition iandρ∈Li\\Lj, then we have that ¯Xj\nv,¯Xj\nu< ρ, souandvare not assigned to partition j. In this\ncase, when partition iis processed uis assigned to iandvis not. (iii) If iis processed before jandρ∈Li, then uis\nassigned to iandvis not. Note that if ρ∈Lj\\Li, both uandvare assigned to i. This is because ρ < ¯Xi\nu,¯Xi\nvsince\nρ /∈Liand all values in Lj\\Liare less than the values in Li. Using the union bound, the probability that edge (u, v)\nis in cut-set Cis as follows:\nPr\u0000\n(u, v)∈C\u0001\n≤1\n2Pr(ρ∈Li∪Lj) +1\n2Pr(ρ∈Li)\n≤1\n2×2d¯X(u, v) +1\n2d¯X(u, v)\n≤3\n2d¯X(u, v);\nin other words, the cut value is 1.5times the objective value of the optimal solution to LP 3, in expectation. □\n5.2 Bicriteria Algorithm\nWe describe our bicriteria algorithm in the next theorem.\nTheorem 1.4 For any 1/2< γ < 1, there exists a polynomial time (1\n1−γ,5\n2γ−1)-approximation randomized algorithm\nforr-move k-partitioning problem, where the first criterion is the number of nodes moved and the second criterion is\nthe cut value.\nProof. The first step of the bicriteria algorithm is to solve LP 2. Let ¯Xbe an optimal solution to this LP. Similar\nto the rounding of LP 3 by [ 11], we can form a graph G∗and extend the solution ¯Xon it, where G∗is formed by\nsubdividing the edges of Gsuch that for every two nodes u, vthat are adjacent, ¯Xuand¯Xvdiffer in at most two entries.\nNote that the values of the optimal cuts in graphs GandG∗are the same. The subdivision process is depicted in\nAlgorithm 3. Remark that the move constraint of at most rnodes is only on the original nodes, i.e., nodes of graph G,\nand not those added to graph G∗by the edge subdivision algorithm.\nLetZdenote the integer solution for the r-move k-partitioning problem, which will be defined throughout the proof.\nLetλbe a parameter chosen uniformly at random from [γ+1\n3, γ]. For any node v∈V(G∗), if there is an entry isuch\nthat¯Xi\nv≥λ, then let Zv=ei, see Algorithm 4. In other words, if ¯Xi\nv≥λ, then node vgets assigned to partition i.\nObserve that since γ >1\n2thenλ≥γ+1\n3>1/2andP\nt∈{1,...,k}¯Xt\nv= 1. So it is not possible for a node vto have\nmore than one entry ifor which ¯Xi\nv≥λ.\nLetA={v∈V(G)|¯Xℓvv≥λ}be the set of nodes that get assigned to their initial partitions, i.e., Zv=ev. By\nLemma 2.1, |V(G)\\A| ≤r\n1−λ≤r\n1−γ. The above procedure assigns every node vinAtoℓv, so no matter how we\ncontinue the rounding we are guaranteed to move no more than a total of 4rnodes from their initial partitions.\n12Algorithm 3 Edge subdivision algorithm [11]\nRequire: Input Xwhich is a solution to the LP 3.\n1:G∗←G.\n2:while there is an edge (u, v)∈E(G∗)such that XuandXvdiffer in more than two entries do\n3: Subdivide edge (u, v)with new a node w∈G∗.\n4: cuw←cuv,cvw←cuv.\n5: i←arg min t∈{1,...,k},|Xt\nu−Xt\nv|̸=0|Xt\nu−Xt\nv|.\n6: ifXi\nu> Xi\nvthen\n7: Swap uandv.\n8: end if\n9: α←Xi\nv−Xi\nu. Letjbe an entry where Xj\nu−Xj\nv≥α.\n10: Xi\nw←Xi\nu+α,Xj\nw←Xj\nu−α,Xt\nw=Xt\nufor all t̸=i, j.\n11:end while\n12:return G∗and the extension of Xon it.\nLetS={v∈V(G∗)|∃i∈ {1, . . . , k },¯Xi\nv≥λ}be the set of all the nodes that are processed by steps 7-10 of\nAlgorithm 4, and let S′=V(G∗)\\Sbe the rest of the nodes. Random variable ρis then chosen uniformly at random\nfrom interval [0, λ]. Let B(ρ, i)be the set of nodes v∈S′with ¯Xi\nv> ρ. Let σ= (σ(1), σ(2), . . . , σ (k))be one of\nthe two permutations (1,2, . . . , k −1, k)and(k−1, . . . , 1, k)chosen with equal probabilities at random. Nodes are\nassigned to partitions in the order of σ. In particular, for each j < k , all the nodes in B(ρ, σ(j))\\ ∪j−1\ni=1B(ρ, σ(i))are\nplaced in partition σ(j)and, finally, the remaining nodes are placed in partition σ(k) =k. Processing each node takes\nconstant time, so the rounding procedure takes linear time.\nLetCbe the cut-set created by this rounding scheme. Here, we bound the value of C. Consider an edge\n(u, v)∈E(G∗). We provide an upper bound on the probability that edge (u, v)is in cut-set C, i.e., Pr\u0000\n(u, v)∈C\u0001\n.\nFirst, note that if d¯X(u, v)≥2γ−1\n3, then Pr\u0000\n(u, v)∈C\u0001\n≤1≤3\n2γ−1d¯X(u, v)<5\n2γ−1d¯X(u, v). Therefore, we\nassume that d¯X(u, v)<2γ−1\n3. Under this assumption, we have three cases for edge (u, v):\n•Case 1: Nodes u, v∈S. In this case, we compute Pr\u0000\n(u, v)∈C|u, v∈S\u0001\n. Let i, jbe the entries for\nwhich ¯Xi\nv≥λand¯Xj\nu≥λ. Ifi=j, then (u, v)/∈C. Ifi̸=j, then since ¯Xj\nv,¯Xi\nu≤1−λ, we have\nd¯X(u, v)≥1\n2\u0000\n|¯Xi\nv−¯Xi\nu|+|¯Xj\nv−¯Xj\nu|\u0001\n≥2λ−1≥2γ−1\n3. This is a contradiction to our assumption of\nd¯X(u, v)<2γ−1\n3; thus, in this case (u, v)is never in cut-set C.\n•Case 2: Node u∈Sand node v∈S′. In this case, we compute Pr\u0000\nu∈S, v∈S′\u0001\n. By constraint (C4) of\nLP 2, for a node wthere can at most be one entry isuch that ¯Xi\nw≥7\n12. Letjbe such an entry for node u. Since\nu∈Sandv∈S′, We must have ¯Xj\nu≥λ≥max( ¯Xj\nv,7\n12). The probability that such λis chosen is at most\nd¯X(u, v)/(γ−γ+1\n3) =3\n2γ−1d¯X(u, v), concluding that Pr(u∈S, v∈S′)≤3\n2γ−1d¯X(u, v).\n•Case 3: Nodes u, v∈S′. In this case, we compute Pr\u0000\n(u, v)∈C|u, v∈S′\u0001\n. This case is very similar to the\nargument provided in [ 11], which we mention here for the sake of completeness. Suppose that ¯Xuand¯Xvonly\ndiffer in iandjentries. Without loss of generality, suppose that ¯Xi\nu= max( ¯Xi\nu,¯Xi\nv,¯Xj\nu,¯Xj\nv). Subsequently,\nby constraint (C4) of LP 2 we have ¯Xj\nu= min( ¯Xi\nu,¯Xi\nv,¯Xj\nu,¯Xj\nv). Let Li= (¯Xi\nv,¯Xi\nu)andLj= (¯Xj\nu,¯Xj\nv).\nNote that |Li|=|Lj|=d¯X(u, v). In order for (u, v)to be in cut-set C, we must have ρ∈Li∪Lj. Due\nto step 13 of Algorithm 4, an entry 1≤i < k is processed before an entry 1≤j < k with probability 1/2.\nSuppose that entry jis processed before i. Ifρ∈Lj, then node vgoes to partition jandudoes not. If\nρ∈Li\\Lj, then we have that ¯Xj\nv,¯Xj\nu< ρ. Therefore, uandvare not assigned to partition j, and when\nentry iis processed, node uis assigned to partition iandvis not. So, when entry jis processed before entry\ni, edge (u, v)is in cut-set Cifρ∈Li∪Lj. If entry iis processed before jandρ∈Li, then uis assigned\nto partition iandvis not. If ρ∈Lj\\Li, then both uandvare assigned to partition i. Therefore, in total,\nPr\u0000\n(u, v)∈C|u, v∈S′\u0001\n≤1\n2Pr(ρ∈Li∪Lj) +1\n2Pr(ρ∈Li)≤3\n4λd¯X(u, v)≤9\n4(γ+1)d¯X(u, v)≤2\n2γ−1.\nUsing these three cases we can quantify the probability that an edge (u, v)lies in cut-set Cas follows. Note that we are\n13Algorithm 4 Bicriteria approximation algorithm\nRequire: Input graph Gwith an initial partitioning, parameter γ >1\n2and move parameter r.\n1:¯X←an optimal solution to the LP 2.\n2:Form graph G∗as a super graph of Gand extend ¯XonG∗by Algorithm 3.\n3:Choose λuniformly at random from [γ+1\n3, γ].\n4:Choose ρuniformly at random from [0, λ].\n5:S=∅\n6:forall nodes v∈G∗do\n7: ifthere exists i∈ {1, . . . , k }such that ¯Xi\nv≥λthen\n8: Zv←ei.\n9: S=S∪ {v}.\n10: end if\n11:end for\n12:B(ρ, i)←the set of nodes vnot in Ssuch that ¯Xi\nv> ρ, for each i∈ {1, . . . , k }.\n13:Choose σto be one of the permutations (1,2, . . . , k −1, k)and(k−1, . . . , 1, k)with equal probabilities.\n14:foreachj < k do\n15: forall nodes v /∈Sthat are in B(ρ, σ(j))\\ ∪j−1\ni=1B(ρ, σ(i))do\n16: Zv←eσ(j).\n17: end for\n18:end for\n19:forany node vnot assigned to a partition do\n20: Zv←ek.\n21:end for\n22:return Z.\nconditioning on d¯X(u, v)<2γ−1\n3and so Case 1 does not take place:\nPr\u0000\n(u, v)∈C\u0001\n= Pr\u0000\n(u, v)∈C|u, v∈S\u0001\nPr(u, v∈S)\n+ Pr\u0000\n(u, v)∈C|u∈S, v∈S′\u0001\nPr(u∈S, v∈S′)\n+ Pr\u0000\n(u, v)∈C|u, v∈S′\u0001\nPr(u, v∈S′)\n≤Pr\u0000\n(u, v)∈C|u∈S, v∈S′\u00013\n2γ−1d¯X(u, v) +2\n2γ−1d¯X(u, v) Pr(u, v∈S′)\n≤5\n2γ−1d¯X(u, v).\n□\n6 Approximation Algorithm for k= 2\nIn this section, we describe our approximation algorithm for the r-move 2-partitioning problem. Since there are two\npartitions in this problem, there only exists two terminal nodes. We refer to these terminal nodes as sandtto be\ncompliant with the 2-partitioning and the min s-tcut problems literature. We first give a high level comparison of our\nalgorithm and that of [ 18], who studied a related problem called the Min r-sizes-tcut problem and from whom we\nborrow some ideas for our approximation method. Section 1 provides a formal definition of the Min r-size s-tcut\nproblem and the distinctions between this problem and the r-move 2-partitioning problem.\nMain ideas of [ 18]:Zhang [ 18] uses a problem called the parametric max flow problem introduced in [ 34]. The\nparametric max flow problem is a generalization of the s-tmax flow problem where we think of capacities/edge weights\nas parameters. More formally, the weight of an edge (u, v)can be parameterized by a parameter α, such that the weights\n14of the edges connected to terminal s, i.e., cα(s, v), are non-decreasing functions of αand the the weights of the edges\nconnected to terminal t, i.e., cα(v, t), are non-increasing in α. Finally, the weights of the edges that are connected to\nneither of the terminals, i.e., cα(u, v)for any node u̸=s, v̸=t, are constant in terms of α.\nZhang [ 18] creates the following graphs Gαfor a parameter α≥0: Let the weight of all edges (v, t)beα, for all\nnodes v̸=s. Clearly this fits into the parametric flow definition of [ 34], discussed above. When α= 0, the min s-tcut\nofGαis the same as the min s-tcut of G. LetS0be the set of nodes in the s-side of the min s-tcut of G. Asαgets\nlarger, they show that the s-side of the min s-tcut in Gαcontains fewer and fewer nodes until eventually for some large\nαthes-side contains only s. Zhang [ 18] argues that by considering values of αfrom 0to some large number, there\nareh≤n−1distinct sets S0, . . . , S h={s}corresponding to the s-sides of the min s-tcuts of Gαgraphs, such that\n|Si|>|Si+1|for all i= 0, . . . , h −1. Gallo et al. [ 34] show that one can compute the sets SiinO(mnlog(n2/m))\ntime. Zhang [ 18] proceeds by proving a few properties for Gαgraphs and using the linear program formulation of the\nMinr-sizes-tcut problem they give ak+1\nk+1−k∗-approximation algorithm for it.\nWe can approach the r-move 2-partitioning problem using similar Gαgraphs. More specifically, a graph Gαis\nproduced by adding edges (s, v)and(v, t)with weights αto the original graph G, for each v̸=s, t. These graphs do not\nfit into the definition of parametric max flow problem as cα(s, v) =αandcα(v, t) =αboth grow in the same direction\nand we cannot use the results of [ 34]. Nevertheless, we are able to produce the same results as [ 18] for our setting,\nand essentially show that the approach of Zhang can be generalized to work for the r-move 2-partitioning problem.\nAssuming that r∗≤ris the optimal number of nodes moved in r-move 2-partitioning, we first give a simple (r∗+ 1) -\napproximation algorithm for the r-move 2-partitioning problem without using linear programming. In Subsection 6.2,\nwe include a more sophisticatedr+1\nr+1−r∗-approximation algorithm for the r-move 2-partitioning problem.\n6.1 An (r∗+ 1)-Approximation Algorithm\nLetGbe an instance of the r-move 2-partitioning problem. We first define additional graphs constructed based on G\nand continue by stateing a few observations about them. Suppose that in the initial partitioning, terminal sis in cluster 1\nand terminal tis in cluster 2. For any α≥0, we construct graph Gαas follows: Take graph Gand add edges from\nterminal sto any node vin cluster 1with weight α. If there already exists an (s, v)edge in G, then simply add αto the\nweight of this edge. Similarly, add edges from any node vin cluster 2to terminal twith weight α, and if the edge (v, t)\nexists in G, then add αto its weight*.\nLet the value of the initial cut of Gbec∞. Note that if we set α≥2c∞, then any cut other than the initial cut will\nhave value at least 2c∞, so the min s-tcut for G2c∞is the initial cut. If we let α= 0, then the min s-tcut for G0is the\nmins-tcut for G.\nAny cut in graph Gresults in a set of nodes S⊆V(G)\\ {s, t}being moved from their initial partitions. We say\nthat a cut is induced, formed orproduced by set S⊆V(G)\\ {s, t}if this cut leads to the set of nodes Sbeing moved\nto partitions different from their initial ones. For example, when S=∅, the cut formed by Sis the initial cut of the\ngraph Gand when S={v}, the cut induced by Smoves node v, and only node v, from its initial partition to the other\none. We denote the size of a set Sbyr, that is, |S|=r.\nFor any set S⊆V(G)\\ {s, t}, letδ(S)be the value of the cut formed by Sin the original graph G. Then, if the\nmins-tcut in Gαmoves nodes in set Sα, the value of this cut in Gαiscα=rαα+δ(Sα), where rα=|Sα|. Note that\nthere might be several sets that produce a min s-tcut for Gαand we consider one such set arbitrarily as Sα. Observe\nthat if SandS′are both min s-tcuts in Gαsuch that |S′|=|S|, then δ(S) =δ(S′), i.e., they have the same cut value\ninG. The following lemma will help us in developing our approximation algorithm for the problem.\nLemma 6.1 Suppose the min s-tcuts in graphs GαandGα′move sets of nodes SαandSα′from their initial partitions,\nrespectively. We have α≥α′if and only if |Sα| ≤ |Sα′|. Moreover, if we have |Sα|<|Sα′|then it implies that\nδ(Sα)> δ(Sα′).\nProof. Since SαandSα′produce min s-tcuts for GαandGα′, respectively, we have the following two inequalities:\nα|Sα|+δ(Sα)≤α|Sα′|+δ(Sα′)\n*Note that the αidefined here is different from the αiused in [18].\n15α′|Sα′|+δ(Sα′)≤α′|Sα|+δ(Sα).\nBy reordering and combining these inequalities we get\nα′(|Sα′| − |Sα|)≤δ(Sα)−δ(Sα′)≤α(|Sα′| − |Sα|). (4)\nFrom these two inequalities we have (α−α′)(|Sα′| − |Sα)|)≥0. Thus, α−α′and|Sα′| − |Sα|have the same\nsigns and this proves the first part of the lemma. Moreover, by inequalities (4)we have |Sα|<|Sα′|implies that\nδ(Sα)> δ(Sα′). □\nCorollary 6.1 Letα < α′. IfSinduces a min s-tcut for both GαandGα′, then for any ˆα∈[α, α′]setSforms a min\ns-tcut for graph Gˆα.\nProof. Suppose ˆSforms a min s-tcut for graph Gˆα. SetSproduces a min s-tcut for both GαandGα′; therefore,\nSα=Sα′=Sandδ(Sα) =δ(Sα′) =δ(S). Since ˆα > α′, by Lemma 6.1 we have |Sˆα| ≤ | Sα′|=|S|and\nδ(S) =δ(Sα′)≤δ(Sˆα). Similarly, since ˆα < α , we have |Sˆα| ≥ | Sα|=|S|andδ(S) =δ(Sα)≥δ(Sˆα).\nSubsequently, |Sˆα|=|S|andδ(S) =δ(Sˆα). Therefore, Sinduces a min s-tcut for graph Gˆα. □\nLemma 6.1 states that as αgets larger, the set of nodes that are moved from their initial partitions gets smaller and\nthe value of the associated cut in Ggets larger. Using this intuition, we define breakpoints as the different number\nof nodes moved to partitions different from their initial ones by min s-tcuts on graphs G0, . . . , G2c∞. Definition 6.1\nformally remarks the definition of breakpoints.\nDefinition 6.1 (Breakpoints) For some 0≤h≤n−1, a set{r0, . . . , r h}is called a set of breakpoints with each ri\nbeing one breakpoint if r0> r1> . . . > r h= 0and there are sets S0, S1, . . . , S hsuch that |Si|=ri. Furthermore,\nfor any α≥0, there is an index 0≤i≤hsuch that Siis a min s-tcut of Gα, and for any i, there exists at least an\nα≥0such that Siis a min s-tcut of graph Gα.\nNote that for any graph Gthere might be more than one set of breakpoints. We show that by running O(h)∈O(n)\nmax flow algorithms we can compute a set of breakpoints and their associated sets, i.e., sets of ri’s and Si’s for all\n0≤i≤h, respectively. This algorithm is demonstrated in Algorithm 5.\nAlgorithm 5 Computing breakpoints\n1:c∞←the value of the initial cut of G.\n2:Run the min s-tcut algorithm on Gto get a set of nodes S0that are moved from their initial partitioning.\n3:ifS0=∅then\n4: return {0},{∅}.\n5:end if\n6:R← {0,|S0|}. ▷The set of breakpoints.\n7:R′← {∅ , S0}. ▷The set of sets associated to the breakpoints.\n8:I← {(∅, S0)}. ▷The set of active pairs.\n9:while I̸=∅do\n10: (S′, S)←an active pair from I. Letr=|S|andr′=|S′|.\n11: ¯α←δ(S′)−δ(S)\nr−r′. Run min s-tcut algorithm on G¯α. Let the value of the cut be c¯α.\n12: ifc¯α=r¯α+δ(S)then ▷Equivalently, r¯α+δ(S) =r′¯α+δ(S′).\n13: Remove (S′, S)from I.\n14: else\n15: Let the min s-tcut in G¯αbe produced by set S′′.\n16: AddS′′toR′and|S′′|toR.\n17: Remove (S′, S)from Iand add (S′, S′′)and(S′′, S)toI.\n18: end if\n19:end while\n20:return R, R′.\n16Lemma 6.2 Algorithm 5 finds the breakpoints for an instance Gof the r-move 2-partitioning problem in O(n)Tmin−cut(n),\nwhere Tmin−cut(n)is the running time of a min s-tcut algorithm on an n-node graph.\nProof. First, observe that if the initial partitioning in Gis a min s-tcut of the graph, then S0=∅and for any α≥0\nsetS0induces a min s-tcut of Gα. Note that if the initial partitioning in Gis a min s-tcut, then S0induces the min s-t\ncut for G0andG2c∞. By Corollary 6.1, set S0induces a min s-tcut for any graph Gαforα∈[0,2c∞]. Moreover, for\nanyα≥2c∞it is easy to see that that ∅induces a min s-tcut for Gα. Therefore, in this case there exists only one\nbreakpoint, that is 0. In the rest of this proof, we assume that S0̸=∅.\nThe algorithm starts with an incomplete set of breakpoints R={0,|S0|}, the set of sets associated with these\nbreakpoints R′={∅, S0}and the set of active pairs I={(∅, S0)}. At each iteration of the algorithm, we start with a\npair of sets of nodes (S′, S)that we call an “active pair” such that S′, S∈Rand|S′|<|S|. This active pair is either\nbroken into two active pairs and a new breakpoint is added to Rby the algorithm, or the active pair is removed from the\nset. We show that the set of all values αsuch that SorS′induce a min s-tcut for Gαis a contiguous interval. Recall\nthatS0forms a min s-tcut of G0and for any α≥2c∞, we have that ∅induces a min s-tcut of Gα. Since we start\noff with an active pair (∅, S0), for any α≥0the algorithm returns a set Sαsuch that Sαforms a min s-tcut of Gα.\nWhenever the algorithm adds a new breakpoint to set R, the number of active pairs in Iincreases by one. Note that the\nnumber of breakpoints cannot be larger than n, which means that the total number of active pairs that are processed in\nthe algorithm cannot be larger than n+ 1. So, the running time of the algorithm is O(n)Tmin−cut(n).\nConsider an active pair (S′, S)and let r=|S|andr′=|S′|such that r′< r. Suppose that there exists an α\nand an α′such that SandS′form min s-tcuts in graphs GαandGα′, respectively. Since r′< r, by Lemma 6.1\nwe have we have α′> α. Let ¯α= (δ(S′)−δ(S))/(r−r′). Since set S′forms a min s-tcut in graph Gα′, we\nhaveα′r′+δ(S′)< α′r+δ(S)which infers ¯α < α′. With similar calculations, it can be shown that α < ¯α; thus,\nα <¯α < α′. We then run the min s-tcut algorithm on G¯α. If the value of a min s-tcut in G¯αis smaller than r¯α+δ(S),\nthen let S′′be the set forming the min s-tcut in G¯α. Since α <¯α < α′, using Lemma 6.1, we have that r′<|S′′|< r.\nSo we add |S′′|as another breakpoint to Rand its associated set S′′toR′. We also need to do an iteration of the\nalgorithm on (S′, S′′)and another iteration on (S′′, S). To do that, we add these two pairs to the set of our active pairs\nIand remove (S′, S)from it.\nNow, consider the case that the value of a min s-tcut in G¯αisc¯α=r¯α+δ(S). From the definition of ¯α, we also\nhavec¯α=r′¯α+δ(S′). Letαandα′be two values such that SandS′induce min s-tcuts of GαandGα′, respectively.\nFirst, since r′< r, by Lemma 6.1, we have that α′≥α. We argue that in this case there is no breakpoint between\nrandr′, in other words, for any α′′∈[α, α′]either SorS′induces a min s-tcut of Gα′′. More precisely, since S\ninduces a min s-tcut for both GαandG¯α, ifα′′∈[α,¯α]then by Corollary 6.1 Sinduces a min s-tcut for Gα′′, as\nwell. Similarly, since S′induces a min s-tcut for both Gα′andG¯α, ifα′′∈[¯α, α′], then by Corollary 6.1 S′induces a\nmins-tcut for Gα′′, too. Therefore, there is no need to search for additional breakpoints between randr′and we can\nsafely remove (S′, S)from the set of active pairs.\nWe show that when the algorithm terminates, for any α≥0there is a set Sα∈R′such that Sαproduces a min\ns-tcut of Gα. Ifα≥2c∞, then S=∅has this property. If α= 0, then S=S0satisfies this condition. It only\nremains to show that this claim holds for any 0< α < 2c∞. When the algorithm terminates, let r0> . . . > r h= 0\nbe the breakpoints in Rwith corresponding sets S0, . . . , S h, where ri=|Si|. For each 0≤i≤h, letα′\nibe the value\nfor which Siforms a min s-tcut for Gα′\ni, and let α0= 0 andαh= 2c∞. There is an index 0≤i < h such that\nα′\ni≤α≤α′\ni+1. Since there is no breakpoint between riandri+1, there is a stage of the algorithm where the active\npair(Si+1, Si)gets processed and then gets removed from I. So from the argument in the previous paragraph, either Si\norSi+1produces a min s-tcut for Gα. Finally, note that if the algorithm adds a breakpoint r′′with associated set S′′to\nR, then S′′induces a min s-tcut of G¯αfor some ¯α≥0, concluding that set Rcontains all required breakpoints and\nevery sets associated with the breakpoints are included in R′. □\nBefore proceeding to our algorithm, we prove an important property of the breakpoints.\nLemma 6.3 LetR={r0, . . . , r h}be a set of breakpoints with associated sets S0, . . . , S h. For any 0≤i < h , let\nαi= (δ(Si)−δ(Si+1))/(ri+1−ri). We have that both SiandSi+1induce min s-tcuts for Gαi.\nProof. Since Ris the set of breakpoints, there exist α′\niandα′\ni+1such that SiandSi+1induce min s-tcuts for Gα′\niand\nGα′\ni+1, respectively. Therefore, we have α′\niri+δ(Si)≤α′\niri+1+δ(Si+1)andα′\ni+1ri+1+δ(Si+1)≤α′\ni+1ri+δ(Si),\n17which infer that α′\ni≤αi≤α′\ni+1. Moreover, since Ris a set of breakpoints and sets S0, . . . , S hcorrespond to\nthe breakpoint, there exists an index 0≤j≤hsuch that Sjforms a min s-tcut for Gαi. By Lemma 6.1, from\nα′\ni≤αi≤α′\ni+1we have that ri≥rj=|Sj| ≥ri+1; therefore, j∈ {i, i+ 1}. Note that from the definition of αi, we\nhaveαiri+δ(Si) =αiri+1+δ(Si+1). Putting this together with the fact that the value of the min s-tcut in Gαiis\nαirj+δ(Sj), we have that both SiandSi+1induce min s-tcuts for Gαi. □\nOur algorithm for the r-move 2-partitioning problem works as follows: First, using Algorithm 5, we compute the\nbreakpoints r0> . . . > r h= 0and their associated sets S0, . . . , S hsuch that |Si|=rifor all 0≤i≤h. Letjbe the\nlargest index such that rj≤r. The s-tcut produced by Sjis returned as the approximate r-move 2-partitioning. Our\nalgorithm is represented in Algorithm 6. Note that, even though our algorithm looks similar to that of [ 18], its first step\nis computed differently. To analyze the algorithm, we first state a few lemmas that also appear in [ 18] and we show that\nthey hold in our setting, as well.\nAlgorithm 6 Approximation algorithm for the r-move 2-partitioning problem\n1:Using Algorithm 5, compute breakpoints r0> . . . > r h= 0 and their associated sets S0, . . . , S h, such that\n|Si|=ri.\n2:ifr≥r0then\n3: return S0.\n4:else\n5: j←the index for which rj−1> r≥rj\n6: return Sj.\n7:end if\nLemma 6.4 For graph G, letS0denote the set of nodes moved by a min s-tcut to partitions different from their initials\nones and let r0=|S0|. Moreover, suppose that for some 0≤h≤n−1,{r0, . . . , r h}is a set of breakpoints for\nG, with their associated sets being S0, S1, . . . , S h, such that |Si|=ri. Let r∗denote the number of nodes moved\nfrom their initial partitions by an optimal r-move 2-partitioning. The following properties hold for an optimal r-move\n2-partitioning in G:\n1. Ifr≥r0, then S0provides an optimal r-move 2-partitioning in G.\n2.Ifr < r 0and there exists some 0< i≤hsuch that ri=r∗, then the largest index jfor which rj≤risiand\nthe corresponding set Siinduces an optimal r-move 2-partitioning in G.\n3. Ifr < r 0andr∗/∈ {r0, . . . , r h}, then none of the values of r, r−1, . . . , r∗−1are among the breakpoints, i.e.,\nri’s.\n4. Ifr=rifor some 0≤i≤h, then set Siproduces an optimal r-move 2-partitioning in graph G.\nProof. We denote the set that induces the optimal r-move 2-partitioning for graph GbyS∗, where |S∗|=r∗. In the\nfirst property, it is assumed that r≥r0; therefore, set S0moves at most rnodes from their initial partitions. Thus, S0is\nan optimal r-move 2-partitioning in G.\nTo prove the second property, for the sake of contradiction, suppose that there exists an index jsuch that i < j≤h\nandr∗=ri< rj≤r. LetSiandSjbe the associated sets and let αandα′be the values such that SiandSjinduce\nmins-tcuts of GαandGα′, respectively. In other words, we have Si=Sα,Sj=Sα′,|Si|=riand|Sj|=rj. First,\nfrom the optimality of SiforGαwe have that αri+δ(Si)≤αr∗+δ(S∗). Since ri=r∗, we have δ(Si)≤δ(S∗).\nDue to the optimality of the cut induced by set S∗among all s-tcuts that move r∗nodes to partitions different from their\ninitial ones, we have δ(Si) =δ(S∗). Now since ri< rj, by Lemma 6.1 we have that δ(Si) =δ(Sα)> δ(Sα′) =δ(Sj)\n(Note that δ(Sα)is strictly greater than δ(Sα′)asriis strictly less than rj). Therefore, set Sjforms a smaller cut for G\nby moving at most rnodes, and this contradicts the optimality of the cut induced by set S∗and the second property\nholds.\nTo prove the third property, again for the sake of contradiction, suppose that there exists an r′such that r≥r′> r∗\nandr′=rifor some 0≤i≤h. Let Sibe the set associated to riand let αbe a value for which Siforms a min s-t\n18cut for graph Gα. We have αri+δ(Si)≤αr∗+δ(S∗). Since r′=ri> r∗, we have δ(Si)< δ(S∗). Since ri≤r,\nhaving δ(Si)< δ(S∗)contradicts the optimality of the s-tcut induced by set S∗and the third property holds.\nFor the fourth property, note that if r=r0, then by the first property set S0forms an optimal r-move 2-partitioning\nforG. Ifr=r∗=ri, then the second property gives us the result. If ri=r > r∗, then this contradicts the third\nproperty, so the only way that r=riis that r=r∗orr=r0. □\nTheorem 6.1 There is an (r∗+ 1) -approximation algorithm that solves the weighted instances of the r-move 2-\npartitioning problem, where r∗is the number of nodes moved from their initial partitions by an optimal solution. This\napproximation algorithm runs in time O(n)Tmin−cut(n), where Tmin−cut(n)is the running time of a s-tmin-cut\nalgorithm on an n-node graph.\nProof. We show that Algorithm 6 provides the claimed guarantees. Let Gbe the original graph of n-nodes. Let\n0≤h≤n−1and{r0, . . . , r h}be the set of breakpoints computed in step 1 of the algorithm, with their associated\nsets being S0, S1, . . . , S h, such that |Si|=ri. LetS∗denote the set that induces the optimal r-move 2-partitioning for\ngraph Gwith a cut value of δ(S∗), where |S∗|=r∗.\nIfr≤r0, then from the first property of Lemma 6.4 we have that S0provides an optimal r-move 2-partitioning of\nGwhich is the set that the algorithm outputs. So suppose that r > r 0.\nFrom the second and fourth properties of Lemma 6.4, we have that if r∗orrare among the breakpoints, then the\nalgorithm finds the set inducing the optimal cut. Otherwise, if rjis the largest breakpoint smaller than r, then we have\nrj< r∗≤r < r j−1. This is because if rj+1< r∗< rj≤r < r j−1, then by Lemma 6.1 set Sjforms a smaller cut\nthanS∗inG, contradicting the optimality of the s-tcut induced by S∗. The rest of this proof focuses on the case that\nrj< r∗≤r < r j−1.\nLetα=αj−1as defined in Lemma 6.3, such that for rj−1=|Sj−1|, rj=|Sj|we have αrj−1+δ(Sj−1) =αrj+\nδ(Sj). Since Sj−1andSjproduce min s-tcuts for Gα, we have that αr∗+δ(S∗)≥αrj−1+δ(Sj−1) =αrj+δ(Sj),\nproviding the following two inequalities\nδ(S∗)−δ(Sj−1)≥α(rj−1−r���), (5)\nδ(Sj)≤δ(S∗) +α(r∗−rj). (6)\nSince rj−1> r∗, from inequality (5)we have δ(S∗)−δ(Sj−1)≥α(rj−1−r∗)≥α, providing α≤δ(S∗). Combining\nthis with inequality (6), we have δ(Sj)≤δ(S∗)(1 + r∗−rj)≤δ(S∗)(r∗+ 1)≤δ(S∗)(r+ 1) . Therefore, the output\nof Algorithm 6, i.e., Sj, produces an (r∗+ 1) -approximation solution for the r-move 2-partitioning problem. By\nLemma 6.2, the running time of step 1 of Algorithm 6 is O(n)Tmin−cut(n). The rest of the algorithm works in linear\ntime, providing an overall running time of O(n)Tmin−cut(n). □\n6.2r+1\nr+1−r∗-Approximation\nWe show that Algorithm 6 has an approximation factor ofr+1\nr+1−r∗.\nTheorem 6.2 Given a positive integer r, there is an approximation algorithm that solves the weighted n-node m-edge\ninstances of the r-move 2-partitioning problem with an approximation factorr+1\nr+1−r∗, where r∗is the number of\nnodes moved from their initial partitions by an optimal solution. The algorithm runs in O(n)Tmin -cut(m)time, where\nTmin -cut(m)is the running time of the s-tmin-cut algorithm on an m-edge graph.\nNote that when k= 2, in LP 2 we can have one variable xvinstead of a vector Xv, and if the terminals are\nrepresented by sandt, then we let xv= 1stand for the case that node vis assigned to the same partition as terminal s,\nandxv= 0otherwise (i.e., the case that node vis allocated the same partition as terminal t). By defining yuv=|xu−xv|\nwe have d(u, v) =yuv. Let cluster 1be the cluster that terminal sis in and cluster 2be the cluster that terminal tis\nin. Using these notation we can simplify LP 2 to LP 4, see Table 4. Note that if node vis initially in cluster 1, that is,\nℓv= 1, then (1−xv)is1ifvhas moved to cluster 2and it is zero otherwise. If node vis initially in cluster 2, that is,\nℓv= 2, then xvis1ifvhas moved to cluster 1and zero otherwise.\nNext, using a non-negative parameter α′we formulate the Lagrangian relaxation of LP 4 to obtain LP 5, see Table 5.\nWe can show that LP 5 has integer optimal solutions by showing that it is in fact the min s-tcut LP for graph Gα′.\n19Minimize:P\n(u,v)∈E(G)cuvyuv\nSubject to:\nyuv≥xu−xv ∀(u, v)∈E(G)(C1)\nyuv≥xv−xu ∀(u, v)∈E(G)(C2)\nxs= 1 (C3)\nxt= 0 (C4)\n0≤xv ∀v∈V(G) (C5)P\nv∈V(G),ℓv=1(1−xv) +P\nv∈V(G),ℓv=2xv≤r (C6)\nTable 4: r-move 2-partitioning LP (LP 4)\nRecall that as discussed in Section 6, to obtain a graph Gαwe add αweighted edges (s, v)and(t, u)for each vandu\nwithℓv= 1andℓu= 2. If for each vanduwe let ysv= 1−xv=|xs−xv|andytu=xu=|xt−xu|, respectively,\nthen we can rewrite LP 5 as LP 6, see Table 6. Since LP 6 has integer optimal solutions [ 35,18], LP 5 also has integer\noptimal solutions.\nThe rest of the proof follows similar to [ 18], which we state here for completeness. Recall that S∗is an optimal\nsolution to the r-move 2-partitioning problem with |S∗|=r∗andδ(S∗)is the value of cut induced by S∗in the original\ngraph G.\nMinimize: α′(P\nv∈V(G),ℓv=1(1−xv) +P\nv∈V(G),ℓv=2xv) +P\n(u,v)∈E(G)cuvyuv\nSubject to:\nyuv≥xu−xv ∀(u, v)∈E(G)(C1)\nyuv≥xv−xu ∀(u, v)∈E(G)(C2)\nxs= 1 (C3)\nxt= 0 (C4)\n0≤xv ∀v∈V(G) (C5)\nTable 5: Lagrangian of LP 4 (LP 5)\nMinimize:P\n(u,v)∈E(Gα′)cuvyuv\nSubject to:\nyuv≥xu−xv ∀(u, v)∈E(Gα′)(C1)\nyuv≥xv−xu ∀(u, v)∈E(Gα′)(C2)\nxs= 1 (C3)\nxt= 0 (C4)\n0≤xv ∀v∈V(Gα′) (C5)\nTable 6: Min s-tcut LP for Gα′(LP 6)\nLemma 6.5 LetSjbe the set of nodes that Algorithm 6 outputs. If for some λ >0we have |Sj−1| ≥1\n1−λr∗, then\nδ(Sj)≤1\nλδ(S∗).\nProof. By Lemma 6.3 we have that for α′:=αj−1=δ(Sj)−δ(Sj−1)\nrj−1−rj, both SjandSj−1produce min s-tcuts for graph\nGα′. LP 5 and LP 6 provide integer optimal solutions to the min s-tcut problem on graph Gα′. Therefore, the s-t\ncuts produced by sets SjandSj−1are both optimal solutions of LP 5 and LP 6 for parameter α′. Let the LP 5 integer\nsolutions associated with sets Sj−1andSjbe(x−, y−)and(x+, y+), respectively. Any linear combination of these\n20two solutions is also an optimal solution of LP 5. We choose γ∈(0,1)such that\n(1−γ)(X\nv,ℓv=1(1−x−\nv) +X\nv,ℓv=2x−\nv) +γ(X\nv,ℓv=1(1−x+\nv) +X\nv,ℓv=2x+\nv) =r∗. (7)\nNote that since rj−1=P\nv,ℓv=1(1−x−\nv) +P\nv,ℓv=2x−\nv,rj=P\nv,ℓv=1(1−x+\nv) +P\nv,ℓv=2x+\nvandrj≤r < r j−1,\nsuchγexists. Let (x∗, y∗)be the solution obtained by taking a linear combination of (x−, y−)and(x+, y+)with\nweight γ. More formally,\n(x∗, y∗) = (1 −γ)(x−, y−) +γ(x+, y+) (8)\nr∗=X\nv,ℓv=1(1−x∗\nv) +X\nv,ℓv=2x∗\nv (9)\nSince both (x−, y−)and(x+, y+)are optimal solutions of LP 5, it follows that (x∗, y∗)is also an optimal solution\nto LP 5. We show that (x∗, y∗)is an optimal solution to LP 4, as well. To see this, suppose that (x′, y′)is an optimal\nsolution to LP 4, while (x∗, y∗)is not an optimal solution to LP 4. Then, we haveP\nv,ℓv=1(1−x′\nv)+P\nv,ℓv=2x′\nv≤r∗\nandP\ne∈E(G)cey′\ne<P\ne∈E(G)cey∗\ne. Combining this with equation (9) we have\nα′(X\nv,ℓv=1(1−x′\nv) +X\nv,ℓv=2x′\nv) +X\ne∈E(G)cey′\ne< α′(X\nv,ℓv=1(1−x∗\nv) +X\nv,ℓv=2x∗\nv) +X\ne∈E(G)cey∗\ne,\nimplying that (x′, y′)is a solution to LP 5 with a better objective value than (x∗, y∗), which is a contradiction. Thus,\n(x∗, y∗)is an optimal solution to LP 4 and we have that\nX\ne∈E(G)cey∗\ne≤δ(S∗). (10)\nBy equation (7), we have that |Sj−1|=P\nv,ℓv=1(1−x−\nv) +P\nv,ℓv=2x−\nv≤r∗/(1−γ). By the lemma’s assumption,\nwe have |Sj−1| ≥r∗/(1−λ); therefore, it holds that γ≥λ. Using equation (8), we have that y∗\ne≥γy+\ne≥λy+\ne, and\nso\nδ(Sj) =X\ne∈E(G)cey+\ne≤1\nλX\ne∈E(G)cey∗\ne≤1\nλδ(S∗),\nwhere the last inequality uses inequality (10). □\nTo finish the proof of Theorem 6.2, let λ= 1−r∗\nr+1. Then |Sj−1| ≥1\n1−λr∗=r+1, which holds by the construction\nof Algorithm 6. Therefore, by Lemma 6.5 we have δ(Sj)≤1\nλδ(S∗). Note that1\nλ=r+1\nr+1−r∗, concluding the proof of\nthe theorem.\n7 HARDNESS RESULTS\nIn this section, we state our hardness results, including lower bounds on the integrality gap of LP 2 and our W[1]-\nhardness findings.\n7.1 Integrality Gap of the r-Move k-Partitioning Linear Program\nThe following lemma discusses the integrality gap of the r-move k-partitioning linear program (i.e., LP 2).\nLemma 7.1 LP 2 has an integrality gap of at least r+ 1.\nProof. LetP1: (v1, . . . , v r+2)be a path in graph G= (V, E)such that node v1is in partition 1and the rest of the\nnodes on P1belong in partition 2, see Figure 1. Let the weight of the edge (v1, v2)beϵand the weight of all of the\nother edges on P1be one. Let P2: (vr+3, . . . , v n)be a path with edges all of weight one. All nodes of P2belong in\n21s1=v1 v2v3 vr+2 vr+3vr+4\nvn=s2\nP2 P1... ...\n/epsilon1Figure 1: The example graph for the proof of Lemma 7.1.\npartition 2. Letv1andvnbe the terminals for partition 1and2, respectively. The defined partitions introduce a cut that\nhas a weight of ϵ. Moving any node from P2to partition 1only increases the cut value, so any optimal integer solution\nkeeps all the nodes in P2in their initial partition. Moving any proper subset of nodes of P1to partition 1increases the\ncut value to at least one. Furthermore, one cannot move all of the nodes in P1to partition 1without violating the move\nconstraint. So an optimal integer solution to LP 2 does not move any node in this graph and has a cut value of ϵ.\nOn the other hand, any (fractional) optimal solution of LP 2 on this graph has a cut value of at mostϵ\nr+1: For any\nnodevi,i∈ {2, . . . , r + 2}, this is achieved by setting X1\nvi=r\nr+1andX2\nvi=1\nr+1. Let the nodes in P2stay in partition\n2, i.e., X1\nvi= 0, X2\nvi= 1fori∈ {r+ 3, . . . , n }. The value of the cut in this solution isϵ\nr+1. So the integrality gap is at\nleastr+ 1. Note that this proof can be generalized to any k >2by adding dummy partitions with singletons in them. □\nSince the construction of Lemma 7.1 is in fact a Min r-sizes-tcutconstruction, we have the following corollary.\nCorollary 7.1 The linear program of the Min r-sizes-tcut problem stated in [ 18] has an integrality gap of at least\nr+ 1.\n7.2 The r-Move k-Partitioning Problem is W[1]-hard\nWe begin by introducing two new notations here that will be useful for the proofs of this section and then discuss\nthe computational complexity of the r-move k-partitioning problem. For a graph Gand any subsets of its nodes\nC1, C2⊆V(G), letE(C1, C2)refer to the set of edges between subsets C1andC2. For any node v∈C1, we define\ndegC1(v)as the number of neighbors of node vthat are in C1; that is, the number of nodes in C1that share an edge\nwith node v.\nTheorem 7.1 Given an n-node graph Gas an instance of the densest r-subgraph problem, there is an O(n2)-node\ngraph G′with initial partitions {A, B}such that the densest r-subgraph of Ghasm∗edges if and only if the optimal\nsolution of the r-move 2-partitioning problem on G′reduces the initial cut value by 2m∗.\nProof. We reduce the densest r-subgraph problem to the r-move 2-partitioning problem. Let Gbe an instance\nof the densest r-subgraph problem that has nnodes, medges and unit edge weights. Using G, we construct a graph\nG′as an instance of the r-move 2-partitioning problem. Graph G′consists of two subgraphs AandB, where Ais\nan exact copy of graph G(with nnodes and medges) and Bis a clique of size 2n2+n. Furthermore, mnodes in\nBare reserved for the medges of A; in other words, for each edge e∈E(A)there exists a node e∈V(B). Then,\nthe endpoints of each edge e= (u, v)∈E(A)gets connected to node e∈V(B). Next, we add a terminal node tto\nsubgraph Aand connect it to all nodes in A. Similarly, a terminal node sis added to subgraph Band it gets connected\nto all nodes in V(G′). Note that after the addition of node s, subgraph Bbecomes a 2n2+n+ 1clique. This finishes\nthe construction of graph G′, see Figure 2. Let the initial cut induced on G′split the graph into two partitions AandB.\nThis cut has a value 2m+n < n2+ 1.\nFirst, we show that if we move a subset of nodes X⊆V(A)\\ {t}(along with all edges incident to the nodes in\nX) to subgraph B, then the cut value reduces by 2\f\fE(X)\f\f. Note that, if we move a set of nodes X⊆V(A)\\ {t}to\nsubgraph B, then the cut value changes by\f\fE\u0000\nX, V(A)\\X\u0001\f\f−\f\fE\u0000\nX, V(B)\u0001\f\f. Since subgraph induced by V(A)\\{t}\nis exactly the same as graph G, we have\n\f\fE\u0000\nX, V(A)\\X\u0001\f\f=\f\fE\u0000\nX,{t}\u0001\f\f+\f\fE\u0000\nX, V(A)\\(X∪ {t})\u0001\f\f\n=\f\fX\f\f+X\nv∈X\u0002\ndegG(v)−degX(v)\u0003\n.\n22u\nv\ne\nt\ns\nA\nB\nClique\ncopy of G\njE(G)jnodes\nG0\neFigure 2: Reduction graph G′created from a densest r-subgraph instance G. The solid-line edges do not depend on G.\nFor each edge e= (u, v)inG, nodes uandvinAare connected to a node einB.\nNext, we have\n\f\fE\u0000\nX, V(B)\u0001\f\f=\f\fE\u0000\nX,{s}\u0001\f\f+\f\fE\u0000\nX, V(B)\\ {s}\u0001\f\f\n=\f\fX\f\f+X\nv∈XdegG(v);\nthus,\f\fE\u0000\nX, V(A)\\X\u0001\f\f−\f\fE\u0000\nX, V(B)\u0001\f\f=−P\nv∈XdegX(v) =−2\f\fE(X)\f\f.\nNow, let X∗be the densest r-subgraph of G. If we move the copy of X∗inAfrom AtoB, then the cut value\nreduces by 2m∗= 2\f\fE(X∗)\f\f, as shown above. To prove the other direction, we show that if there exists a set of nodes\nX⊆V(G′)such that |X| ≤rand moving each node in Xto a partition different from the initial one it was assigned\nto reduces the cut by at least 2m∗, then there exists an r-subgraph in Gwithm∗edges. To see this, first note that X\ncannot have any nodes from subgraph B. This is because the resulting cut would have a value at least 2n2+ 1which is\nbigger than the value of the initial cut; consequently, X⊆V(A). As shown earlier in this proof, by moving XfromA\ntoBthe cut value is reduced by 2\f\fE(X)\f\f, so\f\fE(X)\f\f≥m∗. Therefore, the equivalent set of XinGhas at least m∗\nedges, hence m∗=\f\fE(X)\f\f. Finally, the time it takes to build graph G′isO\u0000\f\fV(G′)\f\f+\f\fE(G′)\f\f\u0001\n⊆O(n4). □\nTheorem 1.1 Ther-move k-partitioning problem with parameter ris W[1]-hard.\nProof. For the r-move 2-partitioning problem, this comes from Theorem 7.1 and the W[1]-hardness of the densest\nr-subgraph problem [ 21]. For the r-move k-partitioning problem with k >2, by making the following modifications to\ngraph G′we can make the same reduction as that of the proof of Theorem 7.1 work: adding k−2dummy partitions to\ngraph G′, each containing one terminal that is not connected to any other node. Then, if Yis the set of nodes that are\nmoved to one of these dummy partitions P, then by moving Yto one of the two main partitions the cut value will not\nincrease. This is because Yhas no edges to the terminal node in P. Therefore, all solutions consist of moving nodes to\none of the two main partitions. □\nHardness of the Min r-sizes-tcut problem: We now slightly change the above construction to fit to the Min r-size\ns-tcut problem.\nTheorem 7.2 Given an n-node graph Gas an instance of the densest r-subgraph problem, there is an O(n2)-node\ngraph G′and a value c(G′)such that the densest r-subgraph of Ghasm∗edges if and only if the optimal solution of\nthe Min r-sizes-tcut problem on G′has a value of c(G′)−2m∗.\n23u\nv\nt\nA\ncopy of G\ne\ns\nG0Figure 3: Reduction graph G′created from a densest r-subgraph instance G.\nProof. LetAbe a copy of the graph G. Add a terminal node ttoAand connect it to all nodes in Awith unit-weight\nedges. Partition Bconsists of only one node, terminal s. For each v̸=tinA, connect node vto terminal swith\nan edge of weight 1 + degG(v), see Figure 3. Let c(G′)be the cut value of partitions {A, B}which is equal to\n2\f\fE(G)\f\f+\f\fV(G)\f\f.\nHere, we show that moving any subset of nodes X∈V(A)\\ {t}from partition AtoBreduces the value of the cut\ninduced by partitions {A, B)}by2\f\fE(X)\f\f. To see this, note that after moving nodes Xfrom AtoB, the cut value is\nreduced by\f\fE\u0000\nX, V(A)\\X\u0001\f\f−\f\fE\u0000\nX, V(B)\u0001\f\f=\u0000\n|X|+P\nv∈XdegG(v)−degX(v)\u0001\n−\u0000P\nv∈X(degG(v)+1)\u0001\n=\n−2\f\fE(X)\f\f. Suppose X∗is the densest r-subgraph in graph Gandm∗=\f\fE(X∗)\f\f. Moving X∗to partition Breduces\nthe cut by 2m∗. If the optimal solution to the Min r-sizes-tcut problem on G′is to move a set Xfrom AtoB, then\nthis solution has a cut value of c(G′)−2\f\fE(X)\f\f, and so we must have that\f\fE(X)\f\f≥m∗. Since |X| ≤randX∗is\nthe densest r-subgraph in G, we must have\f\fE(X)\f\f=m∗. Note that if |X|< r, then we can add arbitrary nodes to X\nto make it have size rnodes without reducing its number of edges. □\nFrom the above theorem and the W[1]-hardness of the densest r-subgraph problem we have that the weighted Min\nr-sizes-tcut problem is W[1]-hard, thus, we have proved the following Corollary.\nCorollary 7.2 The Min r-sizes-tcutproblem is W[1]-hard.\nHardness of the r-move k-partitioning problem without terminals: In this paper, we mostly consider the r-move\nk-partitioning problem as a variant of the Multiway cut problem, i.e., we assume that the input graph has terminals that\ncannot be moved. However, one might wonder what the computational complexity of the r-move k-partitioning problem\nwithout terminals is. All our algorithmic results carry over to the r-move k-partitioning problem without terminals,\nsince one can reduce the r-move k-partitioning problem without terminals to the r-move k-partitioning problem by\nadding dummy terminals for each partition as singletons. However, the main question in considering the r-move\nk-partitioning problem without terminals is whether it can be solved faster than the r-move k-partitioning problem,\nsince in many partitioning problems the “with terminal” version of the problem is harder than the “without terminal”\nversion. We show that our reduction works for the r-move k-partitioning problem without terminals as well, so this\nproblem is also W[1]-hard. Thus, one cannot expect the complexity to change drastically by removing the terminals.\nTheorem 7.3 Ther-move k-partitioning problem without terminals is W[1]-hard.\nProof. Given a densest r-subgraph instance G, our reduction graph G′is the same as that of Theorem 7.1 except\nthat we do not add terminal nodes sandtto the graph. The key observation here is that in the proof of Theorem 7.1\nhaving terminal nodes sandtdo not provide us with any specific benefit. We give a high level overview of the proof\nand the details can be easily derived from the proof of Theorem 7.1. We argue that moving a set of nodes Xfrom\npartition AtoBreduces the cut value by 2|E(X)|. Then we can show that if the cut value is reduced by 2m∗after\nmoving at most rnodes, then these nodes must all be in A. This is because the nodes in Bcreate a clique and moving\nany of them to Aincreases the cut value. Moreover, the set of nodes that are moved must induce a densest r-subgraph\ninGin order for them to be an optimal solution to the r-move k-partitioning problem in G′. □\n24Figure 4: Performance of rounding (Algorithm 2) and greedy algorithms with respect to the solution of LP 2 for the\nexplained stochastic block graphs.\n8 Numerical Evaluation\nWe conduct a simple empirical assessment of our rounding algorithm (Algorithm 2) of Theorem 1.3 and FPTAS of\nTheorem 1.2 We remark that this experiment is notan extensive empirical evaluation of the algorithms, and the focus of\nit is on the results that are not well explained by theory, namely that the approximation factor of Algorithm 2 in practice\nis better than what Theorem 1.3 demonstrates. This suggests that there could be alternative analysis of our algorithm\nwhich results in better approximation guarantees for certain classes of graphs.\n8.1 Experiments on Algorithm 2\nSet-up: We generate our sample graphs using the stochastic block model on 90nodes as follows: First the 90nodes\nare divided into three equally sized clusters. Then, between any two nodes in the same cluster we add an edge with\nprobability pH. Similarly, between any two nodes in different clusters we put an edge with probability pL. We let\npH= 0.3andpL= 0.1. Note that this partitioning is very close to the optimal k-partitioning of the graph for k= 3;\nhence, we re-partition the constructed graph into three partitions uniformly at random. These new random partitions are\nthen set as the initial partitioning of the graph. Following these steps, we make 100random such graphs in total. As our\nbenchmark, we consider the following simple greedy algorithm: The greedy algorithm has at most rrounds. At each\nround, the algorithm moves the node that decreases the value of the 3-cut by the largest amount. If at any point there is\nno such node, then the greedy algorithm halts. For each graph, we run Algorithm 2 for 30different values of parameter\nρand take the 3-cut with the smallest value as the output.\nResults: For each of the 100 random graph and each of the two algorithms (greedy and Algorithm 2), we compute the\nfollowing ratios: The output of the algorithm divided by the objective value of LP 2. For each value of rin[45,60],\nwe compute the average value of this ratio over all graphs, see Figure 4. For smaller values of r, we observe that both\nalgorithms output similar cut values and they demonstrate similar performances. This could be due to the small size of\nour sample graphs and we believe the difference between the performances of these two algorithms is more evident\nwith a larger sample size of graphs. While the greedy algorithm proves to perform reasonably well when the move\nparameter is bounded, we show that our FPTAS algorithm can beat greedy in this case. As rapproaches 60and above,\nthe LP solution is integer in most instances; hence, the Algorithm 2 does not play a significant role*.\n*One can see that if ˆris the number of nodes needed to be moved in order to get the optimal k-cut solution (in the absence of any budgetary\nconstraints), then E[ˆr]is near 60.\n25Figure 5: Performance of the FPTAS with two different values of αand greedy algorithm with respect to the solution of\nLP 2 for the email-Eu-core network graph.\n8.2 Experiments for FPTAS\nSet-up: The graph in this set of experiments is based on email data from a large European research institution [ 36].\nThe graph has 1005 nodes and each node belongs to one of the 42 departments at the research institute. There is an edge\nbetween to nodes in the graph if those corresponding employees sent at least one email to each other. That is, emails\nare used as a proxy for communication between employees. There are 25,571 edges in the network. To downscale the\nsize of the experiments, we take the induced sub-graph of the three largest departments. We take each department as\na partition, thus the induced graph comes with a 3-partitioning. We then choose 5 to 10 nodes nodes of this graph,\nuniformly at random, and move them to a random partition. We perform this procedure 20 times, i.e., 20 different\ngraphs are generated from the initial graph with 3-partitioning. On each of these graphs we run the greedy algorithm\nand the FPTAS to find the cut size resulted by each of these policies. We report the averaged the cut size for each policy\nas the cut size obtained from that method. Figure5 shows the performance of the greedy algorithm and the FPTAS for\ntwo values of αwith respect to the optimal solution of the problem which is computed by solving an Integer Program\n(which relaxes to LP 2).\nResults: We observe that for small values of r(that is, r≤3) the greedy algorithm has better performance of either\nFPTAS’s. However, as rincreases, the performance of the greedy algorithm deteriorates, while the performance of\nthe FPTAS’s stays steady or even improves. This is not surprising as with small values of r, the greedy algorithm\nhas a limited number of moves to make and acting myopically most likely results in a good outcome. However, with\nlarger values of rit becomes more likely that a myopic initial move take the greedy algorithm farther from the optimal\nsolution. Furthermore, as expected, we see that smaller αprovides a better performance for the FPTAS, which comes at\nthe expense of a higher running time.\n9 CONCLUSION\nThis paper studies the r-move k-partitioning problem. We show that this problem is W[1]-hard and give simple and\npractical approximation algorithms for it. These algorithms are: an FPTAS for constant r, a3(r+ 1)-approximation\nalgorithm and an (O(1), O(1)) bicriteria algorithm for general r. Our main results focus on LP rounding techniques.\n26There remains several interesting open problems to understand the complexity of the r-move k-partitioning problem,\nsome of them are listed below.\n(1) Is there an approximation algorithm for the r-move k-partitioning problem whose running time and approx-\nimation factor are independent of r? Recall that there exists an O(log(n))-approximation algorithm for the Min\nr-size s-tcut problem using tree decomposition techniques. Could one generalize this algorithm to the r-move\nk-partitioning problem?\n(2) Can one find any approximation lower bound for this problem? Note that similar partitioning problems, such as\nthe MinSBCC problem [33], do not have any approximation lower bounds yet.\nReferences\n[1]Sven Dickinson, Marcello Pelillo, and Ramin Zabih. Introduction to the special section on graph algorithms in\ncomputer vision. IEEE Transactions on Pattern Analysis & Machine Intelligence , 23(10):1049–1052, 2001.\n[2]Bruce Hendrickson and Tamara G Kolda. Graph partitioning models for parallel computing. Parallel computing ,\n26(12):1519–1534, 2000.\n[3]Thorsten Joachims. Transductive learning via spectral graph partitioning. In Proceedings of the 20th international\nconference on machine learning (ICML-03) , pages 290–297, 2003.\n[4]Andrew B Kahng, Jens Lienig, Igor L Markov, and Jin Hu. VLSI physical design: from graph partitioning to\ntiming closure . 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SIAM Review , 25(3):424, 1983.\n[36] Jue Leskovec. email-Eu-core network dataset.\n29"    },    {        "title": "2402.15508v1.The_α_element_enrichment_of_gas_in_distant_galaxies.pdf",        "content": "Astronomy &Astrophysics manuscript no. arxiv_Velichko ©ESO 2024\nFebruary 26, 2024\nTheα-element enrichment of gas in distant galaxies\nAnna Velichko1,2, Annalisa De Cia3,1, Christina Konstantopoulou1, Cédric Ledoux4, Jens-Kristian Krogager5, and\nTanita Ramburuth-Hurt1\n1Department of Astronomy, University of Geneva, Chemin Pegasi 51, 1290 Versoix, Switzerland e-mail:\nanna.velichko@unige.ch\n2Institute of Astronomy, Kharkiv National University, Sumska 35, Kharkiv, 61022, Ukraine\n3European Southern Observatory, Karl-Schwarzschild Str. 2, 85748 Garching bei München, Germany\n4European Southern Observatory, Alonso de Córdova 3107, Casilla 19001, Vitacura, Santiago, Chile\n5Centre de Recherche Astrophysique de Lyon, Univ. Claude Bernard Lyon 1, 9 Av. Charles André, 69230 St Genis Laval, France\nReceived date /Accepted date\nABSTRACT\nContext. The chemical evolution of distant galaxies cannot be assessed from observations of individual stars, in contrast to the case of\nnearby galaxies. On the other hand, the study of the interstellar medium (ISM) o ffers an alternative way to reveal important properties\nof the chemical evolution of distant galaxies. The chemical enrichment of the ISM is produced by all the previous generations of stars\nand it is possible to precisely determine the metal abundances in the neutral ISM in galaxies. The chemical abundance patterns in the\nneutral ISM are determined by the gas metallicity, presence of dust (the depletion of metals into dust grains), and possible deviations\ndue to specific nucleosynthesis, for example, α-element enhancements.\nAims. We aim to derive the metallicities, dust depletion, and α-element enhancements in the neutral ISM of gas-rich mostly-metal-poor\ndistant galaxies (Damped Lyman- αabsorbers, DLAs). Furthermore, we aim to constrain the distribution of α-element enhancements\nwith metallicity in these galaxies.\nMethods. We collected a literature sample of column density measurements of O, Mg, Si, S, Ti, Cr, Fe, Ni, Zn, P, and Mn in the\nneutral ISM of DLAs at redshifts of 0 .60<z<3.40. We used this sample to define a golden sample of DLAs with constrained\nobservations of Ti and at least one other α-element. By studying the abundance patterns, we determined the amount of dust depletion,\nsolely based on the observed relative abundances of the α-elements. We then used the abundances of Fe-peak elements to determine\nthe overall metallicity of each system, after correcting for dust depletion. In addition, we studied the deviations from the basic (linear)\nabundance patterns. We divided our sample into two groups of galaxies based on the widths of their absorption lines ( ∆v90above or\nbelow 100 km s−1), which may be considered as a proxy for their dynamical mass. We characterised the distribution of the α-element\nenhancements as a function of metallicity for the galaxy population as a whole, by fitting a piecewise function (plateau, decline,\nplateau) to the data.\nResults. We observed systematic deviations from the basic abundance patterns for O, Mg, Si, S, Ti, and Mn, which we interpreted as\nα-element enhancements and a Mn underabundance. The distribution of the α-element enhancements with metallicity is di fferent in\nthe high- ∆v90and low- ∆v90groups of galaxies. We constrained the metallicity of the α-element knee for the high- ∆v90and low- ∆v90\ngroups of galaxies to be −1.02±0.15 dex and −1.84±0.11 dex, respectively. The average α-element enhancement at the high-plateau\nis [α/Fe]=0.38±0.07 dex. On the other hand, Mn shows an underabundance in all DLAs in the golden sample of −0.36±0.07 dex, on\naverage.\nConclusions. We have constrained, for the first time, the distribution of the α-element enhancement with metallicity in the neutral\nISM in distant galaxies. Less massive galaxies show an α-element knee at lower metallicities than more massive galaxies. This can\nbe explained by a lower star formation rate in less massive galaxies. If this collective behaviour can be interpreted in the same way\nas it is for individual systems, this would suggest that more massive and metal-rich systems evolve to higher metallicities before the\ncontribution of SN-Ia to [ α/Fe] levels out that of core-collapse SNe. This finding may plausibly be supported by di fferent SFRs in\ngalaxies of di fferent masses. Overall, our results o ffer important clues to the study of chemical evolution in distant galaxies.\nKey words. ISM: abundances – dust, extinction – quasars: absorption lines – galaxies: evolution\n1. Introduction\nThe outcome of the evolutionary history of galaxies is recorded\nin the interstellar abundances of chemical elements. Observa-\ntions of the interstellar medium (ISM) in galaxies of various\ntypes, di ffering in terms of mass, size, metallicity, and their re-\nspective evolutionary stages, may provide a key to understand-\ning the processes taking place in galaxies (Maiolino & Mannucci\n2019).\nIn the Milky Way (MW) and nearby galaxies, one of the\nways to trace their evolutionary process is from observations of\nthe detailed abundances of chemical elements in individual stars(McWilliam 1997; Tolstoy et al. 2009). Although it is impossible\nto obtain information about individual stars in distant galaxies,\nthe chemical properties of their interstellar medium (ISM) can\nbe analysed in great detail. The chemical abundance patterns in\nthe ISM is the result of enrichment produced by all the previous\ngenerations of stars. These patterns make up a snapshot of the\nevolutionary history of the galaxy.\nOne of the techniques to determine element abundances in\nthe gas of faint high-redshift galaxies is based on the detection of\nits imprint on the spectrum of a background sources which may\nbe a quasi-stellar object (QSO) or gamma-ray burst (GRB) after-\nArticle number, page 1 of 22arXiv:2402.15508v1  [astro-ph.GA]  23 Feb 2024A&A proofs: manuscript no. arxiv_Velichko\nglow (Prochaska & Wolfe 1999; Prochaska et al. 2007; Ledoux\net al. 2002a; Vreeswijk et al. 2004; De Cia et al. 2012). Damped\nLyman-αabsorbers (DLA) are defined as systems with high col-\numn densities of neutral hydrogen (N(H I) ≥2×1020cm−2) that\nappear in absorption in QSOs or GRBs spectra.\nThe use of DLAs to study galaxies is important for several\nreasons. First, due to the fact that they are observed in absorp-\ntion, there is no selection e ffect with respect to luminosity of the\nDLA-galaxies and presumably masses. Hence, it is possible to\ndetect in this way both the faintest low-mass and bright high-\nmass galaxies. In our study, it is important to cover a wide range\nof galaxies as much as possible in terms of masses.\nSecondly, due to the fact that hydrogen in DLAs is mainly\nin a neutral state (HI), the HI atoms shield the gas against ioni-\nsation (e.g. Viegas 1995), and other elements are also mainly in\ntheir dominant states, while the other states are negligible. For\nexample, it is known that under such conditions, oxygen and ni-\ntrogen, like hydrogen, are neutral (OI, NI), while elements such\nas C, Mg, Si, Fe, and S are predominantly singly ionised (CII,\nMgII, etc., see, e.g. Hernandez et al. 2020). Therefore, there is\nno need to apply ionisation corrections, which, in turn, can in-\ntroduce large uncertainties. This makes it possible to accurately\ndetermine the element abundances from the column densities.\nThird, unlike emission spectra, in which only very strong\nspectral lines can be detected, absorption spectra provide very\ndetailed chemical composition even for faint high-z galaxies\nsince, in this case, it is possible to make measurements from very\nweak line profiles.\nTheα-element enhancement, [ α/Fe], and its evolution with\nthe metallicity [M /H] is a strong diagnostic of the chemical evo-\nlution of galaxies (e.g. Tinsley 1979; Hill et al. 1995; Chiappini\net al. 1997; Tolstoy et al. 2009; Kobayashi et al. 2020a; Mat-\nteucci 2021). More details about [ α/Fe] evolution with metal-\nlicity are discussed in Section 5.1. Until recently, the [ α/Fe]\nhas mostly been observed in stars in the local group (e. g.\nMcWilliam 1997; Tolstoy et al. 2009; Matteucci 2021). Cullen\net al. (2021) observed [ α/Fe] in nearby galaxies from a combina-\ntion of stellar and gas metallicities. In the gas, the measurements\nof [α/Fe] are additionally complicated by the presence of dust,\nwhich dramatically alters the observed abundances. Large frac-\ntions of refractory metals are missing from the gas-phase and\nare instead incorporated into dust grains, a phenomenon called\ndust depletion (e.g. Field 1974; Savage & Sembach 1996; Jenk-\nins 2009; De Cia et al. 2016). When dust depletion is negligi-\nble, it has been possible in the past to observe α-element en-\nhancement in a few least dusty, typically metal-poor DLA sys-\ntems (Dessauges-Zavadsky et al. 2002; Ledoux et al. 2002b;\nDessauges-Zavadsky et al. 2006; Cooke et al. 2011; Becker\net al. 2012; De Cia et al. 2016). Recently, Ramburuth-Hurt et al.\n(2023) studied the depletion patterns of individual gas compo-\nnents within DLA systems and discovered α-element enhance-\nments for systems with a wide range of dust depletions. In this\nstudy, we analyse the abundance patterns of a sample of DLAs\nout to z ∼4, correct for dust depletion, and measure the α-element\nenhancements in the gas in the DLA galaxies. Furthermore, we\nsplit our sample in a group of lower-mass galaxies and one of\nhigher-mass galaxies to investigate the evolution of the [ α/Fe]\ndistribution with metallicity for these groups.\nWe refer to metal relative abundances [X /Y] defined as:\n[X/Y]=logN(X)\nN(H)−logN(X) ⊙\nN(H) ⊙, (1)where N(X) ⊙, N(Y) ⊙are column densities from Asplund et al.\n(2021), following the recommendations of Lodders & Palme\n(2009), as reported in Table 1 of Konstantopoulou et al. (2022).\n2. Features of the nucleosynthesis of the elements\nunder study\n2.1. Theα-elements\nThe study of the distribution of [ α/Fe] with metallicity, [M /H],\nis fundamental to understand the chemical evolution of galaxies\n(e.g. Tinsley 1979; Hill et al. 1995; Chiappini et al. 1997; Tol-\nstoy et al. 2009; Kobayashi et al. 2020a; Matteucci 2021) and\ncan provide important information on the star-formation history\n(SFH) in a galaxy. It is well known that α-elements (i.e. those\nthat can be created by the α-process, e.g. O, Mg, Si, S, Ca, Ti)\nare mainly produced inside massive stars (M ⩾8M⊙) with life-\ntimes lower than 30 Myr. They are subsequently released into\nthe interstellar medium (ISM) via core-collapse Type II super-\nnovae. As a result, the ejected matter is enriched in α-elements\nrelative to Fe, until after about 1 billion years most SNe Ia begin\nto explode and produce more iron (Tinsley 1979).\nSchematically, in simplified form, this behaviour can be plot-\nted in coordinates [ α/Fe] – [M /H], as shown in Fig. 1. During the\nfirst phase, when only core-collapse SNe (CCSNe) contribute to\nthe abundance of elements, [ α/Fe] is about constant and forms\na plateau. The level of the high- αplateau depends on the initial\nmass function (IMF, McWilliam 1997) because a larger number\nof massive stars produce more α-elements. In addition, the IMF\nmay become more \"top-heavy\" with increasing star formation\n(Köppen et al. 2007).\nAlthoughα-elements enrichment is dominated by CCSNe,\nSNe Ia also can partially contribute to the abundances depend-\ning on the element. For instance, O and Mg are produced almost\nexclusively by CCSNe, while Si, S, Ca and Ti in the given or-\nder have an increasing contribution from SNe Ia, so that about\n1/3 of titanium contained in the Sun has been produced in SNe\nIa (Andrews et al. 2017). In addition, according to Woosley &\nWeaver (1995); Cayrel et al. (2004), O is mainly produced by\ncentral helium burning with some contribution of neon burning;\nMg is mostly product of hydrostatic carbon burning in a shell\nand explosive neon burning; Si and Ca are formed in both the\noxygen burning and neon burning shells; Ti by explosive oxy-\ngen burning. As a result of these di fferences in the formation of\nα-elements, the level of the high- αplateau can systematically\ndiffer by 0.15 dex, if oxygen is not taken into account (Cayrel\net al. 2004; Petitjean et al. 2008).\nAfter SNe Ia start to explode, the abundance of α-elements\nrelative to iron gradually decreases, and the transition point from\none regime to another is commonly referred to as the ’ α-element\nknee’ (McWilliam 1997). The position of the knee is sensitive to\nthe early star formation rate (SFR). Observations at di fferent red-\nshifts show that the SFR increases with galactic mass (Katsianis\net al. 2016). If the SFR is high, then more metals have time to\nform during the first phase, resulting in a knee located at a higher\n[M/H] (e.g. Tinsley 1979; McWilliam 1997). The position of the\nα-element knee has been measured in stars in local galaxies, and\nit seems to depend on the mass of the galaxy, i.e. the knee is at\nlower metallicities for lower stellar masses (de Boer et al. 2014).\nBecause [α/Fe] are decreasing after the knee, the plateau is fol-\nlowed by a decay with a certain slope. According to Andrews\net al. (2017), the latter can be sensitive to the gas outflow rate.\nThen, after a certain period of time, the third phase of the\n[α/Fe] evolution begins, when the contribution in the chemical\nArticle number, page 2 of 22Anna Velichko et al.: The α-element enrichment of gas in distant galaxies\nyields of the SNe Ia compensates those of the CCSNe. At this\npoint, the [α/Fe] reaches a plateau at a value close to zero. In\nthe case of the MW, the transition point is at [M /H]=−0.2\n(McWilliam 1997), and the low- αplateau level is at [ α/Fe]=\n0.05. To distinguish between the two knees mentioned above,\nwe call them high- αand low-αknee for transition points from\nthe high-αplateau to the decay and from the decay to the low- α\nplateau, respectively.\nFigure 1 shows the curves [ α/Fe] – [M /H] we obtained us-\ning the release DR17 of the Apache Point Observatory Galac-\ntic Evolution Experiment (APOGEE, Majewski et al. (2017))\ndata for satellite galaxies of the MW: Large and Small Magel-\nlanic Clouds (LMC and SMC, respectively), Sagittarius Dwarf\nGalaxy (Sgr), Fornax (Fnx), and the now fully disrupted Gaia\nSausage /Enceladus (GSE) system. We selected stars belonging\nthe galaxies according to the lists from Hasselquist et al. (2021).\nFor comparison, the curve for the MW given by McWilliam\n(1997) is also shown. For details on APOGEE DR17 data fitting,\nwe refer to Appendix C.\nOf course, this piecewise model is simplified compared to\nthe natural behaviour. In fact, the evolution of galaxies of di ffer-\nent types, sizes, and masses have their own peculiarities. For ex-\nample, dwarf galaxies often have several separate bursts of star\nformation in their evolutionary history, in contrast to the more\nor less stable star formation in large galaxies such as the MW\n(Tolstoy et al. 2009; Atek et al. 2022). Such a discontinuous star\nformation manifests itself in the [ α/Fe]–[M /H] behaviour in the\nform of bump on a low- αplateau (see e.g. the modelled tracks\nin Fig. 4 from Hasselquist et al. 2021). We see such a bump (or a\nhint of it) in the behaviour of [ α/Fe] for the all considered dwarf\ngalaxies (see dark gray curves in Fig. C.1). The abundances of\nα-elements follow each other fairly well, that is, they make up a\nfairly homogeneous group; however, there are small di fferences\nin their production, such as the location of synthesis and depen-\ndencies on stellar mass and metallicity.\n2.2. Zinc\nZinc is a volatile element that is often used as a tracer of metallic-\nity in the ISM of various objects, assuming that it is not captured\nin dust grains. In reality, the depletion of Zn is non-negligible.\nTherefore, the evaluation of metallicity using Zn can be erro-\nneous, especially in highly dusty environments.\nZinc is often considered to be an ideal tracer of iron-peak el-\nements, because in the Solar neighborhood [Zn /Fe] is about 0.0\nfor [Fe /H] between −2 and 0 dex (e.g. Nissen et al. 2004; Saito\net al. 2009). However, the nucleosynthetic nature of Zn is com-\nplex since it is neither an iron-peak nor an α-element. At [Fe /H]\nless than −2 dex [Zn /Fe] increases towards lower metallicities\nup to ≈0.5 dex (Nissen et al. 2007). For the red giants in the\nMW bulge, at [Fe /H]>0 dex [Zn /Fe] shows a decrease down\nto≈ −0.5 dex (Barbuy et al. 2015). In dwarf galaxies the be-\nhaviour [Zn /Fe] versus [Fe /H] is di fferent from that observed in\nthe MW due to the di fferent star formation history. Instead of a\nplateau, dwarf galaxies show a constant decreasing [Zn /Fe] over\nthe entire observed metallicity range (Skúladóttir et al. 2017; Hi-\nrai et al. 2018). This di fference between the behaviour of [Zn /Fe]\nin massive and dwarf galaxies can be explained by the compli-\ncated production of Zn.\nAccording to (Bensby et al. 2003; Nissen & Schuster 2011;\nMishenina et al. 2011; Barbuy et al. 2015; Du ffau et al. 2017),\nthe relation [Zn /Fe] versus [M /H] tends to behave partially like\nan [α/Fe] but with a smaller amplitude. Sitnova et al. (2022)\nshowed from non-LTE calculations for the MW stars that Zn andMg do not strictly follow each other. Nevertheless, [Zn /Fe] is de-\ncreasing with metallicity because of various contributions from\ncore-collapse SNe and SNe Ia, and their di fferent timescales (as\nforα-elements).\nDe Cia et al. (2024) analysed element abundances in the neu-\ntral ISM of the Small and Large Magellanic Clouds and observed\na potential tendency of a deviation of Zn with respect to Fe up\nto 0.2 dex at lower metallicities and down to −0.2 dex at higher\nmetallicities. If confirmed, this might be in agreement with the\nbehaviour derived from stellar measurements described above.\nHowever, this still needs to be verified with further investiga-\ntions with more data.\n2.3. Phosphorus\nPhosphorus is thought to be mainly produced in massive\nstars during their hydrostatic carbon and neon burning phases\n(Woosley & Weaver 1995). SNe Ia produce less significant\namount of P compared to massive stars (Leung & Nomoto 2018).\nLittle or no P is produced in AGB stars (Karakas & Lugaro\n2016). According to Maas et al. (2019, 2022), the behaviour of\nP with metallicity in the MW is most similar to that of the α-\nelements, especially Mg. Over the [Fe /H] range from −1.0 to 0.2\ndex [P /Fe] decreases from ∼0.6 dex to ∼ −0.2. This is in agree-\nment with the assumption that P is likely mostly produced in\nCCSNe.\n2.4. Iron group elements Cr, Fe, and Ni\nIron-group elements such as Cr, Fe, Ni are mostly produced by\nSNe Ia (Nomoto et al. 1997; Kobayashi et al. 2020a,b). From\nobservations of the MW stars [Cr /Fe] increases with metallicity\nfrom∼ −0.5 to ∼0 in the range ∼ −4.2<[Fe/H]<∼ −1.5 (Frebel\n2010; Xing et al. 2019; Cayrel et al. 2004). From the APOGEE\ndata, [Cr /Fe] and [Ni /Fe] are around 0 within the range of [Fe /H]\nfrom∼ −1.5 to ∼0.5 dex (Lim et al. 2022). Bensby et al. (2003)\nfind for thin and thick disk stars in the Solar neighborhood a\nslight overabundance of Ni with respect to Fe at metallicities\nbelow 0 and a small increase of [Ni /Fe] up to 0.1 dex above\n[Fe/H]=0.\nIn nearby dwarf galaxies, LMC, SMC, Fnx, Sgr, GSE and\nSculptor, Ni follows the MW trend in the metal-poor regime\nwhile for [Fe /H]≥ −1.5 it becomes underabundant down to −0.2\ndex (e.g, Tolstoy et al. 2009; Van der Swaelmen et al. 2013;\nLemasle et al. 2014; Hasselquist et al. 2021). From the abun-\ndance measurements in the neutral ISM of LMC and LMC, Ni is\nslightly underabundant relative to Fe at lower metallicities and\nincreases up to 0 or a bit above at higher metallicities, while\n[Cr/Fe] is about 0 dex in the entire range of [Fe /H] (De Cia et al.\n2024).\n2.5. Manganese\nManganese is an iron-group element. Despite extensive studies\nby many authors (e.g. Seitenzahl et al. 2013; Eitner et al. 2020),\nthe production of Mn remains uncertain. [Mn /Fe] has a di ffer-\nent evolution compared to both other iron-peak elements and\nα-elements (Mishenina et al. 2015). Compared to α-elements,\ncore-collapse SNe produce much less Mn. Hence, at low metal-\nlicities, Mn deficiency with respect to Fe (or Mn underabun-\ndance) is observed. When type Ia SNe start exploding and pro-\nducing more Mn (Nomoto et al. 1997), the underabundance\ngradually disappears with an increasing [Fe /H] (e.g. Mishen-\nArticle number, page 3 of 22A&A proofs: manuscript no. arxiv_Velichko\nFig. 1: Piecewise approximation of the α-element behaviour in\nnearby dwarf galaxies SMC, LMC, GSE, Fnx, and Sgr from the\nAPOGEE data (see details in Fig. C.1).\nina et al. 2015; Prantzos 2005; Mikolaitis et al. 2017, for the\nMW).\nMoreover, according to Bergemann & Gehren (2008), in stel-\nlar atmospheres Mn abundances may be a ffected by deviations\nfrom the LTE in such a way that the measured values may be\nunderestimated by up to 0.5–0.6 dex. However, Mishenina et al.\n(2015) did not confirm these deviations which is explained by\nthe lack of high-quality atomic data and, hence, the inability to\nbuild an adequate model of Mn atoms. Nevertheless, this should\nnot affect the values obtained from ISM.\n3. Data and sample\n3.1. Column densities\nFor the analysis, we utilised the column density measurements\nof O, Mg, Si, S, Ti, Cr, Fe, Ni, Zn, P, and Mn for the QSO-\nDLA sample taken from the work by Konstantopoulou et al.\n(2022). After removing duplicate entries Konstantopoulou et al.\n(2023) finalised a sample of 108 DLAs with a measurement of\nZn. Among them, 37 QSO-DLAs are from De Cia et al. (2016),\nwhich is a homogeneous sample of high-resolution ( R=λ/∆λ\n=35000 – 58000) spectra taken with the Very Large Telescope\n(VLT) Ultraviolet and Visual Echelle Spectrograph (UVES). The\nsignal-to-noise ratios (S /N) of these spectra are in the ranges\n14–33 pixel−1and 54–10 pixel−1in the blue and red spectro-\nscopic arm of UVES, respectively. The data were supplemented\nby Konstantopoulou et al. (2022), wherever possible, with new\ncolumn density measurements of Ti and Ni from UVES /VLT\nspectra using the V oigt-profile fit method (Krogager 2018). 71\nQSO-DLAs are from the large compilation by Berg et al. (2015a)\npublished during the period between 1994 and 2004, with a min-\nimum resolution restriction of R>10000, but typically R∼\n40000. The S /N for the spectra is up to ∼50 pixel−1, with the typ-\nical spectrum having an S /N of∼10 pixel−1(Berg et al. 2015a,b).\nIn order to homogenise the measurements taken from di ffer-\nent sources, Konstantopoulou et al. (2022) corrected the column\ndensities to the newest possible oscillator strengths for both of\nthe samples.\nThe sample described above has been supplemented with one\nQSO-DLA (Q1210 +175) from De Cia et al. (2016), which isnot included the work of Konstantopoulou et al. (2022) due to\nthe lack of measurements for zinc; however UVES data cov-\nering absorption lines of Ti were indeed available. In addition,\nthere is one QSO-DLA (Q1232 +082) from the list provided by\nKonstantopoulou et al. (2022), for which there was no Ti mea-\nsurement, but it turned out to be possible. In this system, the Ti\nIIλ1910 line can be slightly blended, but we consider the col-\numn density to be reliable. In this paper we measure Ti and Ni\nfor these additional two systems, using the method described in\nKonstantopoulou et al. (2022). In additional, we provide column\ndensities of Ti and Ni, where is possible, for other DLAs from\nthe list from De Cia et al. (2016) (see Table F.1). Velocity pro-\nfiles are shown in Figs. F.1, F.2. In total, our sample contains 110\nQSO-DLAs.\nFrom the list of 110 QSO-DLAs described above, we have\nselected 24 objects for which there are column density measure-\nments of Ti and at least one more α-element (Mg, Si, S, and O).\nWe did not take into account those objects for which only limits\nhave been calculated. We refer to this subsample as the \"golden\"\nsample. The need for this selection is explained in Section 4. 15\nQSO-DLAs in the golden sample are from the list of De Cia et al.\n(2016), while the remaining 9 QSO-DLAs are from the compi-\nlation by Berg et al. (2015a) (see references in Table 1). All the\ndata for the golden sample is high quality, with resolution of at\nleast 35000 and high S /N.\nThe results of this work are based on the golden sample. For\nthe rest of the sample, we do a minimal analysis and report the\nabundance patterns in the appendix.\n3.2. Velocity widths\nIn addition to the column densities, our analysis uses information\nabout the velocity widths of the absorption lines, ∆v90, namely,\nthe velocity intervals encompassing 90% of the integrated opti-\ncal depths. This quantity is computed as c[λ(95%) −λ(5%)]/λ0,\nwhereλ(5%) andλ(95%) are the wavelengths corresponding to\nthe 5th and 95th percentiles of the apparent optical depth distri-\nbution, respectively, and λ0is the first moment of this distribu-\ntion, as defined by Ledoux et al. (2006).\nOne of the advantages of the DLA absorption spectrum\nis that it is produced by low-ionisation atoms dominated\nby galactic-scale motions governed by gravity (unlike highly\nionised atoms which can make up the hot ejected gas). There-\nfore, these spectra are well suited for determining the dynamical\nvelocity, which can be a proxy for their mass (Ledoux et al. 2006;\nProchaska et al. 2008; Christensen et al. 2014; Arabsalmani et al.\n2018). Thus, by measuring ���v90in DLAs, we can get an in-\nsight in a statistical sense on the mass of the galaxies responsible\nfor the absorption lines. In this work, we adopted the values of\n∆v90from Ledoux et al. (2006); Herbert-Fort et al. (2006); No-\nterdaeme et al. (2008); Jorgenson et al. (2010); Rafelski et al.\n(2012); Neeleman et al. (2013); Berg et al. (2015a) (see Table.\n1).\n4. Method\nThe major challenge in properly determining the total (gas +\ndust) element abundances in the ISM is to correct account for\nthe depletion by dust. To measure the total ISM metallicity, it\nis highly important to estimate the fraction of atoms of di fferent\nelements withdrawn from the gas phase and incorporated into\ndust grains.\nDe Cia et al. (2016) have developed an e ffective method to\ncharacterise the dust depletion in the ISM of the MW and QSO-\nArticle number, page 4 of 22Anna Velichko et al.: The α-element enrichment of gas in distant galaxies\nDLAs based on the analysis of the relative abundances, that al-\nlows to derive the total (gas +dust) metallicity. Based on this,\nDe Cia et al. (2021) have built the so-called \"relative\" method\nto derive the total metallicty in the neutral ISM in the MW. The\ntotal abundance [X /H]totcan be derived as follows:\n[M/H]tot=[X/H]−δX, (2)\nwhere [X /H] is the observed abundance of element X, and δXis\nits dust depletion. The ’minus’ sign in the formula 2 is due to the\nfact that the value of δXis negative in its classical notation (e.g.\nDe Cia et al. 2016).\nThe method is aimed at deriving the total metallicity [M /H]tot\nand overall strength of depletion [Zn /Fe] fitfrom the distribution\nof element abundances [X /H] depending on their refractory in-\ndex, B2X. The latter represents how strongly an element is in-\ncorporated into the dust. This linear dependence is defined as\nfollows:\ny=a+bx, (3)\nwhere\nx=B2X, (4)\ny=logN(X)−logN(H)−X⊙+12−A2X∼[X/H], (5)\na=[M/H]tot, (6)\nb=[Zn/Fe] fit. (7)\nThe coe fficient A2Xis the normalisation of the depletion se-\nquences of each element after subtracting the correction for nu-\ncleosynthesis e ffects such as α-element enhancement or Mn un-\nderabundance. In general it could be assumed to be zero, im-\nplying no dust depletion at [Zn /Fe]=0 (De Cia et al. 2016). The\ndepletion factor [Zn /Fe] fitrepresents the overall strength of de-\npletion and it is derived from several di fferent metals.1\nTheA2XandB2Xcoefficient have been adopted from Kon-\nstantopoulou et al. (2022). To derive these coe fficients, these au-\nthors used a large collection of column density measurements of\n18 metals with di fferent refractory properties located in various\nenvironments such as MW, the Magellanic Clouds, and DLAs.\nA few remarks should be made. First, due to their nucleosyn-\nthesis peculiarities, some elements may have systematic devi-\nations from abundance patterns that we expect from depletion\neffects. The best-known and most prominent examples are α-\nelement enhancement and Mn underabundance (e.g. McWilliam\n1997; Mishenina et al. 2015).\nSecond,δXis a linear function of the overall strength of de-\npletion characterised by the parameter [Zn /Fe] fitin the relative\nmethod. The di fference from the observed parameter [Zn /Fe] is\nthat [Zn /Fe] fitis derived from the abundances of all available\nmetals or their subset.\nHere, we consider the application of the method on the ex-\nample of the QSO-DLA source Q0405–443c (see Fig. 2). The\nleft panel shows the linear fit to all the element abundances. For\nthe linear fit, we took into account error bars on both axes and\ndid not include elements for which only the limits have been\nmeasured. The values of [X /H] have a fairly large scatter rela-\ntive to the best fit curve, which could not be explained by the\n3σconfidence interval (shown as the gray area in Fig. 2). This\nscatter is likely caused by the nucleosynthesis peculiarities of\nα-elements and Mn. This becomes evident from the right panel\n1Despite using all available metals, [Zn /Fe] still has a somewhat priv-\nileged role in the analysis, i.e. the relations of δXwith [Zn /Fe] (Kon-\nstantopoulou et al. 2022).of Fig. 2, where the elements have been divided into three sub-\nsets:α-elements (Ti, Si, S), elements of the iron group (Fe, Ni,\nCr) and others. We do not include Zn to the iron group since\nits abundances does not always follow those of Fe, Ni and Cr,\nespecially in dwarf galaxies (see Hirai et al. 2018).\nDespite small di fferences in the nucleosynthetic origin of\nvariousα-elements (see Sect. 1, Tolstoy et al. 2009), their abun-\ndance pattern is fairly uniform, unlike other metals (Prantzos\n2005, see also the discussion in Sect. 5.1). It is important to sep-\narateα-elements from other metals for the further analysis. With\nthis assumption, the best way to derive the slope [Zn /Fe] fitof the\nlinear fit to the abundance pattern is from the total amount of\nα-element measurements available.\nFor the correct application of this method, it is important to\nhave an abundance measurement of titanium and at least one\notherα-element. This provides su fficient dynamical range in the\nx-axis for a more confident fit because titanium is the most re-\nfractory among other α-elements. In six cases among the golden\nsample systems, the preferred slope of the linear fit to the abun-\ndance patterns would be negative. However, a negative slope is\nnot physical because dust depletion only removes metals from\nthe gas-phase. We limited the possible slopes to be non-negative,\nusing the same methodology in De Cia et al. (2024).\nTo obtain the total metallicity of the system, we shifted the\nlinear fit obtained solely from the α-elements to the weighted\nmean of the Fe, Cr, and Ni abundances, and derived the y-\nintercept. This is shown in Fig. 2 with a red line. The value of\nthe shifted straight line at B2X=0 is the total metallicity [M /H]tot\nof the system. In this example, for Q0405-443c, we derived\n[M/H]tot=−1.12±0.12.\nWe derived the uncertainty on the slope [Zn /Fe] fitfrom the\nPython procedure we used for the linear fit to the abundance pat-\nterns. To calculate the uncertainty of the normalisation [M /H]tot,\nwe took into account the uncertainty, σαfit, derived from the fit-\nting procedure, the uncertainty of the weighted mean of the Fe-\ngroup element abundances, σtheFe, and the uncertainty of hydro-\ngen,σHtotas follows:\nσ[M/H]tot=q\nσ2αfit+σ2\nFe+σ2\nHtot. (8)\nIn addition to the metallicity and amount of dust depletion,\nwe can further study any peculiarities due to nucleosynthesis by\ncharacterising any deviations of the observed abundances from\nthe main trend (red curve). We derived the relative abundances\nof different elements [X /H]nucldue to nucleosynthesis, after cor-\nrecting for dust depletion as follows:\n[X/H]nucl=[X/H]−y(B2 X). (9)\nHence, we obtain:\n[X/Fe] nucl=[X/H]nucl−[Fe/H]nucl. (10)\n5. Results and discussion\nWe analysed the abundance patterns of 110 DLAs, in particular,\n24 DLAs in the golden sample. In most cases, we find a satis-\nfactory fit to the data. These are shown in Figs. A.1 and B.1.\nTables 1 and A.1 contain the resulting data for the golden sam-\nple: total (gas +dust) metallicities, [M /H]tot, the overall strength\nof dust depletion, [Zn /Fe] fit, and the relative abundances due to\nnucleosynthesis, [X /Fe] nucl, which are the deviations from the\nabundance patterns after taking dust depletion into account.\nArticle number, page 5 of 22A&A proofs: manuscript no. arxiv_Velichko\nFig. 2: Linear fitting over the entire data set of [X /H] (left panel). The elements (right panel) are divided into three groups: α-\nelements (blue triangles), Fe-group elements (red diamonds), and other metals (purple circles). The blue dashed line shows linear\nfitting over solely the α-element abundances, the red solid line is the blue dashed line shifted to the value averaged over Fe-group\nelements. The gray filled areas show 3 σconfidence intervals. The total metallicity [M /H]totis shown by intersection of gray dashed\nlines.\nFig. 3: Distribution of the golden sample by velocity widths.\nWe divided the golden sample into two parts by velocity\nwidths: ∆v90<100km s−1and∆v90≥100km s−1which we\nrefer to as low- and high- ∆v90subsamples, respectively. Fig-\nure 3 shows the distribution of QSO-DLAs by ∆v90. The divid-\ning point has been chosen in such a way that the two subsam-\nples roughly contains a comparable number of systems: there\nare 15 and 9 QSO-DLAs in low- and high- ∆v90subsamples, re-\nspectively. Using 70 DLA or sub-DLA systems Ledoux et al.\n(2006) have shown a correlation between metallicity and veloc-\nity widths which is probably the consequence of an underly-\ning mass-metallicity relation (Christensen et al. 2014). Thus, the\nlow-∆v90sub-sample is expected to have overall lower galaxy\nmasses than the high- ∆v90sub-sample. According to Arab-\nsalmani et al. (2018), the threshold velocity width of 100 km s−1\ncorresponds to the stellar mass of the galaxy M∗of about 109M⊙.\nFigure 4 shows the distribution of the golden sample by the\ntotal metallicity. We see that [M /H]totvaries from −2.0 to −0.5.\nThe metallicity ranges for the two subsamples mostly overlap,\nFig. 4: Distribution of the golden sample by metallicity [M /H]\nfor low (shown by blue) and high (orange) ∆v90subsamples.\nalthough the distributions are di fferent: vertical lines show the\nweighted mean which is −1.36 and −0.97 for the low- and high-\n∆v90subsamples, respectively.\nThe strength of dust depletion [Zn /Fe] fitshown in Fig. 5 is in\nthe range from 0.0 to 0.55, with the two subsamples displaying\nonly a very small systematic di fference in distribution of 0.01\n(see vertical lines in Fig. 5 that show the weighted mean).\nFigure 6 displays a broad correlation between the total metal-\nlicity and the strength of dust depletion with quite a large scat-\nter. Similar relations have been derived by other authors (De Cia\net al. 2024; Noterdaeme et al. 2008; Ledoux et al. 2002a). Hence,\nthe amount of dust in DLAs overall depends on the gas metal-\nlicity. This is in agreement with an assumption that significant\namount of dust is built through grain-growth in the ISM depend-\ning on the gas metallicity, density, and temperature (Dwek 2016;\nMattsson et al. 2014; De Cia et al. 2013). From Fig. 6, it might\nappear that low- ∆v90systems show larger [Zn /Fe] fitcompared to\nArticle number, page 6 of 22Anna Velichko et al.: The α-element enrichment of gas in distant galaxies\nFig. 5: Distribution of the golden sample by dust depletion\n[Zn/Fe] fitfor low (shown by blue) and high (orange) ∆v90sub-\nsamples.\nFig. 6: Relation between the total metallicity, [M /H]totand the\nstrength of dust depletion, [Zn /Fe] fit, in the neutral ISM of\nDLAs. Large blue and orange triangles show the data for the\ngolden sample with low- and high- ∆v90, respectively. The small\ngray diamonds are the data for the non-golden sample. The solid\nline shows a linear fit to all the data including uncertainties on\nboth axes, [Zn /Fe] fit=0.33±0.03×[M/H]tot+0.69±0.03 . The\nshaded area represents the 3 σconfidence interval.\nhigh- ∆v90galaxies. In fact, this is a shift in metallicity (see Fig.\n4), while [Zn /Fe] fitvalues are similarly distributed in low- and\nhigh- ∆v90galaxies, as shown in Fig. 5. The data for the MW do\nnot strongly follow the dust-metallicity relation showing steeper\nslope than DLAs. The possible cause of this is that the MW disk\nhas higher pressure and higher density of colder gas compared\nto more di ffuse warm neutral medium in DLAs (see De Cia et al.\n2024). As a result, dust is expected to grow more easily in the\nMW.\nWe compared our most solid derivation of the total metal-\nlicity, [M /H]tot, for the golden sample with alternative ways of\nestimating the metallicity, to test their robustness. First, we com-\npared our results to the total metallicity based on the fit of the\nFig. 7: Total metallicity [M /H]totin the ISM of QSO-DLAs from\nthe golden sample. Large purple diamonds are the values derived\nfrom solely α-elements (this work, the basic result). Small green\ntriangles show the values derived from fitting the iron-group el-\nements +Zn. Small orange crosses and small blue stars are the\ndeterminations given by De Cia et al. (2016) and De Cia et al.\n(2018), respectively.\nabundance pattern of only the Fe-group elements and Zn (green\ntriangles in Fig. 7). This reveals that in most cases, the estimates\nbased only on the Fe-group are slightly lower than the fiducial\nvalue, on average by 0.06 dex. The values of total metallicity de-\ntermined by De Cia et al. (2016) and De Cia et al. (2018) (yellow\ncrosses and blue stars in Fig. 7) generally demonstrae a greater\nsimilarity to the metallicity we determined using only the Fe-\ngroup elements and Zn. This reflects the fact the methodologies\nArticle number, page 7 of 22A&A proofs: manuscript no. arxiv_Velichko\nof De Cia et al. (2016) and De Cia et al. (2018) based their re-\nsults more heavily on Fe-group elements and Zn. In this work,\nhowever, the methodology is refined to separate the e ffects of\nmetallicity, dust depletion, and nucleosynthesis, and thus more\nrobust. One outstanding example is the case of Q1232 +082, for\nwhich we measure a total metallicity, which is 1 dex lower than\nthe estimate of De Cia et al. (2016). This is due to high α-element\nenhancement (0.5 dex, see Fig. A.1) in this case, combined with\nthe absence of measurements of Ti in the work of De Cia et al.\n(2016). As a result, the slope of the linear fitting is steeper which\nleads to the zero intercept at a higher value of [M /H]tot.\n5.1.α-elements\nThe totalα-element abundances [ α/Fe] nuclfor each object has\nbeen obtained by calculating an average weighted by the uncer-\ntainties over all available individual α-element (Ti, Si, S, Mg,\nand O) values [X /Fe] nucl. Looking at the α-element enhancement\n[α/Fe], as calculated from the fitting described in Sect. 4, for our\nentire sample does not yield any obvious relation as seen for\nindividual galaxies in Fig. 1. This is expected as the DLA popu-\nlation probes a large range of di fferent galaxies (e.g. Fynbo et al.\n2008; Krogager et al. 2017) that will have heterogeneous enrich-\nment histories (see also Dvorkin et al. 2015). However, when\nsplitting our sample into bins of ∆v90, a correlation is apparent.\nIn Fig. 8, we show [ α/Fe] for our golden sample as a func-\ntion of total metallicity in the low- and high- ∆v90subsamples\n(as blue and orange points, respectively). Since ∆v90is a proxy\nof galactic mass, it is reasonable to expect some di fference be-\ntween the high- αknee positions for these two groups of sys-\ntems (see discussion in Section 5.1). It is evident that the orange\npoints tend to cluster at larger metallicities than the blue points,\nas expected from the correlation between ∆v90and metallicity\n(Ledoux et al. 2006). Yet, the two subsamples also seem to fol-\nlow a similar anti-correlation between [ α/Fe] versus [M /H]tot.\nThis is illustrated in the simplest case by a linear fit to the two\nsubsamples (Case 4). For both subsamples, we find consistent\nslopes of −0.19±0.15 and −0.16±0.22 for the low- and high-\n∆v90subsamples, respectively, with an o ffset in metallicity of\n∼0.7 dex.\nBoth of the inferred slopes are slightly lower than the slope\ninferred for the MW ( −0.3, see McWilliam 1997), yet consistent\nwithin the considerable uncertainties that are dominated by scat-\nter in the data. Upon further inspection, there tends to be a slight\nflattening of the two relations: the relation for the low- ∆v90sub-\nsample appears reach a low-alpha plateau at higher metallicities,\n≳−1, whereas the high- ∆v90subsample tends towards a high- α\nplateau at low metallicity, ≲−1.2. This behaviour follows the\nexpectation from the MW and local dwarf galaxies (see Fig. 1\nand McWilliam 1997). The presence of a plateau in the data\nwould indeed bias the slopes towards lower values. We consider\nfour possibilities for fitting the data, which are shown in Fig. 8\nand reported in Table 2:\n–Case 1: Motivated by the behaviour of [ α/Fe] in the MW and\nlocal dwarf galaxies, we fit our data with a three-piecewise\nfunction (high- αplateau +decline +low-αplateau) with five\nparameters. Due to the strong spread of values and a small\nnumber of points, an attempt to set all parameters as free\nleads to a large uncertainty in fitting. As a first guess, we\nleft the position of the α-element knee free to vary and we\nfixed the other parameters according to the values defined for\nthe MW by McWilliam (1997): the level of high- αplateau[α/Fe] nuclis 0.35, the slope is −0.3, and the level of low- α\nplateau is 0.05 (see Table 2).\n–Case 2: We fixed the slope and the low- αplateau at the val-\nues of McWilliam (1997) and fit the data with the three-\npiecewise functions to find the high- αknee positions and the\nhigh-αplateau levels.\n–Case 3: Since the paucity of observational data of [ α/Fe] nucl\nhampers our capability to constrain all the five parameters of\na three-piecewise function, we tested the results by applying\na two-piecewise function (high- αplateau +decline) with all\nthree parameters remaining free.\n–Case 4: Fitting the data with a linear function of the form:\n[α/Fe]=α0+slope ×[M/H]tot.\nBecause of the large scatter of points, the values of all the\nparameters vary from case to case, but they are consistent within\nthe error bars (see Table 2). For the high- ∆v90subsample, we can\nclearly determine the high- αplateau, high- αknee, and slope.\nHowever, the low-alpha knee and low-alpha plateau cannot be\nconstrained due to the lack of points at metallicities lower than\n−0.5 dex. For the low- ∆v90subsample, the plateau seems to be\nless clearly constrained, unlike the high- ∆v90one (see also Fig.\nD.1 in the appendix to assess the variations between di fferentα-\nelements). Overall, all the models show systematic di fferences\nbetween the low- and high- ∆v90subsamples in the [ α/Fe] nucl\n– [M/H]totplane. These are probably due to di fferent chemical\nevolution phases and properties (e.g. SFR and SFH) of galaxies\nwith di fferent stellar masses.\nTo compare these four cases, we computed the reduced χ2\n(χ2\nν) values and report them in Table 2. From the values, it was\nimpossible to choose a preferred model because the di fference in\nχ2\nνshould be larger than ∼1 (or∼2σ) to state that a given model\nis better over another.\nFor a comparison with the results shown above, we applied\nthe same fitting technique to the APOGEE stellar data in nearby\ndwarf galaxies. This is shown in Figs 1 and C.1. It is only for Sgr\nthat the data allow for full three-piecewise function to be traced\nand the high- αknee is at [M /H]=−1.92±0.05. From the data, it\nis impossible to determine the position of the high- αknee for the\nSMC, LMC, and Fnx galaxies. Nidever et al. (2020) constrained\nit to be at [Fe /H]≲−2.2 for both LMC and SMC, despite the fact\nthat LMS is 10 times more massive than SMC. The abundances\nderived by Van der Swaelmen et al. (2013) from high-resolution\nspectra at VLT with the FLAMES /GIRAFFE multifibre spectro-\ngraph show higher α-element enhancement for the LMC. Also,\nthere is some hint that the knee is at higher metallicity of ≈ −1.5.\nHendricks et al. (2014) reported that the high- αknee for Fnx\ndwarf galaxy is at [Fe /H]≈ −1.9, which was determined from\nabundance measurements of Mg using VLT /GIRAFFE spectra.\nThe position of the knee is less clearly defined for Si and Ti, but\napparently it is more clearly seen at metallicities below −1.8.\nFrom the APOGEE data, there are no signs of the presence of\nhigh-αplateau at metallicities above −2.0 (see Figs. 1 and C.1).\nWe derived the position of the high- αknee for GSE to be at\n[M/H] of −1.21±0.01.\nThe high-αknee is challenging to determine even in local\ngalaxies because it requires measuring reliable α-element abun-\ndances for a statistically significant number of faint metal-poor\nstars. Furthermore, there are some discrepancies among di fferent\nobservations (Nidever et al. 2020; Hendricks et al. 2014; Van der\nSwaelmen et al. 2013) that causes additional confusion in try-\ning to interpret the results. Compared to GSE and Sgr, the high-\nαknee in LMC is more metal-poor (Nidever et al. 2020) even\nthough LMC is more massive than the progenitors of GSE and\nArticle number, page 8 of 22Anna Velichko et al.: The α-element enrichment of gas in distant galaxies\nTable 1: Collection of data for QSO-DLAs in the golden sample. The values of logN(HI) were taken from Konstantopoulou et al.\n(2022) or De Cia et al. (2016).\nQSO zabs logN(HI) [M /H]tot [Zn/Fe] fit ∆v90 Ref.\nQ0058 −292 2.671 21.10 ±0.10 −1.36±0.23 0.32 ±0.10 34a1,2\nQ0100 +130 2.309 21.35 ±0.08 −1.62+0.22\n−0.210.09+0.10\n−0.0937a1,2\nQ0102 −190a 2.370 21.00 ±0.08 −2.00+0.23\n−0.120.00+0.11\n−0.0017a1,2\nQ0405 −443a 1.913 20.80 ±0.10 −0.78±0.21 0.27 ±0.07 98a1,2\nQ0405 −443c 2.595 21.05 ±0.10 −1.12±0.20 0.25 ±0.07 79a1,2\nQ0528 −250a 2.141 20.98 ±0.05 −1.62+0.17\n−0.090.00+0.08\n−0.00105a1,2\nQ0528 −250b 2.811 21.35 ±0.07 −0.85±0.17 0.39 ±0.07 304a1,2\nQ0841 +129a 1.864 21.00 ±0.10 −1.58+0.22\n−0.190.04+0.08\n−0.0432a1,2\nQ1117 −134 3.350 20.95 ±0.10 −1.60+0.25\n−0.150.00+0.10\n−0.0044a1,2\nQ1157 +014 1.944 21.80 ±0.10 −1.49±0.20 0.24 ±0.07 89a1,2\nQ1223 +178 2.466 21.40 ±0.10 −1.48±0.20 0.16 ±0.07 91a1,2\nQ2206 −199a 1.921 20.67 ±0.05 −0.53±0.14 0.21 ±0.06 136a1,2\nQ2243 −605 2.331 20.65 ±0.05 −1.03±0.15 0.09 ±0.07 173a1,2\nSDSS1249 −0233 1.781 21.45 ±0.15 −1.15±0.29 0.18 ±0.11 152b3\nQ2230 +025 1.864 20.83 ±0.05 −0.72±0.22 0.29 ±0.10 148a3\nHE1104 −1805 1.662 20.85 ±0.01 −1.02±0.20 0.44 ±0.10 50c3\nQ0027 −1836 2.402 21.75 ±0.10 −1.80±0.19 0.33 ±0.07 44d3\nQ1210 +17 1.892 20.63 ±0.08 −0.74±0.20 0.30 ±0.09 62a3\nFJ0812 +32 2.067 21.00 ±0.10 −1.57+0.23\n−0.140.00+0.09\n−0.0026e,∗3\nJ1135 −0010 2.207 22.05 ±0.10 −0.97±0.24 0.55 ±0.09 168f3\nQ2231 −00 2.066 20.53 ±0.08 −1.23+0.25\n−0.140.00+0.11\n−0.00145a3\nJ1200 +4015 3.220 20.65 ±0.15 −0.66±0.26 0.12 ±0.07 127g3\nQ1210 +175 1.892 20.70 ±0.08 −0.60±0.20 0.52 ±0.07 62a⋆,1\nQ1232 +082 2.338 20.90 ±0.08 −1.65±0.23 0.15 ±0.10 85a⋆,1\nNotes: The values of ∆v90have been taken froma– Ledoux et al. (2006);b– Herbert-Fort et al. (2006);c– Neeleman et al. (2013);d–\nNoterdaeme et al. (2008);e– Jorgenson et al. (2010);f– Christensen et al. (2019);g– Berg et al. (2015a). Symbol∗designates a very poorly\nstudied system, for which it is not clear what the extent of the metal-line profile is. Only lower limit has been determined. Column densities\nare taken from (1) De Cia et al. (2016), (2) Konstantopoulou et al. (2022), (3) Berg et al. (2015a) (corrected by Konstantopoulou et al. 2022,\nto the newest oscillator strengths, see Sec. 3.1), and ( ⋆) this work.\nTable 2: Parameters of the piecewise function fitting α-element\nenhancement in the low- and high- ∆v90subsamples. Case 1 is\nshown in Fig. 8.\nCaseαknee Plateau level Slope χ2\nν\n[M/H]tot, dex [α/Fe] nucl, dex\nHigh ∆v90\n1 −1.03±0.15 0.35∗−0.30∗0.39\n2 −1.01±0.28 0.34 ±0.05 −0.30∗0.45\n3 −0.82±0.66 0.31 ±0.04 −0.26±0.81 0.60\n4 ... ... −0.16±0.22 0.37\nLow∆v90\n1 −1.87±0.11 0.35∗−0.30∗0.58\n2 −1.80±0.32 0.32 ±0.09 −0.30∗0.60\n3 −1.56±0.38 0.25 ±0.04 −0.21±0.11 0.71\n4 ... ... −0.19±0.15 0.62\n∗– fixed parameters\nSgr (see Table 3). This can be explained by the di fferences in the\nearly evolution of the Magellanic Clouds from that of many otherTable 3: Stellar M ∗and dynamical M dmasses of nearby galaxies\nor their progenitors in cases of GSE and Sgr. The data are fromO\n– D’Onghia & Fox (2016),L– Limberg et al. (2022),W– Walker\net al. (2006),B−C– Bermejo-Climent et al. (2018).\ngalaxy M ∗, (M ⊙) M d, (M ⊙)\nSMC 3 ×108,O2.4×109,O\nLMC 3 ×109,O1.7×1010,O\nGSE 1.3 ×109,L...\nSgr ∼109,B−C...\nFnx ∼107,B−C108– 109,W\nlocal galaxies (Nidever et al. 2020). Obviously, Local Group\ngalaxies do not fully fit the simplified picture where faint, low-\nmass galaxies show lower enrichment e fficiencies. Their evolu-\ntionary histories depend on several factors such as the total stel-\nlar mass, amount of gas, interaction with the MW, e fficiency in\nproducing metals, and so on. However, it is not the goal of this\npaper to reconcile the stellar observations in nearby galaxies.\nFigure 10 shows the distribution of [X /Fe] nucl for di ffer-\nentα-elements in QSO-DLAs from the golden sample – except\nfor oxygen, for which only one measurement is available. The\nArticle number, page 9 of 22A&A proofs: manuscript no. arxiv_Velichko\nFig. 8:α-elements abundances for low- ∆v90(blue open triangles) and high ∆v90(orange open diamonds) QSO-DLA systems. Case\n1: Data have been approximated by three-piecewise function: high- αplateau +decline +low-αplateau. The five parameters of the\nfunction are: levels of high- and low- αplateaus, and the slope and positions of the high- and low- αknees. During the fitting, only\nthe position of the high- αknee is considered to be free, while the other parameters are fixed and set to values given for the MW.\nThe gray dashed curve shows behaviour for the MW as determined by McWilliam (1997). Case 2: The level of the high- αplateau\nand position of the high- αknee are allowed to vary. Case 3: Fitting the data with a two-piecewise function. Case 4: Fitting the data\nwith a linear function. Two filled points at the lower right panel indicate median values in [M /H]totand average values in [ α/Fe] nucl\nshow the systematic di fference between the two subsamples. All the parameters and their uncertainties are shown in Table 2.\nweighted averages of [X /Fe] nucl(gray vertical lines in Fig. 10)\nvary between the elements within 0.05 dex with standard de-\nviation (filled areas in Fig. 10) between 0.1 and 0.2. Thus, the\nabundance patterns of di fferentα-elements are fairly uniform\nand a splitting sample based on available elements is not ex-\npected to produce su fficiently di fferent [α/Fe] nuclversus [M /H]tot\nbehaviour. This also can be seen from Fig. D.1, where we\nshow the fitting of [X /Fe]–[M /H]totfor Ti, S, and Si (if avail-\nable), with three-piecewise functions. For the high- ∆v90sub-\nsample, the high- αplateau varies from [ α/Fe] tot=0.31±0.03 to\n0.36±0.02, while the position of the knee is within [M /H]tot=\n−1.09±0.13 –−0.80±0.18. For the low- ∆v90subsample, the\nparameter determination is less stable due to lack of data, espe-\ncially at low metallicities. From [Si /Fe] nucl, the parameters are\nnot constrained, while from Ti and S the plateau is 0.39 ±0.02\nand 0.25 ±0.02 and the knee is −1.84±0.44 and −1.34±0.11.5.2. Manganese\nThe distribution of [Mn /Fe] nuclwith metallicity is shown in\nFig. 11. The top panels shows the results for the golden sample.\nThere is a clear Mn underabundance for all DLAs with weighted\nmean values of −0.33±0.07 and −0.39±0.07 for the high- and\nlow-∆v90subsamples, respectively. We do not see a hint of an\nincrease of [Mn /Fe] nuclwith metallicity up to [Fe /H]tot=−0.53\ndex. This is contrary to our expectations because it is clear from\nthe behaviour of α-elements in Fig. 1 that in some DLAs, Type\nIa SNa should contribute to the nucleosynthesis, which, in turn,\nshould lead to an increase in the abundance of manganese rel-\native to iron. In addition, observations of stellar abundances\nof Mn both in the MW and in nearby galaxies show increas-\ning [Mn /Fe] nucltoward higher metallicity starting from [Fe /H]∼\n−1.5 dex (see gray dashed curve in Fig. 11 for the averaged\nbehaviour in the MW). From the analysis of elemental abun-\ndances in the neutral ISM of the SMC, De Cia et al. (2024) ob-\nArticle number, page 10 of 22Anna Velichko et al.: The α-element enrichment of gas in distant galaxies\nFig. 9:α-element overabundance with respect to Fe as a function\nof the total metallicity for the non-golden sample. Two points are\nout of the range of the figure at ([M /H]tot, [α/Fe])=(−1.93±0.15,\n0.82±0.09) and ( −0.76±0.13,−0.63±0.08).\nFig. 10: Distribution of [X /Fe] nuclfor Ti, Si, S, and Mg for the\ngolden sample. Gray vertical lines show the weighted average,\nwhile the filled areas correspond to the standard deviation.\nserved consistent [ α/Fe] nuclenhancement and [Mn /Fe] nuclunder-\nabundance at an approximately constant level throughout the\nwhole range of metallicity, which can be explained by contri-\nbution from recent core-collapse SNe.\nFor the non-golden sample, a hint of increase of [Mn /Fe] nucl\nis seen starting from [M /H]tot≈ −0.7 dex (lower panel of Fig.\n11). For the consistency of this comparison, the figure has been\nsupplemented with the data for the golden sample, but in this\ncase the [Mn /Fe] nucland [M /H]totvalues has been determined ac-\ncording to the methodology used for the non-golden sample. We\napproximated the entire data set with a two-piecewise function\nFig. 11: Manganese abundances in the ISM of DLAs. Upper\npanel: Mn abundances for the golden sample for low- ∆v90(blue\nopen triangles) and high- ∆v90(orange open diamonds). Horizon-\ntal color lines show the weighted average values of −0.39±0.10\nand−0.33±0.04 for the low- and high- ∆v90subsamples, re-\nspectively. Lower panel: Mn abundances for the non-golden\nsample (magenta open diamonds) as well as for the golden\nsample (symbols as on the upper panel) with [Mn /Fe] nucland\n[M/H]totbeing derived according to the methodology used for\nthe non-golden sample. Two points are out of the range of\nthe figure at ([M /H]tot, [Mn /Fe])=(−1.32±0.22, −0.63±0.16),\nand (−1.34±0.08, −0.66±0.15). In both panels, the gray dashed\ncurve shows typical behaviour for the MW taken from Mishen-\nina et al. (2015).\nconsisting of a plateau and an increase. The best-fitting parame-\nters are the following: the plateau is at the level of [Mn /Fe] nucl=\n−0.35±0.02, the knee position is at [M /H]tot=−0.69±0.13 and\nthe slope is 0.34 ±0.10.\n5.3. Phosphorus\nKonstantopoulou et al. (2023) found a most reliable value of re-\nfractory index B2Pfor DLAs equal to −0.26±0.08, which we\nused in this study. For Q1232 +082, we find that the column den-\nsity measurement for the bluest spectral component of the PII\nλ1152 line profile is unreliable because of strong contamination\nArticle number, page 11 of 22A&A proofs: manuscript no. arxiv_Velichko\nFig. 12: Phosphorus abundances in the ISM of DLAs for low-\n∆v90(blue open triangles) and high ∆v90(orange open dia-\nmonds).\n(De Cia et al. 2016). Thus, we excluded it from the calculation\nof the total N(PII). The new total column density is 12.88 ±0.28,\nwhich is lower than the previous value by 0.35 dex.\nAmong six DLAs for which there are measurements of P,\nfour do not show significant enhancement ([P /Fe] nucl≤0.30 dex),\nwhile the rest have values higher than 0.45 dex. However, in all\nthese cases the PII column densities were measured only from\none line (λ1152Å or 963Å), which is located in the Ly- αfor-\nest (De Cia et al. 2016). In general, it is necessary to fit multi-\nple lines of the same ion simultaneously to obtain reliable mea-\nsurements of the column density and avoid contaminations from\nother sources. In particular, contamination from HI lines are very\nlikely within the Ly- αforest (bluer than Ly- α). In addition, the\nsystems with high ∆v90are even more likely to be contaminated,\nbecause they have a broader velocity profile. Thus, we did not\nfind the current data to be reliable enough to robustly assess the\n[P/Fe] nucl. Nevertheless, the methodology in this paper shows the\npotential for discovery in this field.\n5.4. Zinc\nZn and Fe are often assumed to follow each other in nucle-\nosytnhesis in studies of DLAs with metallicities between 1 /100th\nof Solar and Solar, so that the observed gas-phase [Zn /Fe] is\ninterpreted as a pure tracer of dust (e.g. De Cia et al. 2016;\nDe Cia 2018; Konstantopoulou et al. 2022). However, accord-\ning to (Bensby et al. 2003; Nissen & Schuster 2011; Mishenina\net al. 2011; Barbuy et al. 2015; Du ffau et al. 2017), the rela-\ntion [Zn /Fe] versus [M /H] in stars in this metallicity range tends\nto behave partially like an [ α/Fe], but with a smaller amplitude.\nSitnova et al. (2022) showed from non-LTE calculations for the\nMW stars that Zn and Mg do not strictly follow each other.\n[Zn/Fe] is decreasing with metallicity because of various contri-\nbutions from core-collapse SNe and SNe Ia, and their di fferent\ntimescales (as for α-elements).\nOur measurements of [Zn /Fe] nuclare reliable only for the\ngolden sample because Zn is not included in the fit for to the\nabundance patterns. Figure 13 shows the [Zn /Fe] nuclwith metal-\nlicity for the golden sample, where the strength of depletion\nis calculated only from α-elements, without involving Zn. The\nFig. 13: Zinc abundances in the ISM of DLAs for the golden\nsample for low- ∆v90(blue open triangles) and high ∆v90(orange\nopen diamonds).\ndistribution partially resembles the behaviour of [ α/Fe] nucl, but\nwith smaller amplitudes ([Zn /Fe] nuclup to 0.2 dex). We observe\na potential decrease of [Zn /Fe] nucl, which is not unlike that of\nSitnova et al. (2022). The low- and high- ∆v90subsamples form\nseparate sequences: one at lower and one at higher metallicities.\nFor the non-golden sample, it is not meaningful to analyse the\n[Zn/Fe] nuclbecause it is distributed around zero by construction\n(see Fig. E.1).\n5.5. Additional caveats\nThis work is based on the assumption that we know well the\nrefractory behaviour of di fferent metals, namely, according to\nthe refractory index B2X. These coe fficients were calculated by\nKonstantopoulou et al. (2022) and De Cia et al. (2016) based\non the observed depletion sequences, that is, how the depletion\nvary with the overall amount of dust. This was derived after\ntaking an assumption on the α-element enhancement and Mn\nunderabundance to focus on and characterise the properties\nof dust depletion. Our results on the α-elements and Mn are\nindependent from the initial assumptions of Konstantopoulou\net al. (2022) and De Cia et al. (2016), for the following reasons.\nFirst, this paper uses additional data with respect to Konstan-\ntopoulou et al. (2022) and De Cia et al. (2016). Second, the\nenhancement of α-elements and deficiency of Mn are observed\nfor DLA systems with [Zn /Fe]∼0 (e.g. Fig. 3 of De Cia et al.\n2016), so there is no assumption in this regime (the α-plateau\nat low metallicities). Third, the assumed α-knee results of\nKonstantopoulou et al. (2022) and De Cia et al. (2016) were\nat the same metallicity as for the MW ([M /H]∼ −1.2) and the\nsame for all systems. These works made an overall statistical\ncorrection for α-elements to the whole sample, without studying\nthe details of individual systems. Eventually, the α-knee that we\nfind are at [ M/H]tot∼ −1.0 and [ M/H]tot∼ −1.8, different than\nprevious assumptions. Finally, we measured di fferentα-knees\nfor di fferent samples of galaxies, which make physical sense.\nArticle number, page 12 of 22Anna Velichko et al.: The α-element enrichment of gas in distant galaxies\n5.6. Testing the robustness of the results on the α-elements.\nTo derive the depletion coe fficients B2Xforα-elements, Kon-\nstantopoulou et al. (2022) applied corrections to account for\nthe effect ofα-element enhancement. To do so, they adopted\nthe shape of a standard nucleosynthetic curve taken from De\nCia et al. (2016) (see also Section 5.1). To demonstrate that\nthe assumption made by De Cia et al. (2016) and Konstan-\ntopoulou et al. (2022) has no (or minimal) impact on our present\nresults, we tested our calculations using an alternative set of\nB2Xcoefficients; for instance, considering that no assumption\nonα-element nucleosynthesis had ever been made. For this pur-\npose, we adopted the B2Xcoefficients of Konstantopoulou et al.\n(2022), but before the correction for α-element enhancement was\nmade. The test coe fficients should not be considered as more\nsolid, because they describe dust depletion without taking any\npotentialα-element enhancements into account. Nevertheless,\nthey are useful to check the robustness of our results and test\nthat the assumptions that went into the calculation of the B2X\ndoes not substantially a ffect our results.\nWith the test calculations, the values of B2Xturn out to be\nsystematically shifted downward by 0.11–0.20 depending on the\nelement (S, Mg, O, Si, and Ti). With the test coe fficients theα-\nelement enhancement is either the same or slightly higher com-\npared to the reference values. However, the di fference can be-\ncome more prominent, up to 0.10 dex, if the slope of the dust de-\npletion [Zn /Fe] fitis steep. In average the di fference of [α/Fe]nucl\ncalculated with the test and standard B2Xcoefficients is 0.03 for\nboth low- and high- ∆v90subsamples. Despite these di fferences,\nwe reproduced the same α-element behaviour as in Fig. 8, but\nwith slightly di fferent fitting parameters: in case of test calcu-\nlations, the position of the high- αknee is at [M /H]totlower by\n0.02 and 0.09 dex for low- and high- ∆v90subsamples, respec-\ntively. The α-plateau becomes slightly higher compared to the\nreference values by 0.1 and 0.05 dex for low- and high- ∆v90sub-\nsamples, respectively. The two subsamples are still clearly sepa-\nrated and the assumption used by Konstantopoulou et al. (2022)\nhas hardly any impact on our results.\nFor the golden sample, measurements of Ti are derived from\nthe weak TiII λ1910 absorption line. If there is a contamina-\ntion of the line, the column density of Ti could be overestimated\nand this may lead to an underestimation of the metallicity. We\ntest our main results by deriving [M /H]totand [Zn /Fe] fitfor\nthe golden sample in the same way as for the non-gold sample,\nnamely, based on only iron-group elements (Fe, Ni, Cr) and Zn\nand without using Ti and other α−elements. There are some dif-\nferences in metallicities (0.15 dex in average) only for the high-\n∆v90subsample, while for the low- ∆v90subsample the estimates\nare the same. Thus, our conclusions are holding regardless of the\nTi measurements.\n6. Conclusions\nIn this paper, we study abundance patterns of the neutral ISM\nin 110 QSO-DLA systems. We characterise the strength of de-\npletion [Zn /Fe] fitand measure the total (dust +gas) metallici-\nties [M /H]tot. In addition, from the deviations of the observed\nabundances from the linear fits to the abundance patterns, we\nderive [X /Fe] nuclfor O, Mg, Si, S, Ti, Cr, Fe, Ni, Zn, P, and Mn.\nThese are relative abundances with respect to Fe, after taking\ndust depletion into account and, thus, directly comparable to stel-\nlar relative abundances. We analyse the behaviour of [ α/Fe] nucl,\n[Mn/Fe] nucl, and [Zn /Fe] nucldepending on the total metallicity.The main analysis is based on 24 QSO-DLAs (the golden\nsample) for which there are measurements of column densities\nof Ti and at least one other α-element. We separated our golden\nsample into two groups, one with ∆v90<100 km s−1(low∆v90)\nand the other with ∆v90>100 km s−1(high ∆v90). This separa-\ntion is aimed at creating two subsamples of galaxies: one with a\nhigher and one with lower average stellar mass. In addition, we\nmade a minimal analysis for a non-golden sample for which the\ndepletion by dust and the total metallicity have been obtained\nfrom a linear fitting abundances of Zn and the iron-group ele-\nments Fe, Cr, and Ni.\nFor the golden sample, we found that less massive galaxies\nshow anα-element knee at lower metallicities than more mas-\nsive galaxies. If this collective behaviour can be interpreted as\nfor individual systems, this would suggest that more massive and\nmetal-rich systems evolve to higher metallicities before the con-\ntribution of SN-Ia levels out the [ α/Fe] enhancement created by\ncore-collapse SNe. This is possibly explained by di fferent SFR\nin galaxies of di fferent masses.\nFor the golden sample, there is a clear manganese under-\nabundance at about constant level up to [M /H]tot=−0.53 dex\nwith average values of [Mn /Fe] nuclequal to −0.33±0.07 and\n−0.39±0.07 for the high- and low- ∆v90subsamples, respectively.\nThis is not fully consistent with our expectations since (accord-\ning to the behaviour of α-elements) in some QSO-DLAs, SNe\nIa contribute to abundance pattern that, in turn, should lead to\nincrease of [Mn /Fe] nucl. It is only for the non-golden sample that\n[Mn/Fe] nuclincreases starting from [M /H]tot=−0.69 dex.\nWe traced slight e ffects in the behaviour of [Zn /Fe] nuclwith\nthe total metallicity that resemble the behaviour of [ α/Fe] nucl\nbut with a smaller amplitude. This is in agreement with several\nworks on stellar relative abundances (Bensby et al. 2003; Nis-\nsen & Schuster 2011; Mishenina et al. 2011; Barbuy et al. 2015;\nDuffau et al. 2017). A systematic bias by [M /H]totbetween high-\nand low- ∆v90subsamples is observed. These e ffects can be ob-\ntained only for the golden sample. Otherwise, by construction of\nthe method [Zn /Fe] nuclshould be approximately Solar.\nMeasurements of phosphorus should be treated with caution,\nsince in all cases the column densities were measured only from\none line (λ1152Å or 963Å), which is located in the Ly- αfor-\nest (bluer than Ly- α). Probably, because of this, all DLAs in the\nhigh- ∆v90subsample have inexplicably high values of [P /Fe] nucl,\nhigher than 0.45 dex. A wider velocity profile is more likely\nto be contaminated and lead to erroneous measurement. The\nlow-∆v90DLAs do not show significant enhancement of P, with\n[P/Fe] nucl≤0.15 dex.\nStudying the chemical properties of the neutral ISM in a\nsample of QSO-DLAs enables us to trace the collective be-\nhaviour of element abundances that can be interpreted as for in-\ndividual systems. A homogeneous set of α-element abundances\n(Ti and at least one other α-element) provides a powerful tool\nfor estimating the strength of depletion [Zn /Fe] fitand the total\nmetallicity [M /H]totin the ISM of QSO-DLAs, as well as the\nmore subtle deviations in the relative abundances due to the nu-\ncleosynthesis of specific stellar populations. This opens a new\nwindow into the study of the chemical evolution of distant galax-\nies.\nAcknowledgements. We thank the anonymous referee for the useful and con-\nstructive comments that improved this manuscript. A.V ., A.D.C., C.K., J.K.K.\nand T.R.H. acknowledge support by the Swiss National Science Foundation un-\nder grant 185692 funding the “Interstellar One” project.\nArticle number, page 13 of 22A&A proofs: manuscript no. arxiv_Velichko\nReferences\nAndrews, B. H., Weinberg, D. H., Schönrich, R., & Johnson, J. A. 2017, ApJ,\n835, 224\nArabsalmani, M., Møller, P., Perley, D. A., et al. 2018, MNRAS, 473, 3312\nAsplund, M., Amarsi, A. 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A. 1995, ApJS, 101, 181\nXing, Q.-F., Zhao, G., Aoki, W., et al. 2019, Nature Astronomy, 3, 631\nArticle number, page 14 of 22Anna Velichko et al.: The α-element enrichment of gas in distant galaxies\nAppendix A: Element abundances in the golden\nsample\nAppendix B: Element abundances in the\nnon-golden sample\nAppendix C: Dwarf galaxies\nAppendix D: Fitting the data\nAppendix E: Abundances of zinc for the non-golden\nsample\nAppendix F: New column densities\nArticle number, page 15 of 22A&A proofs: manuscript no. arxiv_Velichko\nTable A.1: Element abundances derived after taking into account the dust depletion.\nQSO [O /Fe] nucl[Mg/Fe] nucl[Si/Fe] nucl [S/Fe] nucl [Ti/Fe] nucl [Cr/Fe] nucl [Ni/Fe] nucl [P/Fe] nucl [Mn/Fe] nucl [Zn/Fe] nucl\nQ0058 −292 ... 0.26 ±0.04 0.23 ±0.05 0.27 ±0.04 0.29 ±0.12 0.07 ±0.05 0.00 ±0.09 ... ... −0.09±0.05\nQ0100 +130 ... 0.31 ±0.07 ... 0.32 ±0.04 0.32 ±0.12 0.01 ±0.04 ... −0.05±0.06−0.48±0.07 0.04 ±0.05\nQ0102 −190a ... ... ... 0.17 ±0.04 0.37 ±0.13 0.06 ±0.05 −0.10±0.08 ... −0.49±0.07 0.01 ±0.09\nQ0405 −443a ... ... 0.04 ±0.04 ... 0.04 ±0.06 0.00 ±0.05 −0.10±0.05 ... −0.41±0.07−0.25±0.06\nQ0405 −443c ... ... 0.19 ±0.04 0.19 ±0.04 0.18 ±0.08 0.03 ±0.05 −0.09±0.05−0.02±0.06−0.38±0.07 0.01 ±0.05\nQ0528 −250a ... ... 0.41 ±0.04 0.33 ±0.04 0.45 ±0.09 0.03 ±0.05 0.05 ±0.05 ... −0.30±0.08 0.25 ±0.06\nQ0528 −250b ... 0.05 ±0.04 0.33 ±0.04 0.10 ±0.04 0.05 ±0.08−0.07±0.04 0.01 ±0.05 ... −0.31±0.06−0.03±0.05\nQ0841 +129a ... ... ... 0.13 ±0.06 0.13 ±0.08 ... −0.10±0.05 ... −0.43±0.07−0.14±0.12\nQ1117 −134 ... ... 0.20 ±0.04 ... 0.29 ±0.08−0.03±0.06 <−0.10 ... ... 0.14 ±0.07\nQ1157 +014 ... 0.25 ±0.05 0.22 ±0.04 ... 0.23 ±0.06 0.02 ±0.04 −0.09±0.05 0.15 ±0.07 −0.36±0.06 0.06 ±0.05\nQ1223 +178 ... 0.41 ±0.06 0.12 ±0.04 0.17 ±0.04 0.20 ±0.08−0.03±0.04−0.16±0.07 ... −0.48±0.07−0.19±0.05\nQ2206 −199a ... 0.14 ±0.06 0.25 ±0.04 0.23 ±0.04 0.22 ±0.06−0.04±0.04 0.01 ±0.05 0.46 ±0.07 −0.38±0.07−0.01±0.05\nQ2243 −605 ... ... 0.31 ±0.04 0.34 ±0.04 0.34 ±0.07−0.06±0.05 0.06 ±0.05 0.54 ±0.06 −0.26±0.07 0.17 ±0.05\nSDSS1249 −0233 ... ... ... 0.27 ±0.06 0.27 ±0.12 0.13 ±0.05 0.01 ±0.07 ... −0.40±0.07 0.24 ±0.07\nQ2230 +025 ... ... 0.22 ±0.06 0.22 ±0.06 0.22 ±0.11−0.10±0.06 0.13 ±0.07 ... −0.31±0.07 0.06 ±0.07\nHE1104 −1805 0.32 ±0.20 ... 0.30 ±0.04 ... 0.30 ±0.09−0.00±0.04−0.15±0.05 ... −0.18±0.15 0.01 ±0.05\nQ0027 −1836 ... 0.73 ±0.06 0.44 ±0.05 0.35 ±0.04 0.39 ±0.08 0.09 ±0.04 −0.07±0.05 ... −0.29±0.07 0.23 ±0.05\nQ1210 +17 ... ... 0.08 ±0.05 0.07 ±0.04 0.06 ±0.10−0.04±0.05−0.14±0.07 ... −0.44±0.06−0.17±0.07\nFJ0812 +32 ... ... 0.20 ±0.04 ... 0.25 ±0.07 0.03 ±0.04 −0.09±0.05 ... −0.32±0.07 0.06 ±0.05\nJ1135 −0010 ... ... 0.36 ±0.05 ... 0.36 ±0.07 0.08 ±0.04 0.12 ±0.05 ... −0.35±0.06 0.06 ±0.06\nQ2231 −00 ... ... 0.37 ±0.05 0.62 ±0.15 0.44 ±0.10−0.15±0.06−0.11±0.07 ... −0.32±0.07 0.21 ±0.07\nJ1200 +4015 ... ... ... 0.31 ±0.06 0.31 ±0.07−0.10±0.06 0.06 ±0.07 ... ... 0.17 ±0.07\nQ1210 +175 ... 0.28 ±0.09 ... 0.02 ±0.08 0.11 ±0.06 ... −0.02±0.05 ... −0.33±0.08 ...\nQ1232 +082 ... 0.60 ±0.09 0.53 ±0.08 0.39 ±0.08 0.46 ±0.10 ... −0.03±0.06 0.26 ±0.18 −0.14±0.13 ...\nQSO zabs logN(FeII) log N(NiII) log N(TiII)\nQ0102 −190b 2.92648 13.80 ±0.02<13.05<12.38\nQ0112 +030 2.42299 14.85 ±0.02 13.57 ±0.06 12.48 ±0.16\nQ0112 −306a 2.41844 13.42 ±0.04 ... <12.28\nQ0112 −306b 2.70163 14.81 ±0.03 13.65 ±0.01 ...\nQ0135 −273a 2.10735 14.58 ±0.03<13.35<12.50\nQ0135 −273b 2.80004 14.77 ±0.03 13.46 ±0.06<12.51\nQ0336 −017 3.06209 14.79 ±0.02 13.42 ±0.05<11.92\nQ0450 −131 2.06658 14.24 ±0.02 13.34 ±0.04<12.65\nQ0913 +072 2.61843 13.10 ±0.02<12.42<12.02\nQ1036 −229 2.77779 14.76 ±0.01 13.71 ±0.02<12.16\nQ1108 −077b 3.60767 13.88 ±0.02<12.35<12.25\nQ1210 +175 1.89177 14.90 ±0.02 13.67 ±0.02 12.31 ±0.03\nQ1232 +082 2.33771 14.52 ±0.01 13.30 ±0.03 12.43 ±0.09\nQ1337 +113a 2.50792 13.40 ±0.02<12.85<12.55\nQ1337 +113b 2.79584 14.33 ±0.02 13.20 ±0.06 ...\nQ1340 −136 3.11835 13.93 ±0.02 12.89 ±0.05 ...\nQ1409 +095b 2.45593 13.74 ±0.02<12.66<12.32\nQ1451 +123b 2.46921 13.38 ±0.02<13.22<12.58\nQ1451 +123c 3.17112 13.35 ±0.09 13.30 ±0.06<13.11\nQ2059 −360a 2.50734 13.50 ±0.03<12.79<12.29\nQ2059 −360b 3.08261 14.48 ±0.02 13.02 ±0.06<12.29\nQ2152 +137b 3.31599 14.50 ±0.02<12.88<12.51\nQ2206 −199b 2.07622 13.34 ±0.01 12.24 ±0.09<11.59\nQ2230 +025 1.86427 15.35 ±0.01 14.04 ±0.07<12.45\nQ2231 −002 2.06615 14.92 ±0.01 13.61 ±0.01 ...\nQ2332 −094b 3.05722 14.37 ±0.01 13.19 ±0.11<12.51\nQ2344 +125 2.53787 14.01 ±0.01 12.58 ±0.08<12.30\nQ2348 −011a 2.42695 14.83 ±0.01 13.74 ±0.02<12.52\nQ2348 −011b 2.61473 14.65 ±0.01 13.34 ±0.05 ...\nQ2359 −022b 2.15368 13.78 ±0.04<13.22<12.60\nTable F.1: Column densities for the DLAs that are absent in the list compiled by Konstantopoulou et al. (2022).\nArticle number, page 16 of 22Anna Velichko et al.: The α-element enrichment of gas in distant galaxies\nFig. A.1: Abundance patterns for the golden sample of QSO-DLA (see Sec. 3). Blue dashed curve is fitting to the α-elements,\nshown by blue open triangles. Red curve is shifted blue one to the value averaged through Ni, Cr, and Fe. The intersection of the\ngray dashed lines at B2X=0 shows the metallicity [M /H]totof each system. Gray areas show 1 σconfidence intervals.\nArticle number, page 17 of 22A&A proofs: manuscript no. arxiv_Velichko\nFig. B.1: Abundance patterns for the non-golden sample of QSO-DLA (see Sec. 3). Total metallicity [M /H]totand the dust depletion\n[Zn/Fe] fithave been derived from fitting iron-group elements and Zn.\nArticle number, page 18 of 22Anna Velichko et al.: The α-element enrichment of gas in distant galaxies\nFig. B.1: continued.\nArticle number, page 19 of 22A&A proofs: manuscript no. arxiv_Velichko\nFig. B.1: continued.\nFig. C.1: Chemical abundances of α-elements taken from the APOGEE DR17 catalog for dwarf galaxies (color points with x,yerror\nbars), with Hasselquist et al. (2021) selection. Color solid curves show the approximation of the data points by piecewise functions\nconsisting of two or three straight lines representing the average behaviour of α-elements relative to iron. Dark gray solid curves\nhave been obtained by averaging the data in each metallicity bin. Typically, the bin width is equal to 0.1 dex, but in some ranges\nwhere there are too few stars, it is increased up to 0.2 dex. For comparison, the behaviour typical for the MW is shown by gray\ndashed curve (McWilliam (1997)).\nArticle number, page 20 of 22Anna Velichko et al.: The α-element enrichment of gas in distant galaxies\nFig. D.1: [X /Fe] nuclfor threeα-elements Ti, Si and S. The data are fitted by three-piecewise functions as for Case 2 to find the level\nof the high- αplateau and the position of the high- αknee.\nFig. E.1: Zinc abundances in the ISM of QSO-DLAs for the non-golden sample.\nArticle number, page 21 of 22A&A proofs: manuscript no. arxiv_Velichko\n200\n 100\n 0 100 2000.00.51.0Normalized Flux FeII1125\n200\n 100\n 0 100 2000.00.51.0\nFeII1143\n200\n 100\n 0 100 2000.60.81.0Normalized Flux NiII1317\n200\n 100\n 0 100 2000.60.81.0\nTiII3067\n200\n 100\n 0 100 200\nRel.velocity(kms1)\n0.60.81.0Normalized Flux TiII3073\n200\n 100\n 0 100 200\nRel.velocity(kms1)\n0.51.0\nTiII3384\nFig. F.1: Velocity profiles of selected low-ionisation transition\nlines from the DLA system at zabs=1.892 towards Q 1210 +175.\n200\n 100\n 0 100 2000.51.0\nFeII1608\n200\n 100\n 0 100 2000.80.91.0Normalized FluxFeII1611\n200\n 100\n 0 100 2000.81.0\nNiII1741\n200\n 100\n 0 100 200\nRel.velocity(kms1)\n0.51.0Normalized FluxSiII1808\n200\n 100\n 0 100 200\nRel.velocity(kms1)\n0.500.751.00\nTiII1910\nFig. F.2: Velocity profiles of selected low-ionisation transition\nlines from the DLA system at zabs=2.338 towards Q 1232 +082.\nArticle number, page 22 of 22"    },    {        "title": "2402.15596v1.An_experimental_scheme_for_determining_the_Berry_phase_in_two_dimensional_quantum_materials_with_a_flat_band.pdf",        "content": "An experimental scheme for determining the Berry phase in two-dimensional\nquantum materials with a flat band\nLi-Li Ye,1Cheng-Zhen Wang,2and Ying-Cheng Lai1, 3,∗\n1School of Electrical, Computer and Energy Engineering,\nArizona State University, Tempe, Arizona 85287, USA\n2Wave Transport in Complex Systems Lab, Department of Physics,\nWesleyan University, Middletown, CT-06459, USA\n3Department of Physics, Arizona State University, Tempe, Arizona 85287, USA\n(Dated: February 27, 2024)\nExperimentally feasible methods to determine the Berry phase, a fundamental quantity charac-\nterizing a quantum material, are often needed in applications. We develop an approach to detecting\nthe Berry phase by using a class of two-dimensional (2D) Dirac materials with a flat band, the\nα-T3lattices. The properties of this class of quantum materials are controlled by a single parameter\n0≤α≤1, where the left and right endpoints correspond to graphene with pseudospin-1/2 and\nthe dice lattice with pseudospin-1 Dirac-Weyl quasiparticles, respectively, and each specific value of\nαrepresents a material with a unique Berry phase. Applying a constant electric field to the α-T3\nlattice, we calculate the resulting electric current and find a one-to-one correspondence between the\ncurrent and the Berry phase in both the linear and nonlinear response regimes. In the linear (Kubo)\nregime, the main physics is the Zitterbewegung effect. In the nonlinear regime, the Schwinger mech-\nanism dominates. Beyond the nonlinear regime, Bloch-Zener oscillations can arise. Measuring the\ncurrent thus provides an effective and experimentally feasible way to determine the Berry phase for\nthis spectrum of 2D quantum materials.\nI. INTRODUCTION\nThe Berry phase (the geometric phase or the\nPancharatnam-Berry phase) [1–3] of the electronic wave\nfunction is a fundamental characteristic of quantum ma-\nterials and can have significant effects on material prop-\nerties and physical phenomena such as polarization, mag-\nnetism, and quantum anomalous spin Hall effects [4].\nThe Berry phase arises when a parameter of the sys-\ntem completes a cycle of adiabatic changes: even as the\nparameter returns to its initial value, the wave function\ngains an extra phase of purely geometric origin. The\nvalue of the Berry phase depends on the nature of the\nquasiparticles which, in turn, depends on the specific\nquantum material. Given a family of quantum mate-\nrials, the Berry phase is effectively a unique identifier\nof each material in the family. For example, monolayer\ngraphene hosting a pair of Dirac cones and pseudospin-\n1/2 quasiparticles, the Berry phase is ±πassociated, re-\nspectively, with the electronic states in the two Dirac\ncones [5, 6]. For bilayer graphene, the Berry phase is 2 π,\nwhich leads to unconventional quantum Hall effect [7].\nFor pseudospin-1 Dirac-Weyl materials, the Berry phase\nis zero [8, 9]. In recent years, various two-dimensional\n(2D) Dirac materials have been discovered at a rapid\npace [10–12], each carrying a unique type of quasipar-\nticles with a unique value of the Berry phase. Given a\nnew quantum material, knowing the Berry phase is thus\nof theoretical, experimental, and applied interests.\nIn principle, the phenomenon of Aharonov-Bohm in-\nterference provides an approach to assessing the Berry\n∗Ying-Cheng.Lai@asu.eduphase [13–15]. Take graphene as an example. For a cir-\ncular graphene p-njunction resonator, due to the ±π\nBerry phase of the quasiparticles, as the strength of an\nexternal magnetic field is tuned, a sudden change in\nthe energy of the angular-momentum states can occur,\nproviding an indirect way to ascertain the value of the\nBerry phase [14, 15]. For photonic crystals, their analogy\nwith graphene was exploited to measure the geometric\nBerry phase by removing the dynamical phase [16]. For\na general family of 2D Dirac-Weyl materials (the α-T3\nlattices), the semiclassical dynamics of a chaotic cavity\nmade of such a material were explored to infer its Berry\nphase [9]. In particular, by applying a gate voltage to\ngenerate a quasi-confinement of a certain geometric shape\nthat generates chaos in the classical limit, a one-to-one\ncorrespondence between the exponential rate of particles\nescaping from the cavity and the Berry phase was identi-\nfied. Despite the theoretical appeal of this semiclassical\nphenomenon, experimentally monitoring the decay of an\nensemble of quasiparticles from a cavity of certain quan-\ntum material is not feasible at present.\nIn this paper, we present theoretical calculations lead-\ning to an experimentally feasible approach to detecting\nthe Berry phase for the α-T3lattice family whose ma-\nterial properties are controlled by a single parameter:\n0≤α≤1. An α-T3lattice has the honeycomb lat-\ntice as its base with an additional atom at the center\nof each hexagonal unit cell. In the tight-binding ap-\nproximation, the center atom couples with any of the\nhexagonal atoms with the energy αtϵ, where tϵis the\nnearest-neighbor coupling energy of the honeycomb lat-\ntice. Because of the center atom, an α-T3lattice with\nα > 0 possesses three distinct energy bands: a pair of\nDirac cones and a flat band through the contact point ofarXiv:2402.15596v1  [cond-mat.mes-hall]  23 Feb 20242\nthe two Dirac cones. As αincreases from zero, a contin-\nuous spectrum of 2D Dirac-Weyl materials is generated:\nfrom the graphene ( α= 0) to the dice lattice ( α= 1),\nand the corresponding Berry phase can change from π\nto zero. As a result of the continuous decrease in the\nBerry phase, a number of pertinent physical phenomena\nchange their characteristics. For example, the orbital\nmagnetic response [17] at the Dirac point changes from\ndiamagnetic ( α= 0) to paramagnetic ( α= 1) and the\nnature of the Hall quantization [8] switches from rela-\ntivistic to nonrelativistic. Moreover, the patterns of op-\ntical response [18] and magneto-optical modulation [19]\nchange because they depend on the interband transitions\namong the three bands. Further, optical conductivity\nquantization and higher-order harmonic generation were\nobserved [20], so was the effect of a broken flat-band on\nthe integer quantum Hall effect by the disorder or stag-\ngered lattice potential [21]. Experimentally, the α-T3lat-\ntice has been realized in the critical doping material [22]\nHg1−xCdxTe. For α= 1, the dice lattice described by\nthe pseudospin-1 Dirac-Weyl Hamiltonian can be grown\nin the transition-metal oxide [23] SrTiO 3/SrIrO 3/SrTiO 3\nor in graphene-In 2Te2bilayer [24].\nWe focus our calculations on the electric current den-\nsity (or simply the current) produced when a constant\nelectric field is applied to the α-T3lattice. In the clas-\nsical Drude picture, when driven by a constant electric\nfieldE, the momentum of the electrons in ballistic trans-\nport will increase with time: q=eEt. Nevertheless,\nDirac electrons will be excited instantaneously to the\nFermi velocity (pinned to the “light cone”) [25], where\nthe excitation mechanism is described by the Schwinger\neffect [26] or the Landau-Zener dynamics [27, 28] that\noccur where there are two avoided-crossing energy lev-\nels under the adiabatic evolution induced by the elec-\ntric field. Another relevant phenomenon is Bloch oscilla-\ntions [29, 30] in the time evolution of the electronic states\nin a single energy band. When multiple bands with-\nout crossings are present, Bloch-Zener oscillations [31–\n33] can take place. For the α-T3lattice, irregular Bloch-\nZener oscillations [34] can arise, due to the mixed interfer-\nence of the quantum states in multi-bands. In addition,\nthe mass term associated with the Dirac electrons or a\nweak disorder can render a nonzero minimal conductiv-\nity that depends on the value of α[35]. Other relevant\ntransport phenomena in the α-T3lattice includes the lin-\near response in graphene with the chiral anomaly and\nnonlinear response when the perturbation theory breaks\ndown [36], as well as nonlinear conductivity with THz-\ninduced charge transport [37]. Nonequilibrium dynamics\nbeyond the linear response in 3D Weyl semimetals [38]\nand nodal loop semimetals [39] have also been studied.\nThe main physical considerations behind our calcula-\ntions of the current are as follows. On different time\nscales, the transport properties and the physical mecha-\nnisms are distinct. In particular, in the Kubo regime [25]\nunder the weak field approximation, the average current\ndensity is saturated and dominated by the Zitterbewe-gung effect [40] originated from the interference between\nthe energy bands, which defines the regime of linear re-\nsponse. A strong electric field places the system in the\nSchwinger regime [26], where the electrons are excited by\nthe Schwinger mechanism in which the vacuum field loses\nenergy to produce electron-positron pairs. The transition\nprobabilities among the energy bands are described by\nthe Landau-Zener dynamics [41], where the quasiparti-\ncles adiabatically evolve and transitions occur about the\npoint at which the two levels are closest to each other\nbut without crossing. In this regime, the current is pro-\nportional to the number of the excited particles, repre-\nsenting a nonlinear response. When the product of the\nelectric field and time is comparable to the lattice con-\nstant, Bloch oscillations [29, 30] become important in the\nLandau-Zener dynamics, leading to Bloch-Zener oscilla-\ntions [31–33]. The main finding is a monotonic depen-\ndence of the current on the materials parameter αin\nboth the linear and nonlinear response regimes, implying\na one-to-one correspondence between the current and the\nBerry phase, thereby providing a possible experimental\nscheme to determine the latter.\nII. CURRENT AND BERRY PHASE\nCALCULATION FOR α-T3LATTICE\nA. Zero-field effective Hamiltonian\nTo calculate the current and the Berry phases in an\nα-T3lattice, we begin with the zero-field lattice Hamil-\ntonian. The basic lattice structure is shown in Fig. 1 (a),\nwhere there are three distinct atoms in a unit cell: A and\nB atoms belonging to the base hexagonal lattice, and C\natom at the center of the unit cell. The tight-binding\nHamiltonian is given by [17]\nH=\n0 fpcosφ 0\nf∗\npcosφ 0 fpsinφ\n0 f∗\npsinφ 0\n, (1)\nwith\nfp=−tϵ\u0010\n1 +e−ip·a1/ℏ+e−ip·a2/ℏ\u0011\n, (2)\nwhere p= (px, py) and tϵis the nearest-neighbor hopping\nenergy between an A and a B atom with the parametriza-\ntion [17]: tan φ=α∈[0,1]. In the position space,\nthe primitive translation vectors are a1=a\u0000√\n3/2,3/2\u0001\n,\na2=a\u0000\n−√\n3/2,3/2\u0001\n, where ais the lattice constant\n(inter-site distance). The base vectors in the recipro-\ncal lattice of the hexagonal Brillouin zone are b1=\u0000√\n3/3,1/3\u0001\n2π/a andb2=\u0000\n−√\n3/3,1/3\u0001\n2π/a. The\neigenenergy spectrum [17] of α-T3lattice is independent\nof the value of α, which consists of a zero energy flat band\nE0= 0 and two linearly dispersive bands Eλ=λ|fp|\nwith the band index λ=±. The structure of the posi-\ntive band is shown in Fig. 1(b). There are two nonequiv-3\nFIG. 1. Structure of α-T3lattice and zero field energy-band\nstructure. (a) The lattice structure, where each unit cell con-\ntains three distinct atoms. The nearest-neighbor hopping en-\nergy between the A and B sites is tϵand that between B\nand C sites is αtϵ. The material parameter αcharacterizes\nthe relative coupling strength between the flat band and the\nDirac-cone bands. (b) The zero-field energy spectrum of the\npositive dispersion band in the 2D momentum space for any α\nvalue, where the red hexagon denotes the first Brillouin zone\nand the zero energy points give two topologically nonequiva-\nlent Dirac points ζKwith the valley index ζ=±1.\nalent Dirac contact points: + K=\u0000\n2/(3√\n3),0\u0001\n2π/aand\n−K=\u0000\n−2/(3√\n3),0\u0001\n2π/a.\nDenoting the momentum vector from a Dirac point ζK\nasqand linearizing the corresponding function fqin the\nHamiltonian as\nfq≈vF(ζqx−iqy), (3)\nwhere vFis Fermi velocity, qx, qyare the momentum\ncomponents measured from Dirac points with the valley\nindex ζ=±1, we obtain the effective Hamiltonian in the\ncontinuum limit at low energy excitation as\nHq≈vF\u0000\nζS′\nx(φ)qx+S′\ny(φ)qy\u0001\n, (4)with\nS′\nx(φ)≡\n0 cos φ0\ncosφ0 sin φ\n0 sin φ0\n, (5)\nS′\ny(φ)≡\n0−icosφ 0\nicosφ 0−isinφ\n0 isinφ 0\n. (6)\nForα= 0, S′\nx(0) and S′\ny(0) become σx⊕0 and σy⊕0,\nrespectively, leading to\nHq|α=0=vF(ζσxqx+σyqy)⊕0, (7)\nHSpin−1\n2=vF(σxqx+σyqy). (8)\nForα= 1, S′\nx(π/4) and S′\ny(π/4) are SxandSy, respec-\ntively, i.e., the components of the spin-1 matrix vector.\nIn this case, we have\nHq|α=1=vF(ζSxqx+Syqy), (9)\nHspin−1=vF(Sxqx+Syqy). (10)\nThe continuum effective Hamiltonian of the α-T3lattice,\nas given by Eq. (4), is a general model that includes the\npseudospin-1 /2 and pseudospin-1 lattices as the two op-\nposite limiting cases. By varying the coupling strength\nα∈[0,1], a continuous spectrum of Dirac-Weyl materials\nwith a flat band can be generated.\nB. Low-excitation continuum effective α-T3\nHamiltonian in a constant electric field\nWe apply a uniform and constant electric field to an α-\nT3lattice in the xdirection starting at time t= 0, repre-\nsented by a time-dependent vector potential [25, 42]. The\ncorresponding continuum effective Hamiltonian around\nthe two nonequivalent Dirac points becomes\nHq(t) =vF\u0000\nζS′\nx(φ)qx(t) +S′\ny(φ)qy\u0001\n,(11)\nwith qx(t)≡qx−eA(t), where A(t) =EtΘ (t) and Θ( t)\nis a unit step function of time. The quantum dynamics\nare governed by\niℏ∂tψq(t) =Hq(t)ψq(t). (12)\nIn the Landau-Zener adiabatic basis [25, 41], under an\ninfinitesimal electric field, the evolution of a quantum\nstate is described by\nU†\nq(t)Hq(t)Uq(t) =Szεq(t), (13)\nwhere Szis the z-component of the spin-1 matrix vector\nandUq(t) is given by\n\n1√\n2cosφ eiθq sinφ eiθq 1√\n2cosφ eiθq\n1√\n20 −1√\n21√\n2sinφ e−iθq−cosφ e−iθq1√\n2sinφ e−iθq\n,(14)4\nwith θq(t) being the phase of fq(t) and\ntanθq(t) =−ζqy/[qx−eA(t)]\nbecause of\nfq(t)≈vF(ζqx(t)−iqy). (15)\nThe eigenstates of positive, zero, and negative energy\nbands can then be written down and be distinguished.\nFor example, the positive eigenenergy spectrum ϵq(t)≡\n+|fq(t)|is given by\nεq(t) =vFq\n(qx−eA(t))2+q2y. (16)\nThe transformed time-dependent Dirac equation be-\ncomes\niℏ∂tΦq(t) =\u0014\nSzεq(t)−eSxℏv2\nFqyeE\nζε2q(t)\u0015\nΦq(t),(17)\nwhere\nΦq(t)≡U†\nq(t)ψq(t),\neSx≡Sxsin 2φ−SLcos 2φ,with SLdefined by\nSL≡\n−1/2 0−1/2\n0 1 0\n−1/2 0−1/2\n. (18)\nThe second term in Eq. (17) arises from the time de-\npendence of the unitary transformation −iℏU†\nq(t)∂tUq(t).\nConsider the initial state in which the lower Dirac cone\nis fully occupied:\nΦq(t= 0) = [0 ,0,1]T, (19)\nthe average current density ⟨Jx⟩q(t) in the momentum\nspace is invariant under the unitary transformation:\n⟨Jx⟩q(t)≡ −e[ψq(t)]†\u0000\n∂qx(t)Hq(t)\u0001\nψq(t), (20)\n=−evFζΦ†\nq(t)\u0002\nU†\nq(t)S′\nx(φ)Uq(t)\u0003\nΦq(t).\nIn the adiabatic basis, Φ q(t) can be expressed as\nΦq(t) =\u0002ξq(t), γq(t), βq(t)\u0003T, (21)\nwhere |ξq|2,|γq|2, and|βq|2are the probabilities of find-\ning the quasiparticle in the upper, flat, and lower band,\nrespectively. The average current density can be decom-\nposed into two parts [25],\n⟨Jx⟩q(t) =⟨Jx⟩intra\nq(t) +⟨Jx⟩inter\nq(t), (22)\nwhich are the intraband and interband currents, respec-\ntively, given by\n⟨Jx⟩intra\nq(t) =−evFζcos[θq(t)]\u0010\n|ξq(t)|2− |βq(t)|2\u0011\n, (23)\n⟨Jx⟩inter\nq(t) =−evFζsin[θq(t)]\u0012\n2 cos[2 φ]ℜ\u0002\niξ∗\nq(t)βq(t)\u0003\n+√\n2 sin[2 φ]ℜ\u0002\niξ∗\nq(t)γq(t) +iγ∗\nq(t)βq(t)\u0003\u0013\n,(24)\nwith\nsin[2φ] = 2α/(1 +α2),\ncos[2φ] = (1 −α2)/(1 +α2).\nThe intraband component represents the current density\nof the electrons and holes in the upper and lower band,\nrespectively, with the opposite signs. The interband com-\nponent depicts the current density due to the interference\nbetween the upper, flat and lower bands, where the ma-\nterial parameter αmodulates contributions to the cur-\nrent density from the coupling between energy bands. In\nparticular, for α= 0, the only contribution to the cur-\nrent is the transition from the lower to the upper band.\nHowever, for α= 1, the current density is due to the\ncoupling between the flat band and the other bands. Forα∈(0,1), the interband current density is a mixture of\nthe two extreme cases.\nUsing the normalization condition, we have\n|ξq(t)|2− |βq(t)|2= 2|ξq(t)|2+|γq(t)|2−1.(25)\nSubstituting Eq. (25) into the intraband current in\nEq. (23), the constant in the third term of Eq. (25) will\nvanish after a momentum integration over the Brillouin\nzone [25]. As a result, the intraband current can be re-\ngarded as the contribution from the flat and upper bands.\nSpecifically, we denote Jintra(t) andJinter(t) as the mo-\nmentum integrals of ⟨Jx⟩intra\nq(t), and ⟨Jx⟩inter\nq(t), respec-\ntively, over the first hexagonal Brillouin zone shown in\nFig. 1(b) with\nJ(t) =Jintra(t) +Jinter(t). (26)5\nWe can decompose the contributions from all the energy\nbands in the intraband and interband current by inte-\ngrating Eqs. (23) and (24) over the momentum space and\nfollowing the term order in Eqs. (23) and (24) to define\nJintra(t) =Jintra\nξ(t) +Jintra\nγ(t),\nJinter(t) =Jinter\nξβ(t) +Jinter\nξγ(t) +Jinter\nγβ(t). (27)\nThese expressions are convenient for treating the contri-\nbutions to the current by the multiple energy bands in\nthe weak field (Sec. III A) and strong field (Sec. III B)\ncases.\nTo streamline numerical calculations, we define a num-\nber of dimensionless physical quantities in the continuum\neffective α-T3model:\net=t/t0,\neqx=qx/q0,\neqy=qy/q0,\neE=E/E0,\neεq(t) =εq(t)/ε0,\neJ(t) =J(t)/J0,\n⟨eJx⟩q(t) =⟨Jx⟩q(t)/⟨J0⟩q, (28)\nwhere t0≡ℏ/tϵ,q0≡tϵ/vF,E0≡t2\nϵ/(eℏvF),ε0≡tϵ,\nJ0≡e2E0/ℏ∼evF/a2,\nand⟨J0⟩q≡evF.\nC. General α-T3lattice Hamiltonian in a constant\nelectric field\nWith a constant electric field switched on at t= 0 in\nthex-direction, the x-component of the momentum ispx(t)≡px−eEt. The general Hamiltonian of the α-T3\nlattice is given by\nH(t) =\n0 fp(t) cosφ 0\nf∗\np(t) cosφ 0 fp(t) sinφ\n0 f∗\np(t) sinφ 0\n,(29)\nwhere\nfp(t) =−tϵ \n1 + 2 exp\u0012\n−i3\n2pya\nℏ\u0013\ncos √\n3\n2px(t)a\nℏ!!\n.\n(30)\nThe eigenenergy spectrum of the positive dispersion band\nin the whole hexagonal Brillouin zone is determined by\nεp(t) = + |fp(t)|:\nεp(t) =tϵq\n1 + 4 cos Xp(t) (cos Yp+ cos Xp(t)),(31)\nwhere Xp(t) =√\n3px(t)a/(2ℏ) and Yp= 3pya/(2ℏ). The\nunitary transformation Up(t) is similar to that in Eq. (14)\nexcept that θq(t) is now replaced by θp(t), which is the\nphase of fp(t) in Eq. (30). The transformed quantum\ndynamics are governed by [34]\niℏ∂tΦp(t) =\u0014\nSzεp(t)−eSxat2\nϵeE\nε2p(t)Cp(t)\u0015\nΦp(t),(32)\nwith the coefficient given by\nCp(t) =√\n3 sinYpsinXp(t).\nThe average current density ⟨Jx⟩p(t) contains two contri-\nbutions: interband and intraband transitions [25], which\ncan generally be written as [34]\n⟨Jx⟩intra\np(t) =J11\nx,p(t)\u0010\n|ξp(t)|2− |βp(t)|2\u0011\n, (33)\n⟨Jx⟩inter\np(t) = 2ℜ\u0002\nJ13\nx,p(t)ξ∗\np(t)βp(t)\u0003\n+ 2ℜ\u0002\nJ12\nx,p(t)ξ∗\np(t)γp(t) +J23\nx,p(t)γ∗\np(t)βp(t)\u0003\n. (34)\nTo gain insights into these contributions to the average\ncurrent density ⟨Jx⟩p(t), we recall the matrix of the cur-\nrent density operator:\nJx,p(t) =−eU†\np(t)∂px(t)H(t)Up(t).\nThe intraband contribution is made by both electrons\nand holes, corresponding to\nJ11\nx,p(t)≡J0\nx,p(t) cos Θ p(t)\nJ33\nx,p(t) =−J11\nx,p(t),respectively. The interband contribution arises from the\ninterference of the transitions from the lower to the flat\nband or the upper band and from the flat to the upper\nband, corresponding to J23\nx,p(t),J13\nx,p(t) and J12\nx,p(t),\nrespectively, which are given by\nJ13\nx,p(t)≡iJ0\nx,p(t) cos[2 φ] sin[Θ p(t)],\nJ12\nx,p(t)≡iJ0\nx,p(t) sin[2 φ] sin[Θ p(t)]/√\n2,\nJ23\nx,p(t) =J12\nx,p(t), (35)6\nwhere Θ p(t)≡θp(t) +YpandJ0\nx,p(t) is the common\nfactor with the dimension of the current density:\nJ0\nx,p(t) =−√\n3 sin[Xp(t)]eatϵ/ℏ.\nWe use J(t) to denote the integration of ⟨Jx⟩p(t) in the\nfirst Brillouin zone, which will be used to characterize the\nBloch oscillations (in Sec. III C).\nFor the general α-T3lattice calculations, the following\ndimensionless quantities are convenient:\nepx=px/p0,\nepy=py/p0,\neE=E/E 0,\neJ(t) =J(t)/J0,\n⟨eJx⟩p(t) =⟨Jx⟩p(t)/⟨J0⟩p, (36)\nwith p0≡ℏ/a,E0≡tϵ/(ea),J0=e2E0/ℏ∼etϵ/(ℏa)\nand⟨J0⟩p=eatϵ/ℏ.\nD. Calculating the Berry phases of the α-T3lattice\nThe Berry phases associated with the conical and flat\nbands can be calculated by assuming that the corre-\nsponding eigenstates adiabatically evolve with time along\nan arbitrarily closed loop around the Dirac points ζKin\nthe momentum space [8]:\nϕn,ζ=−i\nπI\ndp· ⟨ψn|∇p|ψn⟩, (37)\nwhich can be calculated either by the continuum effective\nHamiltonian or by the general lattice Hamiltonian (both\ngiving the same results). For example, from the general\nlattice model, the eigenstate of the flat band is\n|ψ0⟩=\nsinφ eiθp\n0\n−cosφ e−iθp\n, (38)\nand the eigenstates of the conduction and valence bands\nwith λ=±1, respectively, are\n|ψλ⟩=1√\n2\ncosφ eiθp\nλ\nsinφ e−iθp\n, (39)\nwhere θpis the phase of the fpin Eq. (2). The eigenstates\nin the continuum effective model are similar to those in\nthe general lattice model except that θpis replaced by\nθq, the phase of the fqin Eq. (3). The Berry phases of\nthe dispersive conical bands and the dispersionless flat\nband are given by [8, 17]\nϕλ, ζ=πζcos 2φ=πζ\u00121−α2\n1 +α2\u0013\n, (40)\nϕ0, ζ=−2πζcos 2φ=−2πζ\u00121−α2\n1 +α2\u0013\n,(41)respectively. Note that the Berry phases are topological\nbut not πquantized [17] and are distinct in the + Kand\n−Kvalleys except for α= 0,1. Figure 2 shows that\nthe Berry phases is a monotonic function of the material\nparameter α. The average current density in Eqs. (23),\n(24), (33) and (34) also depends on α. If this dependence\nis monotonic, there will be a one-to-one correspondence\nbetween the current and the Berry phases, providing a\nmechanism to determine the Berry phases by measuring\nthe current.\n00.20.40.60.81-2-02Berry phase\nFIG. 2. Berry phases of orbits in different energy bands\naround the Dirac points ±Kversus the material parameter\nα. The dependence of the Berry phases on αis monotonic.\nIII. BALLISTIC TRANSPORT AND BERRY\nPHASE DETECTION\nFor nonequilibrium quantum transport in the α-T3lat-\ntice at zero temperature, depending on the time scale of\nballistic transport, distinct physical behaviors can arise.\nFirst, in the presence of a uniform electric field, if its\nproduct with time is comparable to the quantity ℏ/(ea):\nEtBloch∼ℏ/(ea), the average current density will un-\ndergo Bloch oscillations [29, 30] due to the Bloch band in\nthe periodic Brillouin zone. If, on this time scale, two lev-\nels do not cross each other, Landau-Zener transition will\noccur, leading to Bloch-Zener oscillations [34]. Second,\nif the time scale is much shorter than the Bloch time:\nt≪tBloch, the lattice can effectively be described by the\ncontinuum effective α-T3Hamiltonian. Third, when the\ntime scale is in the Schwinger regime:\np\nℏ/(vFeE)≪t≪tBloch,\nthe transport process becomes nonlinear: J ∝ tE3/2.\nFourth, when the time scale continues to reduce to the\nKubo regime:\nh/W≪t≪p\nℏ/(vFeE),7\n(𝑎) (𝑏)\n(𝑐) (𝑑)\nFIG. 3. Time evolution of the average current density from the effective continuum Hamiltonian in the Kubo regime. Shown\nis the interband current density divided by the electric field: eJ/eEfor 0≤α≤1. (a-d) Saturated currents divided by the\nelectric field: eJ/eE, for the total interband current density eJinter, the interference current between the flat and the upper band\neJinter\nξγ, the current density between the lower and the upper band eJinter\nξβ, and that between the lower and the flat band eJinter\nγβ\nforα= 0,0.2,0.6,1, respectively. For comparison, all currents are divided by 1 /4. Other parameters are: electric field\neE= 0.0004, valley index of the Dirac point ζ= +1, size of momentum space in eqx,eqy∈[−8,8] and step sizes of momentum\nand time deq=det= 0.01. Note the cut width about the Dirac point is eqcut= 0.005 in the momentum space to make valid the\nweak field approximation and the quantum dynamic equation (17).\nwhere Wis the bandwidth, the average electric current\ndensity is saturated and independent of time: J ∝ E .\nFinally, for the ultrashort time transient response: t≪\nh/W , the current behavior becomes full classical: J ∝\nEt.\nTo describe our results unambiguously, it is necessary\nto distinguish the electric field and current in the two\ncases where the α-T3material is described by the effective\ncontinuum model and by the general lattice model. We\nuseEandJto denote the electric field and current in\nthe former, while EandJin the latter.\nA. Kubo Regime\nIn the Kubo regime of the weak electric field, we have\n|q|=q\nq2x+q2y≫eEt, (42)\nfor|q| ̸= 0 (not too close to the Dirac points). In\nEq. (17), the term eEtinεq(t) can then be neglected but\nthe field term in the numerator term −iℏU†\nq(t)∂tUq(t)should be retained. Initially, at t= 0, all electrons stay\nin the lower energy band. For t > 0, a uniform con-\nstant electric field is switched on along the xdirection,\nand electron-positron pairs are created by the Schwinger\nmechanism [25, 26] in the continuum effective model.\nSince only a small number of the particles are excited,\nthe interband (or polarization) contribution from the in-\nterference between the energy bands dominates over the\nintraband (or conduction) contribution. In this regime,\nZitterbewegung governs the small field linear response,\nwhere all electrons propagate with the maximal velocity\nvF, leading to a saturated current independent of time.\nIntegrating the average current density associated with\nthe momentum ⟨Jx⟩inter\nq(t) over the whole momentum\nspace gives\nJinter≡ ⟨J x⟩inter,\n=1\nπ2ℏ2Z∞\n0q dqZ2π\n0dφ⟨Jx⟩inter\nq(t). (43)\nFor pseudospin-1 /2 quasiparticles ( α= 0), the linear8\n𝛼=0 𝛼=1 0<𝛼<1(𝑎)\n(𝑏)\nBerry phase\n𝛼\nFIG. 4. Flat band contribution to the saturated current in the\nKubo regime. (a) Corresponding to the Berry phase diagram\nin Fig. 2, the saturated current [the current at the end of time\nevolution in Fig. 3] changes with the materials parameter α\nfor the total interband contribution eJinter, the interference\ncurrent between the flat and upper bands eJinter\nξγ, the current\nbetween the lower and upper bands eJinter\nξβ, and that between\nthe lower and flat bands eJinter\nγβ. (b) A schematic display of the\nmixing process of interference between the lower and upper\nbands and that between the lower and flat bands for α∈\n(0,1).\nscaling law for the current is [25]\nJinter\nspin-1/2 =e2E/(4ℏ)\nwith the dimensionless relation\neJinter\nspin-1/2 =1\n4eE. (44)\nFor pseudospin-1 quasiparticles ( α= 1), due to the flat\nband, the current saturation value is amplified by a fac-\ntor of two and the corresponding linear scaling law be-\ncomes [42]\neJinter\nspin-1 =1\n2eE. (45)\nFor 0 < α < 1, the saturated value of the current over the\nelectric field eJinter/eEis between 1 /4 and 1 /2, as shownin Fig. 3, where the current is normalized by the constant\n1/4.\nOn the ultrashort time scale, the current exhibits a\nfully classical behavior: eJ/eE ∝et. After a certain time,\nthe current saturates. We tune the material parameter α\nto assess the interplay between the flat band and the sat-\nurated current in the weak field regime. For α= 0, there\nis zero coupling between the flat and the two dispersive\nbands, so the interband current is solely determined by\nthe interference between the lower and the upper conical\nbands as eJinter\nξβ. For the opposite extreme case of α= 1,\nthe saturated current is the result of the interference be-\ntween the lower and the flat band: eJinter\nγβ. For 0 < α < 1,\nthe total interband saturated current is a mixture of the\ninterference contributions between eJinter\nξβ andeJinter\nγβ, as\nshown in Fig. 3. Note that the interference between the\nflat and the upper bands does not directly contribute any\ncurrent for the entire αspectrum, because the combina-\ntion of the interference between the lower and flat bands\nand that between the flat and upper bands is physically\nequivalent to the interference between the lower and up-\nper bands.\nAsαincreases from zero, the saturated current from\nthe interference between the lower and upper bands de-\ncreases, and the current from the interference between\nthe lower and the flat band increases, as shown in Fig.\n4(a). In the regime of weak field, the flat band suppresses\nthe current from the interference between the lower and\nupper bands, and enhances the one from the interfer-\nence between the lower and flat bands for α∈(0,1),\nas shown in Fig. 4(b). In this case, detecting the Berry\nphase through the current probe is feasible since there is\na one-to-one correspondence between the saturated cur-\nrent value and the Berry phase, as shown in Figs. 2 and\n4(a).\nB. Schwinger Regime\nUnder a strong electric field,\n(qx, eEt−qx)≫ |qy|, (46)\ntheα-T3lattice system is in the Schwinger regime. In\nthis regime, electrons are excited from the lower band to\nthe flat and upper bands via the Schwinger mechanism,\nwhere the electric field in the vacuum decays and loses en-\nergy due to the production of the electron-positron pairs.\nThe transition probability to the flat or upper band is the\nsame as the Landau-Zener transition probability, where\nthe finite energy gap between the two avoided-crossing\nlevels induces the nonadiabatic Landau-Zener transition\ndriven by the electric field. In the general α-T3lattice or\nthe corresponding continuum model, Landau-Zener tran-\nsitions occur in the neighborhood of the Dirac points [34]\nbecause the energy gap is comparable with that given\nby the magnitude of the electric field. In the Schwinger\nregime, the electric current is dominated by the intraband9\n(𝑎) (𝑏)\n(𝑐) (𝑑)\n2−12 𝛼=0 𝛼=0.2\n𝛼=0.6𝛼=1\nFIG. 5. Time evolution of the normalized intraband current from the continuum effective model in the Schwinger regime. The\nquantity displayed is eJ/(eE3/2et). (a-d) The total intraband saturated current eJintrafrom the upper band current eJintra\nξ and the\nflat band current eJintra\nγ forα= 0,0.2,0.6,1, respectively. For comparison, all currents are divided by 2 /π2. Other parameters\nare: electric field eE= 0.4096, valley index ζ= +1, size of momentum space eqx,eqy∈[−8,8], and step sizes of momentum and\ntime deq=det= 0.01. The width of the region in the momentum space about the Dirac point in which the continuum effective\nHamiltonian holds is eqcut= 0.0001.\ntransitions. The flat band also contributes to the intra-\nband current even though its group velocity is zero, for\nthe following reasons. First, both electrons and holes con-\ntribute to the current. Second, the Landau-Zener transi-\ntion from the lower to the flat band can create relatively\nmore holes, giving rise to an extra current compared with\nthe case without a flat band. As a result, the intraband\ncurrent is proportional to the number of excited particles\nin both the flat and upper bands.\nIntegrating the current ⟨Jx⟩intra\nq(t) over the whole mo-\nmentum space gives\nJintra(t)≡ ⟨J x⟩intra(t),\n=1\nπ2ℏ2Z∞\n0q dqZ2π\n0dφ⟨Jx⟩intra\nq(t),(47)\nwhich is similar in mathematical form to that of the\ninterband case. However, a key difference is that the\ntime-dependent intraband current displays a nonlinear\nresponse. For pseudospin-1/2 Dirac particles, the dimen-\nsionless form of the intraband current is given by [25]\neJintra\nspin-1/2 (et) =2\nπ2eE3/2et. (48)\nFor pseudospin-1 Dirac-Weyl particles, the intrabandcurrent is [42]\neJintra\nspin-1 (et) =2√\n2\nπ2eE3/2et, (49)\nwhere the flat band contribution is\neJintra\nflat(et) =2(√\n2−1)\nπ2eE3/2et. (50)\nThe continuum effective α-T3Hamiltonian can be used\nto gain insights into the origin of the intraband current\nin the Schwinger regime. For α= 0 with pseudospin-1/2\nDirac particles, there is no coupling between the flat band\nand the two conical dispersive bands, so the only excita-\ntion is one from the lower to the upper band. The intra-\nband current depends only on the Landau-Zener transi-\ntion to the upper band, as illustrated in Fig. 5(a).\nGraphene can serve as a benchmark for comparison\nwith the general α >0 cases. For the opposite extreme\ncase of α= 1 with pseudospin-1 Dirac-Weyl particles,\nthe quantity eJ/(eE3/2et) of the upper band is in principle\nthe same as that for the α= 0 case, with the current\nfrom the Landau-Zener transition to the flat band con-\nverging to the constant√\n2−1, as shown in Fig. 5(d). For\n0< α < 1, the intraband current of the upper band is ap-\nproximately constant and flat-band current is enhanced10\n(𝑎)\n(𝑏)\n𝛼=0 0<𝛼≤1𝛼\nBerry phase\n2\nFIG. 6. Flat band contributions to the saturated current in\nthe Schwinger regime. (a) Corresponding to the Berry phase\ndiagram in Fig. 2, the saturated current (at the end of the\ntime evolution in Fig. 5) varies with the material parameter α\nfor the total intraband current density eJintra, the upper band\ncurrent eJintra\nξ, and the flat band current eJintra\nγ for 0≤α≤1.\n(b) A schematic illustration of the electron-hole excitation and\nthe extra holes from the flat band in comparison with the\ngraphene case. The flat band, despite its zero group velocity,\nhas the ability to enhance the electric current.\nwith increasing α, as shown in Figs. 5 (a-d) and Fig. 6\n(a). In the Schwinger regime, the flat band contributes\nextra holes with positive charges, thereby enhancing the\nintraband current by the factor of√\n2 compared with\nthe graphene benchmark, as schematically illustrated in\nFig. 6(b). Since the converged intraband current depends\nmonotonically on the materials parameter α, it can be\nexploited to assess the Berry phase.\nC. Bloch-Zener oscillations\nThe linear and nonlinear responses are obtained from\nthe continuum effective α-T3model that is valid for low-\nenergy excitations. More experimentally relevant is the\ngeneral lattice model. Here, using this model, we cal-\nculate the currents for the Kubo, Schwinger, and Bloch-\nZener oscillation regimes. Figures 7(a) and 7(b) show\nthat the ultrashort time transient, linear and nonlinear\n(𝑎) (𝑏)\n(𝑐)𝛼=0\n𝛼=0𝛼=1\n(𝑑)\n𝛼=1\nFIG. 7. Time evolution of the total current (interband and\nintraband) calculated from the general α-T3lattice model.\nThe quantity displayed is eJ/eE. (a, b) Ultrashort time tran-\nsient response as well as the linear and nonlinear response\nforeE= 0.002,0.004,0.008 and for α= 0,1, respectively.\nFor comparison, all currents are divided by 1 /4, the saturated\nvalue of the weak field for graphene, and a factor of two due\nto the momentum integration region being the first hexagonal\nBrillouin zone that contains two nonequivalent Dirac points.\n(c, d) Bloch-Zener oscillations for α= 0,1, respectively, for\ndifferent electric fields. Relevant parameter values are: time\nstep size det= 0.01, momentum step size depx=depy= 0.002,\nwidth around the Dirac point epcut= 0.001 in the momentum\nspace.\nBerry phase෩E=0.004 ,0.008 ,0.012 ,0.016 ,0.02,0.03 \n𝜙𝜆,+1\n𝜙𝜆,−1\n𝜙0,+1𝜙0,−1\n2\n𝛼\nFIG. 8. Scaling law of the first peak of Bloch-Zener oscilla-\ntions with the material parameter α. Shown are the scaling\nrelations for different values of the electric field corresponding\nto the Berry-phase plots in Fig. 2 for quantum states from the\nconical and flat bands around the Dirac points ±K.\nresponses generated by the continuum effective model\npersist for the lattice model with the respective time-scale\nincrement factors h/W ,p\nℏ/(vFeE),tBloch∼ℏ/(eaE),11\nFIG. 9. Summary diagram: scaling behaviors, transitions, and physical mechanisms associated with ballistic transport in α-T3\nlattice on different time scales.\nfor any fixed electric field. In the Kubo regime, the to-\ntal current eJ/eEstill saturates. The consistency between\nFigs. 7(a-b) and Fig. 4(a) suggests that the linear re-\nsponse can be used to detect the Berry phase. In the\nSchwinger regime, the current is the result of nonlinear\nresponse\neJ/eE∝eE1/2et, (51)\nas shown in Figs. 7(a-b).\nIn the Kubo regime where the time scale is larger\nthan the classical time h/W , the interference between\nenergy bands begins to contribute to the Landau-Zener\ntransitions. In the Schwinger regime, the nonlinear re-\nsponse is dominated by the Landau-Zener transitions.\nForEtBloch∼ℏ/(ea), Bloch oscillations occur. The\ncombination of the Landau-Zener transition and Bloch\noscillation leads to Bloch-Zener oscillations, as shown in\nFig. 7(c-d) for α= 0,1, where the Bloch time period\nisetB= 4π/(√\n3eE). The decay of the amplitude and\nthe irregular behavior of the Bloch-Zener oscillations in\nα-T3lattice are the result of mixed interference of quan-\ntum states in multiple bands modulated by the geomet-\nric and dynamic phases [34]. For a range of the electric\nfield, the time scales of the ultrashort transient and lin-\near responses can be neglected compared with that of\nthe nonlinear response. In this case, the first peak in the\nBloch-Zener oscillations displays a scaling law, as shown\nin Fig. 8.\nTaken together, the first peak in the Bloch-Zener oscil-\nlations, the nonlinear response, and the saturated current\nassociated with the linear response all depend monotoni-\ncally on the material parameter α. These physical quan-\ntities can then be exploited to detect the Berry phase.\nIV. DISCUSSION\nThe Berry phase in the α-T3lattice varies monoton-\nically with the material parameter α. We investigated\nelectronic transport when an α-T3lattice system is driven\nby a constant electric field and calculated a number ofcurrent densities as a function of αin both the linear\n(Kubo) and nonlinear (Schwinger) response regimes. Re-\nmarkably, the current density also exhibits a monotonic\ndependence on α, implying that the Berry phase as a\nfundamental material characteristic can be determined\nby measuring the current (e.g., using graphene for cali-\nbration).\nThe various experimentally relevant scaling behaviors\nof the current density concerning the electric field and\ntime as well as the underlying state transitions are sum-\nmarized in Fig. 9. Depending on the product ˜E˜tof the\nnormalized electric field and time, five distinct scaling\nregimes arise. For ˜E˜t∼0, the current density is zero. In\nthe transient phase, the current density is proportional\nto˜E˜t. The linear response regime comes after the tran-\nsient phase, in which the current is proportional just to\nthe electric field. In the nonlinear response regime that\nfollows the linear regime, the current is proportional to\n˜E3/2˜t. For much larger values of ˜E˜t, Bloch-Zener oscilla-\ntions arise, whose amplitude can be an irregular function\nof time [34]. While the scenario in Fig. 9 is based on the\neffective continuum Hamiltonian, direct calculations of\nthe lattice Hamiltonian indicate that the ultrashort time\ntransient response, linear and nonlinear responses still\narise. In fact, Landau-Zener transitions begin to occur\nin the linear response regime and become dominant in\nthe nonlinear response regime. When ˜E˜tis comparable\nto a quantity of the same physical dimension determined\nby the lattice constant, Bloch-Zener oscillations occur.\nIn this case, the first peak of the oscillation exhibits a\nscaling law with αby the nonlinear response mechanism\nwhen the ultrashort time transient and linear responses\nare negligible. 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B 81, 041416 (2010)."    },    {        "title": "2402.15614v2.Spatial_Variations_of_Stellar_Elemental_Abundances_in_FIRE_Simulations_of_Milky_Way_Mass_Galaxies__Patterns_Today_Mostly_Reflect_Those_at_Formation.pdf",        "content": "Draft version February 28, 2024\nTypeset using L ATEXtwocolumn style in AASTeX631\nSpatial Variations of Stellar Elemental Abundances in FIRE Simulations of Milky Way-Mass\nGalaxies: Patterns Today Mostly Reflect Those at Formation\nRussell L. Graf\n ,1Andrew Wetzel\n ,1Matthew A. Bellardini\n ,1and Jeremy Bailin\n2\n1Department of Physics & Astronomy, University of California, Davis, CA 95616, USA\n2Department of Physics & Astronomy, University of Alabama, Box 870324, Tuscaloosa, AL 35487-0324, USA\nABSTRACT\nSpatial patterns of stellar elemental abundances encode rich information about a galaxy’s formation\nhistory. We analyze the radial, vertical, and azimuthal variations of metals in stars, both today and at\nformation, in the FIRE-2 cosmological simulations of Milky Way (MW)-mass galaxies, and we compare\nwith the MW. The radial gradient today is steeper (more negative) for younger stars, which agrees\nwith the MW, although radial gradients are shallower in FIRE-2. Importantly, this age dependence\nwas present already at birth: radial gradients today are only modestly ( ≲0.01 dex kpc−1) shallower\nthan at birth. Disk vertical settling gives rise to negative vertical gradients across all stars, but vertical\ngradients of mono-age stellar populations are weak. Similar to the MW, vertical gradients in FIRE-2\nare shallower at larger radii, but they are overall shallower in FIRE-2. This vertical dependence was\npresent already at birth: vertical gradients today are only modestly ( ≲0.1 dex kpc−1) shallower than\nat birth. Azimuthal scatter is nearly constant with radius, and it is nearly constant with age ≲8\nGyr ago, but increases for older stars. Azimuthal scatter is slightly larger ( ≲0.04 dex) today than\nat formation. Galaxies with larger azimuthal scatter have a stronger radial gradient, implying that\nazimuthal scatter today arises primarily from radial redistribution of gas and stars. Overall, spatial\nvariations of stellar metallicities show only modest differences between formation and today; spatial\nvariations today primarily reflect the conditions of stars at birth, with spatial redistribution of stars\nafter birth contributing secondarily.\n1.INTRODUCTION\nThe origin and evolution of elemental abundances en-\ncode vital information about a galaxy’s formation his-\ntory (Tinsley 1980). Through stellar nucleosynthesis\n(Burbidge et al. 1957) and supernovae (Baade & Zwicky\n1934), heavy elements (beyond H and He) are ejected\nback into the interstellar medium (ISM), from which\nsuccessive generations of more metal-rich stars are born.\nThus, the ISM and the stars that recently formed out\nof it reflect the current elemental abundance patterns\nwithin a galaxy. Once formed, stars essentially re-\ntain the abundance patterns they were born with, so\na galaxy’s stellar population encodes its entire enrich-\nment, and thus formation, history.\nImportantly, elemental abundances encode spatial in-\nformation. Searle (1971) first identified negative ra-\ndial gradients in gas-phase metallicity, such that smaller\nCorresponding author: Russell L. Graf\nrlgraf@ucdavis.eduradii within galaxies are more enriched. More recently,\nmany works have quantified spatial variations of met-\nals in gas in the Milky Way (MW), M31, and nearby\ngalaxies (for example, Sanders et al. 2012; S´ anchez-\nMenguiano et al. 2016; Belfiore et al. 2017; Sakhibov\net al. 2018; Kreckel et al. 2019, 2020; Wenger et al.\n2019; Moll´ a et al. 2019; Zinchenko et al. 2019; Hernandez\net al. 2021). The spatial patterns of metallicity in stars\nof different ages today encode the history of a galaxy’s\nISM, subject to the complication that stars can move\nfrom their birth radii and vertical height in the disk via\nradial redistribution and vertical heating (for example,\nSellwood & Binney 2002; Haywood 2008).\nMany observational surveys have advanced our abil-\nity to measure stellar metallicity variations across the\nMW, such as GALAH (Buder et al. 2018), Gaia-ESO\n(Gilmore et al. 2012), LAMOST (Cui et al. 2012), and\nSDSS-APOGEE (J¨ onsson et al. 2020), and soon SDSS-\nV (Kollmeier et al. 2017), 4-m Multi-Object Spectro-\ngraph Telescope (4MOST; de Jong et al. 2019), the\nWHT Enhanced Area Velocity Explorer (WEAVE; Dal-\nton et al. 2012), and the MaunaKea Spectroscopic Ex-arXiv:2402.15614v2  [astro-ph.GA]  27 Feb 20242 Graf et al.\nplorer (MSE; The MSE Science Team et al. 2019) will\nmeasure spectroscopy and precise elemental abundances\nfor tens of millions of stars. Many analyses of these sur-\nveys identified radial and vertical gradients in stars and\nused them to infer the spatial and temporal evolution\nof enrichment in the MW (for example, Vickers et al.\n2021; Lian et al. 2022, 2023; Willett et al. 2023; Anders\net al. 2023; Imig et al. 2023). These works generally find\nthat radial gradients are steeper for younger stars. As\nwe discuss below, the origin and evolution of these radial\ngradients are under active debate. Works measuring ver-\ntical gradients for stars in the MW (for example, Hayden\net al. 2014; Imig et al. 2023) find that vertical gradients\nare negative, such that stars closer to the disk midplane\nare more enriched. These works also find that vertical\ngradients generally decrease in strength with increasing\nradius. Furthermore, observations measured azimuthal\nvariations for stars in the MW (for example, Anders\net al. 2023; Hawkins 2023; Ratcliffe et al. 2023), although\nthe strength of these variations remains debated.\nAccurately measuring these spatial and temporal pat-\nterns in elemental abundance, combined with accurately\nmodeling their origin and how they changed across cos-\nmic time, can reveal the formation history of the MW via\n���chemical tagging”. Introduced in Freeman & Bland-\nHawthorn (2002), chemical tagging aims to use the\n(near) invariance of stars’ elemental abundances to tag\nobserved stellar populations to their birth times and lo-\ncations. A key open question for chemical tagging is how\nhomogeneous the ISM was across a galaxy’s formation\nhistory, which sets the information content of elemental\nabundances in terms of birth location. If we can under-\nstand and model this, then chemical tagging also offers\nthe possibility of measuring the amount of radial and\nvertical redistribution that stars today experienced since\nbirth. If the effects of radial and vertical redistribution\nare small, on average, then the abundance patterns of\nstars today nearly directly tell us about the evolution of\nthe (star-forming) ISM across a galaxy’s history.\nOne can think of two regimes for chemical tagging.\n“Strong” chemical tagging attempts to tie stars to their\nbirth clusters/clouds (for example, Bovy 2016; Price-\nJones et al. 2020; Casamiquela et al. 2021). To be ef-\nfective, this would require that star-forming gas clouds\nbe both internally homogeneous and sufficiently unique\nfrom each other across the galaxy. “Weak” chemical tag-\nging attempts to tie stars to their general birth time and\nbroad birth location in the galaxy (for example, Anders\net al. 2017). In terms of spatial variations, weak chem-\nical tagging requires simply that stars born in a given\npatch (of some size) in radius, vertical height, and/or\nazimuth have the same elemental abundances, and thatthese vary as a function of radius, vertical height, and/or\nazimuth.\nFor chemical tagging to be successful, we must have an\neffective theoretical model for how the spatial trends of\nmono-age stellar populations today relate to the trends\nat their formation (birth), through (semi)analytical\nmodels (for example, Minchev et al. 2018; Sharda et al.\n2021), idealized (non-cosmological) N-body simulations\n(for example, Loebman et al. 2016; Khoperskov et al.\n2023), and/or cosmological simulations (for example,\nGibson et al. 2013; Vincenzo & Kobayashi 2018, 2020;\nBellardini et al. 2021, 2022; Lu et al. 2022; Khoper-\nskov et al. 2023; Buck et al. 2023). As we discuss in\nSection 6.4, among these many works, expectations re-\nmain mixed as to the strength and evolution of spatial\nvariations of metallicity at birth, particularly concern-\ning whether radial gradients evolved over cosmic time to\nbe stronger (more negative), weaker (less negative), or\nexperienced little evolution. Furthermore, while many\nworks explored these metallicity patterns in the ISM or\nfor stars at birth, fewer works analyzing cosmological\nsimulations have compared these patterns at birth to\nthe patterns today, which also enables comparisons with\nobservations of the MW and nearby galaxies.\nIn this work, we use the Feedback In Realistic Envi-\nronments (FIRE) cosmological zoom-in simulations to\nexplore these trends across the formation histories of\nMW-mass galaxies. FIRE-2 simulations are compelling\ntheoretical tools for this study, because they are fully\ncosmological, explicitly and self-consistently model the\nkey processes for metal production (stellar nucleosyn-\nthesis) and turbulent mixing in the ISM, and previous\nworks have shown that they successfully model key fea-\ntures of MW-like disk galaxies: including thick+thin\ndisk morphology (Ma et al. 2017b; Garrison-Kimmel\net al. 2018; Sanderson et al. 2020; Yu et al. 2023; Gur-\nvich et al. 2023), dynamics of young stars (McCluskey\net al. 2023) and old metal-poor stars (Santistevan et al.\n2021), atomic gas scale heights (Gensior et al. 2023),\nstellar bars (Ansar et al. 2023), spiral arms (Orr et al.\n2023), and giant molecular clouds (Benincasa et al. 2020;\nGuszejnov et al. 2020).\nOur work builds on previous works that analyzed spa-\ntial variations of metals in both gas and stars in FIRE\nsimulations. Ma et al. (2017b) showed that a FIRE sim-\nulation, m12i, broadly agrees with the MW in terms of\nradial and vertical metallicity gradients of stars. Ma\net al. (2017a) explored the redshift and mass depen-\ndence of metallicity gradients in the FIRE-1 simulations,\nshowing that strong (negative) gradients only develop in\ngalaxies with rotation-dominated dynamics, vϕ/σv≳1.\nBellardini et al. (2021) used the same suite of MW-Spatial Variations of Metals in FIRE 3\nmass FIRE-2 simulations we use here to study spatial\nvariations in gas-phase metallicity. Consistent with Ma\net al. (2017b), they found that negative radial gradients\nbecome stronger over time, while azimuthal variations\nget weaker, as the galaxies evolve to more rotation-\ndominated dynamics, with a transition in which one\ndominates over the other in the overall metallicity vari-\nations across a galaxy ≈8 Gyr ago. Bellardini et al.\n(2022) showed that these trends persist for stars at birth,\nand they also found weak to no mono-age vertical gradi-\nents in stars at birth. The transition from azimuthally\nto radially dominant metallicity variations ≈8 Gyr ago\ncoincides with the onset of rotationally-dominated mo-\ntion ( vϕ/σv≳1) and disk settling that McCluskey et al.\n(2023) found for these galaxies. Carrillo et al. (2023)\nalso showed that this transition time correlates with\nthe beginning of evidence for inside-out radial formation\nfor younger ( <7 Gyr) and higher-metallicity ([Fe/H]\n>−0.25 dex) stars.\nIn particular, we extend the work of Bellardini et al.\n(2021) and Bellardini et al. (2022), to investigate spa-\ntial variations of stellar elemental abundances both to-\nday and at birth using the FIRE-2 cosmological zoom-in\nsimulations. Importantly, we also compare with various\nobservations of stars in the MW.\n2.METHODS\n2.1. FIRE-2\nWe analyze 11 Milky Way-mass galaxies from two\nsuites of cosmological zoom-in simulations from FIRE-2\n(Hopkins et al. 2018): Latte and ELVIS (Exploring the\nLocal Volume in Simulations). These FIRE-2 simula-\ntions are publicly available (Wetzel et al. 2023). From\nthe Latte suite, we include 5 isolated galaxies (m12b,\nm12c, m12f, m12i, m12m) with halo masses of 1 −2×1012\nM⊙, dark-matter mass resolution of ≈3.5×105M⊙,\nand initial baryonic particle mass of 7070 M ⊙; because\nof stellar mass loss, most star particles have ≈5000\nM⊙. The ELVIS suite consists of 3 Local Group-like\ngalaxy pairs (Romeo & Juliet, Romulus & Remus, and\nThelma & Louise). Their mass resolution is ≈2×bet-\nter than Latte, with initial baryonic particle masses of\n3500−4000 M ⊙. Star and dark matter particles have\nfixed gravitational softenings, with a Plummer equiv-\nalent of ϵstar= 4 pc and ϵdm= 40 pc, comoving at\nz >9 and physical at z <9. Gas cells use fully adap-\ntive gravitational softening, which matches the hydrody-\nnamic kernel smoothing, reaching a minimum of 1 pc.\nWe generated cosmological zoom-in initial conditions\nusing MUSIC (Hahn & Abel 2011). Each zoom-in re-\ngion is embedded within a cosmological box with side\nlengths of 70 .4−172 Mpc. The simulations assume aflat ΛCDM cosmological model with parameters broadly\naligned with Planck Collaboration et al. (2020): h=\n0.68−0.71,ΩΛ= 0.69−0.734,Ωm= 0.266−0.31,Ωb=\n0.0455−0.048, σs= 0.801−0.82,andns= 0.961−0.97.\nEach simulation stores 600 snapshots, spanning z= 99\ntoz= 0, with time spacing ≲25 Myr.\nWe ran these simulations using the Gizmo code\n(Hopkins 2015), with the mesh-free finite-mass (MFM)\nmode for hydrodynamics, a quasi-Lagrangian Godunov\nmethod that provides adaptive spatial resolution while\nmaintaining exact conservation of mass, energy, and\nmomentum, excellent angular momentum conservation,\nand accurate capturing of shocks. For gravity, Gizmo\nuses an improved version of the Tree-PM solver from\nGADGET-2 (Springel 2005).\nOur simulations use the FIRE-2 physics model (Hop-\nkins et al. 2018), with the redshift-dependent, spatially-\nuniform cosmic ultraviolet background from Faucher-\nGigu` ere et al. (2009). FIRE-2 accounts for the\nmetallicity-dependent radiative heating and cooling pro-\ncesses for gas across 10 −1010K. These processes include\nfree-free, photoionization and recombination, Compton,\nphoto-electric and dust collisional, cosmic ray, molecu-\nlar, metal-line, and fine-structure processes, accounting\nfor 11 elements (see below). A gas cell becomes a star\nparticle on a local gravitational free-fall time when it\nbecomes self-gravitating, Jeans-unstable, cold ( T <104\nK), and molecular (Krumholz & Gnedin 2011), inherit-\ning the mass and metallicity of its progenitor gas cell.\nA star particle represents a mono-age, mono-abundance\nstellar population, assuming a Kroupa (2001) initial\nmass function.\n2.2. Metal enrichment in FIRE-2\nOnce stars form, FIRE-2 models their feedback and\nnucleosynthesis. Stellar feedback includes winds, core-\ncollapse and white-dwarf (Ia) supernovae, photoioniza-\ntion, photo-electric heating, and radiation pressure. For\nstellar winds and their yields, FIRE-2 uses Wiersma\net al. (2009), which synthesized a combination of mod-\nels (van den Hoek & Groenewegen 1997; Marigo 2001;\nIzzard et al. 2004). Core-collapse supernova rates are\nfrom STARBURST99 (Leitherer et al. 1999) and yields\nare from Nomoto et al. (2006). White-dwarf (Ia) super-\nnova rates are from Mannucci et al. (2006), with nucle-\nosynthetic yields from Iwamoto et al. (1999). FIRE-2\ntracks 11 elements: H, He, C, N, O, Ne, Mg, Si, S,\nCa, Fe. We quote all elemental abundances scaled to\n(proto) Solar values from Asplund et al. (2009). Impor-\ntant for our analysis, FIRE-2 also explicitly models sub-\ngrid mixing and diffusion of metals in gas via turbulent\neddies (Su et al. 2017; Escala et al. 2018; Hopkins et al.4 Graf et al.\nsimulation Mstar\n90 Rstar\n90 Zstar\n90 ∂[Fe/H]/∂R σ [Fe/H] ∂[Fe/H]/∂|Z| Lookback time\n(<1 Gyr) ( <1 Gyr) ( R= 8 kpc) of disk onset\n[1010M⊙] [kpc] [kpc] [dex kpc−1] [dex] [dex kpc−1] [Gyr]\nm12m110.0 11.9 2.3 −0.031 0 .063 −0.11 9 .21\nRomulus28.0 14.8 2.4 −0.034 0 .078 −0.067 7 .42\nm12b37.3 9.2 1.8 −0.031 0 .053 −0.086 7 .42\nm12f46.9 13.5 2.1 −0.032 0 .084 −0.058 7 .42\nThelma36.3 11.7 3.2 −0.026 0 .063 −0.045 4 .35\nRomeo35.9 14.2 1.9 −0.041 0 .084 −0.14 11 .0\nm12i55.3 10.1 2.3 −0.030 0 .050 −0.069 6 .65\nm12c35.1 9.2 2.0 −0.025 0 .058 −0.045 6 .49\nRemus24.0 12.4 2.2 −0.038 0 .066 −0.14 7 .93\nJuliet33.3 9.5 2.2 −0.049 0 .12 −0.046 4 .35\nLouise32.3 12.6 2.2 −0.047 0 .089 −0.12 7 .16\nmean 5.9 11.7 2.2 −0.035 0 .074 −0.084 7 .22\nTable 1. Stellar properties today of the 11 FIRE-2 galaxies in our analysis, in decreasing order of stellar mass. From left\nto right, the columns are: the name of the simulation; Mstar\n90is the stellar mass within Rstar\n90;Rstar\n90is the radius that encloses\n90% of stars; Zstar\n90is the vertical height that encloses 90% of stars; the radial gradient of young stars (ages <1 Gyr); the\nazimuthal scatter (1 σstandard deviation) of young stars (ages <1 Gyr) at R= 8±0.5 kpc; the vertical gradient of allstars\natR= 8±0.5 kpc; the onset of disk formation, defined in Table 1 of McCluskey et al. (2023) via vϕ/σtot>1 at the time\nof formation. The publication that introduced each simulation is: Hopkins et al. (2018)1, Garrison-Kimmel et al. (2019a)2,\nGarrison-Kimmel et al. (2019b)3, Garrison-Kimmel et al. (2017)4, Wetzel et al. (2016)5.\n2018). As Bellardini et al. (2021) showed, the details\nof this diffusion model do not significantly affect radial\nor vertical gradients in FIRE-2 MW-mass galaxies, but\nthey strongly affect the azimuthal scatter. As Bellar-\ndini et al. (2021) also showed, these FIRE-2 MW-mass\ngalaxies have azimuthal scatter in gas at z= 0 similar\nto that observed in nearby galaxies of similar mass.\n2.3. Milky Way-mass galaxies in FIRE-2\nTable 1 presents key properties of stars for our 11 MW-\nmass FIRE-2 galaxies today. We include only FIRE-2\ngalaxies with stellar masses within a factor of ≈2 of the\nMW. These galaxies experienced a wide range of forma-\ntion histories (Santistevan et al. 2020; McCluskey et al.\n2023), given that their selection was agnostic to any\nproperties beyond dark-matter halo mass, including for-\nmation history, concentration, spin, or satellite/subhalo\npopulation (with the additional constraint of a LG-like\npaired-halo environment for the ELVIS galaxies). For\nkey results, we show the mean, 1 σscatter, and full range\nacross these 11 MW-mass galaxies. Therefore, our re-\nsults should represent trends typical for the formation\nhistories of MW-mass galaxies.\nThat said, we also compare with observations of the\nMW, and many observational analyses suggest that the\nMW’s disk began to form early, ≈12 Gyr ago (for ex-\nample, Belokurov & Kravtsov 2022; Conroy et al. 2022;\nXiang & Rix 2022), which may have influenced its metal-\nlicity patterns. Therefore, throughout we also showkey results separately for Romeo, the galaxy within our\nFIRE-2 sample whose disk began to form the earliest\nand thus nearest in time to the MW. This is likely\nbecause Romeo formed in a denser, LG-like environ-\nment (Garrison-Kimmel et al. 2018; Santistevan et al.\n2020). As McCluskey et al. (2023) showed, Romeo’s\nvϕ/σvrapidly increased 11 −12 Gyr ago, likely within\n≈1 Gyr of the MW.\n2.4. Selecting and measuring stars\nWhen we measure stars today, we include only “in-\nsitu” stars that formed within 30 kpc comoving of each\nMW-mass galaxy, and when we measure stars at forma-\ntion, we include only those that remain within 30 kpc\nof the galaxy today, following Bellardini et al. (2022).\nThese criteria help ensure that we measure similar pop-\nulations of stars today and at birth, to compare them\nmore fairly.\nWe include only stars within a vertical range of |Z|<\n3 kpc, to constrain our analysis to the disk. We also\ntested using |Z|<1 kpc and |Z|<5 kpc and found that\nthis caused negligible ( >0.001 dex kpc−1) differences in\nour results.\nTo calculate the spatial gradient of any profile (ra-\ndial or vertical), we fit a single linear relation using\nnumpy.polyfit. However, the radial gradient is sensitive\nto the radial range. As Bellardini et al. (2022) showed\nfor these FIRE-2 galaxies, the gradient is steeper in the\ninner galaxy and shallower in the outer galaxy, with anSpatial Variations of Metals in FIRE 5\naverage break radius of R≈5.5 kpc. Therefore, quanti-\nfying the galaxy via a single gradient is simplified. How-\never, it allows us to readily examine trends across cosmic\ntime, compare gradients today versus at formation, and\naverage across our galaxy sample. Throughout, we mea-\nsure the radial gradient across R= 0−15 kpc today and\nR= 0−Rstar\n90at formation (birth). We tested the results\nof using different radial ranges: reducing the range to-\nday to R= 0−12 kpc steepens the average gradient for\nyoung ( >1 Gyr) stars from −0.035 dex kpc−1to−0.040\ndex kpc−1, while increasing the range to R= 0−20 kpc\nshallows the average today to −0.028 dex kpc−1.\nIn computing any property for a given galaxy, we mea-\nsure the median across the stellar population at a given\ntime and spatial location on a galaxy-by-galaxy basis.\nWe then compute the mean, 68% scatter, and total scat-\nter across our 11 MW-mass galaxies, which is what we\nshow in figures.\nWe measure all quantities today and at birth using\na star particle’s instantaneous RandZ. Some obser-\nvational (and theoretical) works, including some MW\nanalyses that we compare against, instead attempt to\ninfer a more general orbital radius, such as the “guid-\ning center” radius (under the epicyclic approximation).\nIn principle, this introduces a source of inconsistency in\ncomparing metallicity profiles. However, as Bellardini\net al. (in prep.) shows for these FIRE-2 galaxies, the\nresults for population statistics that we examine, like\nradial gradients, do not depend much on the metric of\norbital radius, including instantaneous radius.\nWe focus on [Fe/H], because Fe is the most commonly\nmeasured elemental abundance in stars. In Appendix B\nwe also show trends for [C/H] (which are particu-\nlarly sensitive to stellar winds) and [Mg/H] (which are\nproduced almost entirely in core-collapse supernovae).\nWhile spatial gradients in [Mg/H] are slightly weaker\nthan in [Fe/H], and gradients in [C/H] are slightly\nstronger, gradients for all three are qualitatively sim-\nilar. Therefore, our key results are consistent for all\nelemental abundances in FIRE-2.\n3.RADIAL PROFILE\n3.1. Results in FIRE-2\nFigure 1 (top) shows the profile of [Fe/H] for all stars,\nregardless of age, versus cylindrical radius, R. We\nshow the profile using both the radii of stars today and\ntheir radii at birth; the results are qualitatively similar.\n[Fe/H] is highest in the galactic center and decreases\nwith Rwithin the inner 2 kpc. [Fe/H] then increases\nbetween 2 −6 kpc, before decreasing with Rbeyond\nthat. Observations of the MW also exhibit this increase\nin [Fe/H] between 2 −6 kpc and decrease at larger R\n0.3\n0.2\n0.1\n0.0[Fe/H]today\nformation\n0 2 4 6 8 10 12 14\nRadius [kpc]0.6\n0.5\n0.4\n0.3\n0.2\n0.1\n0.0[Fe/H] - [Fe/H]R=0\n0-1 Gyr\n4-5 Gyr\n8-9 Gyr\n11-12 GyrFigure 1. Top : Median [Fe/H] of all stars, regardless of\nage, versus cylindrical radius, R. Each line and shaded re-\ngion show the average and 1 σscatter across the 11 galaxies.\nThe solid line shows stars today, while the dashed line shows\nstars at formation (using their birth radii). The radial profile\nfor all stars today is nearly flat, though with a dip at R≈2\nkpc from the high density of older stars there. The shape\ntoday is similar to that using the birth radii, so radial redis-\ntribution of stars does not significantly change this overall\nradial profile. Bottom : Median [Fe/H] versus cylindrical\nradius, R, for stars in bins of age, normalized to [Fe /H] = 0\natR= 0 today. We measure this profile only out to Rstar\n90at\nformation. Radial gradients are always negative at a given\nage, and they steepened over time, being strongest for stars\nthat formed most recently. Because of radial redistribution,\nthese gradients are marginally shallower today than at for-\nmation, but the difference is small. Thus, the metallicity pro-\nfile today and its age dependence primarily reflect the trends\nalready in place when stars formed.\n(Lian et al. 2023). These trends relate to trends in stel-\nlar age and the “inside-out” radial growth histories of\nthese galaxies. As Bellardini et al. (2022) showed (their\nFigure 1), [Fe/H] increases with decreasing stellar age\nin FIRE-2. Furthermore, the inner galaxy contains a\nwide range of stellar ages (both young and old), while6 Graf et al.\nthe outer disk contains a much narrower range of stellar\nages, primarily just young. Therefore, we also measure\nmedian stellar age as a function of R(not shown): young\nstars high in [Fe/H] dominate the galactic center, while\nold stars lower in [Fe/H] dominate the inner regions co-\ninciding with the dip in [Fe/H] in these regions, and\nthen at R≳3 kpc stellar age steadily decreases with R.\nGiven the relatively narrow range of ages in the outer\ndisk, it most directly traces the mono-age radial gradi-\nents that we show below. However, trends at R≲8 kpc\nprimarily reflect age dependence, and as such, the dip\nin [Fe/H] at R≈1−3 kpc primarily reflects a “bump”\nin stellar age.\nBecause metallicity depends strongly on stellar age,\nwe separate the radial profile into mono-age populations,\nto uncover the underlying trends across the galaxies’\nhistories. Figure 1 (bottom) shows the radial profile of\n[Fe/H] for mono-age stars, using age bins of 1 Gyr. We\nnormalize all profiles to [Fe/H] = 0 at R= 0 kpc today,\nto compare the shapes of the profiles. [Fe/H] declines\nmore significantly with Rfor younger stars. This trend\nis statistically monotonic with age across our sample,\nand it is similar for stars based on their radii today and\nat birth.\nThese radial gradients for stars at birth reflect the con-\nditions of the ISM at that time. As Ma et al. (2017a)\nand Bellardini et al. (2021) discussed for FIRE galax-\nies, these radial gradients emerge from two effects, both\nof which must be met to cause a strong radial gradi-\nent. First, to source a metallicity gradient, smaller radii\nmust have higher star-formation efficiency, that is, mass\nin young stars (which source most of the new metals) rel-\native to mass in gas (over which the metals are diluted),\nwhich drives [Fe/H] to be higher there. Indeed, as Bel-\nlardini et al. (2022) showed for these FIRE-2 galaxies,\nthe radial profile of [Fe/H] for young stars today closely\ntracks the profile of Σ star(<1 Gyr) /Σgas. Second, to\nsustain a radial gradient in the ISM, the galaxy must\nhave sufficiently well-ordered rotation (in a disk) to pre-\nvent the radial mixing of metals in gas. As these MW-\nmass galaxies evolved, their ISM developed thinner disks\nwith more ordered rotation (McCluskey et al. 2023),\nthereby reducing radial mixing, leading to a stronger\nradial gradient for younger stars.\nAs Table 1 shows, for young stars (ages <1 Gyr), the\naverage gradient across all 11 FIRE-2 galaxies is −0.035\ndex kpc−1, ranging from −0.026 dex kpc−1(Thelma) to\n−0.049 dex kpc−1(Juliet). Noteworthily, the galaxy\nwith the earliest-forming disk, Romeo, has the third\nsteepest radial gradient for young stars of −0.041 dex\nkpc−1. Bellardini et al. (2022) explored the correlation\nbetween the radial gradient today and various metrics\n0 2 4 6 8 10 12\nAge [Gyr]0.04\n0.02\n0.00[Fe/H]/R [dex kpc1]\ntoday\nformation\nFIRE: RomeoFigure 2. Radial gradient of [Fe/H], ∂[Fe/H]/∂R, versus\nstellar age, for stars today and at formation (using their\nbirth radii). For each galaxy, we measure the gradient across\nR= 0−15 kpc today and across R= 0−Rstar\n90at formation.\nEach line, darker shaded region, and lighter shaded region\nshow the average, 1 σscatter, and min/max across the 11\ngalaxies. Dotted lines show Romeo, the earliest-forming disk\nin our sample. The radial gradient is almost always negative,\nboth at formation and today, becoming steeper for stars that\nformed more recently, and being nearly flat for stars that\nformed ≳10 Gyr ago. Because of radial redistribution af-\nter birth, the gradient is marginally shallower today than\nat formation, but the difference is modest, typically ≲0.01\ndex kpc−1, and the trends with age are unchanged. Thus,\nthe radial gradient of metallicity today primarily reflects the\nconditions of stars at birth.\nof formation history for these FIRE-2 galaxies, finding\na weak correlation (with significant scatter), such that\ngalaxies that form earlier tend to have stronger (more\nnegative) radial gradients in young stars today.\nBellardini et al. (2022) presented these gradients for\nstars at birth in these same FIRE-2 galaxies. Our fo-\ncus here is to present similar results for stars today and\ncompare the two. As Figure 1 (bottom) shows, the pro-\nfiles are generally shallower today than at birth, a result\nof radial redistribution, as Bellardini et al. (in prep.)\nquantifies. However, the differences between birth and\ntoday are small, with a typical difference of ≈0.0053\ndex kpc−1, and the largest average difference being 0 .013\ndex kpc−1for 6 Gyr old stars. Therefore, the radial pro-\nfiles of stellar metallicity today primarily reflect those in\nplace in the ISM from which the stars formed.\nFigure 2 quantifies these radial gradients by show-\ning∂[Fe/H]/∂R as a function of stellar age. Again, we\nmeasure a linear fit across R= 0−15 kpc today and\nR= 0−Rstar\n90at formation. As in Figure 1, this gradi-\nent flattens for older stars, both when using radii today\nand radii at birth. At ages ≲3 Gyr, the radial gradi-\nent at birth was constant with age, as Bellardini et al.Spatial Variations of Metals in FIRE 7\n(2022) showed. As expected, radial redistribution causes\nthe radial gradient today to be systematically shallower\n(smaller) than at formation. Interestingly, this differ-\nence increases with age up to ≈3 Gyr, but the difference\nbetween birth and today, ≈0.01 dex kpc−1, is nearly\nconstant with age for stars older than this. This is con-\nsistent with Bellardini et al. (in prep.) that the amount\nof radial redistribution in these galaxies increases with\nage up to ≈3−4 Gyr but no longer depends much on\nage beyond that.\nIn Figure 2, we also show Romeo, which has system-\natically steeper radial gradients than the mean trend at\nall ages, as a result of its early-forming disk, which we\nfurther discuss in the next section, and its pertinence to\nradial gradients in the MW.\n3.2. Comparison to the Milky Way\nBellardini et al. (2021) found that the typical radial\ngradient of metallicity in gas in these FIRE-2 galax-\nies today is ∂[O/H]/∂R =−0.028 dex kpc−1. This\nagrees well with gradients measured for M31, from Zu-\nrita & Bresolin (2012) and Sanders et al. (2012) ( −0.023\ndex kpc−1and−0.0195 dex kpc−1, respectively), and is\nsteeper than observed in most nearby MW-mass galax-\nies, but this is shallower than most measurements of the\nMW of ≈ −0.046 dex kpc−1(see references in Bellar-\ndini et al. 2021). Bellardini et al. (2022) found similar\nresults for young stars at birth as well. This suggests\nthat the MW has a steeper radial gradient than many\nnearby galaxies of similar mass. We discuss this further\nin Section 6.3.\nSeveral observational works have measured how the\nradial gradient of stellar metallicity in the MW depends\non stellar age. Figure 3 shows ∂[Fe/H]/∂Rversus stellar\nage, for five recent observational analyses of the MW, as\nwe summarize below.\nVickers et al. (2021) used LAMOST and Gaia to cal-\nculate the ages and orbital radii of ≈4 million stars to\ndetermine the radial gradients in [Fe/H] for a sample\nof 1.3 million stars. Figure 13 in Vickers et al. (2021)\nshows the radial gradient as a function of age, which we\nshow in Figure 3. Vickers et al. (2021) found a radial\ngradient for the youngest stars of ∂[Fe/H]/∂R≈ −0.075\ndex kpc−1. This gradient decreases with stellar age but\nremains fairly steep for stars 13 Gyr in age, at ≈ −0.045\ndex kpc−1.\nLian et al. (2022) used APOGEE DR16 to study\nmono-age populations of stars. Using lines of best fit,\nwe convert the radial profiles in Figure A2 of Lian et al.\n(2022) which span 0 kpc ≲R≲13 kpc into radial gra-\ndients as a function of age, and we show the results\nin Figure 3. Lian et al. (2022) found a radial gradient\n0 2 4 6 8 10 12\nAge [Gyr]0.08\n0.06\n0.04\n0.02\n0.00[Fe/H]/R [dex kpc1]\nFIRE t/t=0\nFIRE t/t=10%\nFIRE t/t=25%\nFIRE t/t=10%: Romeo\n MW (Vickers+2021)\nMW (Lian+2022)\nMW (Lian+2023)\nMW (Willet+2023)\nMW (Anders+2023)Figure 3. Radial gradient of [Fe/H] today, ∂[Fe/H]/∂R, ver-\nsus stellar age, assuming various fractional uncertainties in\nstellar age, δt/t. Each line, darker shaded region, and lighter\nshaded region show the average, 1 σscatter, and min/max\nacross the 11 galaxies. The dotted line shows Romeo, the\nearliest-forming disk in our sample, with δt/t= 10%. The in-\ntrinsic radial gradient is monotonically flatter for older stars.\nHowever, the trend with age can reverse to become steeper\nfor older stars for sufficiently large age uncertainties, which\nmixes younger stars with steeper gradients into older age\nbins. We compare with observations of the MW from Vick-\ners et al. (2021), Lian et al. (2022), Lian et al. (2023), Willett\net al. (2023), and Anders et al. (2023). FIRE-2 simulations\nhave weaker radial gradients than the MW at most ages,\nthough the steepest gradients in FIRE-2 are nearly consis-\ntent with Lian et al. (2023). More importantly, the trend\nwith age in FIRE-2 agrees with the MW, and FIRE-2 agrees\nwith Lian et al. (2022), Lian et al. (2023), and Anders et al.\n(2023) that the radial profile becomes nearly flat for the old-\nest stars. In conjunction with Figure 2, these trends imply\nthat the age dependence in the MW primarily reflects the ra-\ndial gradients of stars at birth.\nfor stars <1 Gyr of ∂[Fe/H]/∂R≈ −0.053 dex kpc−1.\nTheir gradient becomes shallower with stellar age and\neven becomes positive for stars older than 9 Gyr.\nLian et al. (2023) used APOGEE and Gaia to deter-\nmine metallicity profiles of mono-age populations. Here,\nthey divided stars into larger bins of age: 0 −4 Gyr,\n4−8 Gyr, and 8 −12 Gyr and measured profiles over\na range of 2 kpc ≲R≲14 kpc. We show the gradi-\nents of the profiles from Figure 1 of Lian et al. (2023)\nin Figure 3. Lian et al. (2023) found a radial gradient8 Graf et al.\nfor stars aged 0 −4 Gyr of ∂[Fe/H]/∂R≈ −0.050 dex\nkpc−1, which flattens to ≈ −0.035 dex kpc−1for stars\n4−8 Gyr old and further flattens to ≈ −0.005 dex kpc−1\nfor stars 8 −12 Gyr old.\nWillett et al. (2023) used APOGEE DR17 and Gaia\nDR3 to investigate how metallicity changes with radius\nin the thin disk using a sample of 668 red-giant stars.\nThey divided stars into age bins of 0 −1 Gyr, 1 −2 Gyr,\n2−4 Gyr, 4 −6 Gyr, 6 −10 Gyr, and >10 Gyr, and they\npresented [Fe/H] profiles across 4 kpc < R < 11 kpc\nin their Figure 4, of which we show the gradients in\nFigure 3. Willett et al. (2023) found a radial gradient\nfor stars aged 0 −1 Gyr of ∂[Fe/H]/∂R≈ −0.059 dex\nkpc−1. This gradient flattens with stellar age but still\nremains significant for stars >10 Gyr at ≈ −0.033 dex\nkpc−1.\nAnders et al. (2023) used APOGEE to estimate stel-\nlar ages and investigate elemental, positional, and kine-\nmatic relationships of stars. Figure 11 of Anders et al.\n(2023) shows the radial gradient of [Fe/H], measured us-\ning Galactocentric distance across 5 kpc < R < 11 kpc,\nversus stellar age, which we show in Figure 3. Anders\net al. (2023) found a radial gradient of ∂[Fe/H]/∂R≈\n−0.061 dex kpc−1for stars aged >1 Gyr, which in-\ncreases to ≈ −0.078 dex kpc−1at ages ≈2.25 Gyr.\nFor ages ≳2.25 Gyr, this gradient generally decreases\nwith stellar age, becoming essentially flat for stars aged\n≈11.75 Gyr.\nFigure 3 compares these observational analyses. They\nall agree that the radial gradient becomes shallower for\nolder stars in the MW. However, they show a signif-\nicant amount of variation at a given age. For stars\nyounger than ≈2 Gyr, the measured gradient varies\nfrom≈ −0.05 dex kpc−1to≈ −0.075 dex kpc−1. These\nworks find even wider variations for the oldest stars.\nLian et al. (2022), Lian et al. (2023), and Anders et al.\n(2023) find that the gradient becomes essentially flat for\nstars older than ≈10 Gyr, while Vickers et al. (2021)\nand Willett et al. (2023) find much weaker age depen-\ndence, such that the gradient remains steep at these\nolder ages. These works use different surveys, with dif-\nferent selection functions, and different approaches to\nmeasuring stellar ages, all of which likely contribute to\nthis scatter.\nFigure 3 also shows ∂[Fe/H]/∂R today versus stellar\nage from our FIRE-2 galaxies. Given that observational\nmeasurements of the ages of stars generally have non-\ntrivial uncertainties, we also assume various fractional\nuncertainties in stellar age ( δt/t) in measuring this for\nour FIRE-2 galaxies, aiming to bracket age uncertain-\nties typical of these observational surveys, ≳10−30%\n(for example, Soderblom 2010; Silva Aguirre et al. 2018;Shen et al. 2024). Furthermore, Figure 3 shows (via a\ndotted line) the trend for Romeo, our earliest-forming\ndisk and likely the closest in time to when the MW disk\nspun up, which shows steeper than average gradients at\nnearly all ages. Compared with measurements of the\nMW, the radial gradients in FIRE-2 galaxies are gener-\nally weaker at a given age. We discuss potential reasons\nfor this in Section 6.3. The steepest gradient in FIRE-\n2 (∂[Fe/H]/∂R =−0.049 dex kpc−1for young stars)\nis consistent with Lian et al. (2023) across most of the\nage range, being 0 .050 dex kpc−1for their youngest age\nbin. However, all other analyses of the MW in Figure 3\nare steeper than all of our FIRE-2 galaxies. That said,\nFIRE-2 agrees with Lian et al. (2022), Lian et al. (2023),\nand Anders et al. (2023) that the radial gradients are es-\nsentially flat for the oldest stars, and, most importantly,\nthe general trend of the radial gradient becoming less\nsteep (flatter) with increasing stellar age agrees with all\nof these observations.\nFigure 3 also highlights the effects of applying uncer-\ntainties in stellar age to FIRE-2 galaxies. As expected,\nlarger uncertainties mix stars of different intrinsic ages\ninto the same “measured” age bin, thus mixing stars\nthat formed with different intrinsic radial gradients. The\neffect of this mixing depends on the slope of the under-\nlying relation between the gradient and age. For stars\nyounger than ≈8 Gyr in FIRE-2, larger age uncertain-\nties act to shallow the gradient at a given “measured”\nage, while for older stars, which formed with little-to-\nno intrinsic gradients, this mixing pulls in younger stars\nand leads to a steeper “measured” gradient. Therefore,\nstellar age uncertainties may contribute to the widely\nvarying gradients observed for old stars in the MW. As\nFigure 3 shows, fractional age uncertainties of 10% do\nnot affect this relation much, but uncertainties of 25%\nor larger bias the relation significantly.\nFinally, because the radial trends in FIRE-2 today\nare broadly similar to those in the MW in Figure 3, and\nthese trends in FIRE-2 were mostly set at birth, this\nsuggests that the radial trends of metallicity in the MW\ntoday were set mostly at birth as well.\n4.VERTICAL PROFILE\n4.1. Results in FIRE-2\nFigure 4 shows vertical profiles of [Fe/H] for all stars\ntoday, regardless of age, at 3 radii ( ±0.5 kpc): 4 kpc\n(inner disk), 8 kpc (Solar annulus), and 12 kpc (outer\ndisk). We show trends both today and using the posi-\ntions (both RandZ) of stars at their birth. Because\nsome of our galaxies have |Z|star\n90<2 kpc, we lighten\nthe vertical profile at |Z|>1 kpc, to emphasize that\nZ <1 kpc is our principle range of interest.Spatial Variations of Metals in FIRE 9\n0.6\n0.4\n0.2\n0.0\nRadius = 4 kpctoday\nformation\n0.5\n0.4\n0.3\n0.2\n0.1\n0.0[Fe/H]\nRadius = 8 kpc\n0.0 0.5 1.0 1.5 2.0\n|Z| [kpc]0.4\n0.3\n0.2\n0.1\nRadius = 12 kpc\nFigure 4. [Fe/H] for all stars versus absolute vertical height\nabove/below the disk midplane, |Z|, both today (using |Z|\nandRtoday) and at formation (using |Z|andRat birth),\nat different R(±0.5 kpc): 4 kpc (inner disk), 8 kpc (Solar\nannulus), and 12 kpc (outer disk). Each line and shaded\nregion show the average and 1 σscatter across the 11 galax-\nies. [Fe/H] decreases with increasing vertical height at all R,\nbut this decrease is weaker at larger R.All of these trends\nare present both today and at formation : they arise from\nthe “upside-down” nature of disk settling, and because stars\nspan a larger range of metallicity (and age) at smaller R.\nAs Appendix A shows, the vertical gradients for mono-age\npopulations are weak.\n[Fe/H] is higher near the midplane of the disk and\ndecreases with absolute vertical height. This trend is\nstrong in the inner disk, weaker at the Solar annulus,\nand even weaker in the outer disk, but it is consistent\nthroughout. As Appendix A shows, the vertical gradi-\nents for mono-age stars are essentially flat, both today\nand at formation. Therefore, the primary driver of this\nvertical profile for all stars today is metal enrichmentover time, combined with disk settling over time. The\nyoungest and most metal-rich stars dominate the mid-\nplane, while older more metal-poor stars formed (and\ncurrently tend to be at) larger vertical height. This re-\nsults in a vertical gradient in both metallicity and age\nfrom “upside down” disk formation (for example, Bird\net al. 2013).\nFigure 5 (left) quantifies the vertical gradient,\n∂[Fe/H]/∂|Z|, of all stars regardless of age across |Z|<\n1 kpc, versus cylindrical radius, R, both today and\nat formation (using birth Zand R). Again, the\nvertical gradient is strongest at smaller R, reaching\n∂[Fe/H]/∂|Z|=−0.44 dex kpc−1atR < 1 kpc today\nand−0.48 dex kpc−1based on birth coordinates. At\nR= 8 kpc, the vertical gradient is −0.084 dex kpc−1\ntoday and −0.15 dex kpc−1using birth coordinates.\nThis gradient weakens with R, becoming negligible at\nR≳10 kpc. We also show Romeo, which has systemat-\nically steeper vertical gradients than the mean trend at\nallR, which we discuss further in Section 4.2.\nThis radial dependence for vertical metallicity gradi-\nents arises because larger radii systematically have a\nnarrower range of ages (primarily just younger stars),\nand therefore a narrower range of metallicities (mostly\njust metal-rich stars), while smaller radii have a wider\nrange of ages that sampled the galaxy across a larger\nfraction of the disk settling.\nVertical gradients are systematically shallower today\nthan at birth, because the combination of vertical heat-\ning and radial redistribution tends to wash out (smooth)\nthe gradients at all R. Nonetheless, as with radial gradi-\nents, the redistribution of stars between formation and\ntoday does not systematically change the overall trends\nin the vertical gradient, with differences between birth\nand today typically being ≈0.073 dex kpc−1, and the\nlargest average difference being 0 .19 dex kpc−1for stars\natR= 5 kpc.\nAs with the radial profiles, we also split these verti-\ncal profiles into mono-age populations to analyze their\ncontributions to these gradients. As Appendix A shows,\nunlike the radial gradients, which can be strongest for\nmono-age populations, the vertical gradients of mono-\nage stars are mostly negligible, being ≲−0.083 dex\nkpc−1today and ≲−0.051 dex kpc−1using formation\ncoordinates. This relative flatness of the vertical gra-\ndient at a given age arises because the height of the\nstar-forming gas disk is largely regulated by turbulence,\nwhich also efficiently mixes metals. Thus, star-forming\ngas is unable to stratify into a metallicity gradient the\nway it does radially, causing stars of a given age to pos-\nsess similar metallicities across the vertical extent.10 Graf et al.\n0 2 4 6 8 10 12 140.6\n0.4\n0.2\n0.0[Fe/H]/|Z| [dex kpc1]\ntoday\nformation\nFIRE: Romeo\n0 2 4 6 8 10 12 14FIRE |Z| < 1 kpc\nFIRE |Z| < 2 kpc\nFIRE: Romeo\nMW (Imig+2023)\nRadius [kpc]\nFigure 5. Vertical gradient in [Fe/H], ∂[Fe/H]/∂|Z|, of all stars regardless of age versus cylindrical radius, R. Each line, darker\nshaded region, and lighter shaded region show the average, 1 σscatter, and min/max across the 11 galaxies. Dotted lines show\nRomeo, the earliest-forming disk in our sample. Left: Measuring the vertical gradient across |Z|= 0−1 kpc, both today and\nat formation (using birth |Z|andR). The vertical gradient is always negative, both at formation and today, and it steepens at\nsmaller R. As Appendix A shows, the vertical gradients for mono-age populations are weak, so these vertical gradients result\nfrom “upside-down” disk settling, and they are stronger at smaller R, where stars span a larger range in metallicity (and age).\nThe vertical gradients have weakened somewhat between formation and today, from vertical heating and radial redistribution,\nbut these key trends were already in place at birth. Right : Same, but only for stars today, across a vertical range of |Z|<1 kpc\nand|Z|<2 kpc, to compare with observations of the MW from Imig et al. (2023). FIRE-2 has weaker vertical gradients than\nthe MW at most R, but the trend of a steepening vertical gradient at smaller Ragrees. This implies that the MW underwent\nmore metal enrichment as its disk settled than in FIRE-2 galaxies, likely because FIRE-2 galaxies settled later, when [Fe/H]\nincreased more gradually, than in the MW.\nThe origin and age dependence of the vertical gradient\nis therefore nearly opposite that of the radial gradient.\nThe radial gradient can be strong for mono-age stars\nbut is nearly flat when combining all stars (Figure 1),\nwhereas the vertical gradient is strong when combining\nall stars but is negligible for mono-age stars.\n4.2. Comparison to the Milky Way\nFigure 5 (right) compares vertical gradients in these\nFIRE-2 galaxies against measurements of the MW from\nImig et al. (2023), which used APOGEE DR17 to mea-\nsure vertical gradients in metallicity using 66,496 red-\ngiant stars. We include the results from their Figure 16,\nwhich shows ∂Fe/H]/∂|Z|for all stars regardless of age\nversus R. Imig et al. (2023) measured the gradient\nacross different vertical ranges at different R. At most\nradii, they used |Z|≲1 kpc, but at R≈7−10 kpc,\nthey used Z≲2 kpc. To bracket these ranges, Figure 5\n(right) simply shows vertical gradients in FIRE-2 today\nacross both |Z|<1 kpc and |Z|<2 kpc. As the profiles\nin Figure 4 show, using a larger vertical range can lead\nto a steeper or shallower vertical gradient, depending on\nR. In FIRE-2, using a larger vertical range leads to a\nshallower vertical gradient at R≲6 kpc, but it leads to\na slightly steeper gradient at larger R.Imig et al. (2023) found that the steepest vertical gra-\ndient in the MW of ∂Fe/H]/∂|Z| ≈ − 0.47 dex kpc−1\noccurs at R= 2−5 kpc, and it decreases at R≳5 kpc\nto a minimum of ≈ −0.07 dex kpc−1atR= 15 kpc. Be-\nyond this, the vertical gradient is negligible. At R≲3\nkpc, Imig et al. (2023) measured this across |Z|≲1 kpc,\nand the vertical gradients in FIRE-2 at R≲4 kpc are\nbroadly consistent with Imig et al. (2023) across this\nrange. At larger R, however, the vertical gradients in\nFIRE-2, regardless of the vertical range, are consistently\nshallower than the MW from Imig et al. (2023).\nGiven our results in Appendix A, that the vertical gra-\ndients of mono-age populations contribute negligibly to\nthis trend, this discrepancy with the MW most likely\nreflects differences in metal enrichment history as it re-\nlates to disk settling history. Specifically, the most rapid\nrate of metal enrichment in these FIRE-2 galaxies occurs\nearly in their formation history (see Figure 1 of Bel-\nlardini et al. 2022). If this enrichment history broadly\nreflects the MW’s, and if the MW’s disk started to set-\ntle earlier than these FIRE-2 galaxies, then the MW\nwould have experienced more vertical settling during\nthis rapid enrichment phase, leading to the MW hav-\ning a more vertically stratified metallicity distribution.\nIndeed, as Figure 5 (right) shows, our galaxy with the\nearliest settling disk, Romeo, has a stronger gradientSpatial Variations of Metals in FIRE 11\nthan our mean trend line, more similar to the MW. From\nFigure 5 (left), another possibility is that the MW expe-\nrienced less vertical heating and/or radial redistribution\nthan in FIRE-2 (as we explore directly in Bellardini et\nal. in prep.), such that its vertical gradient today more\ndirectly reflects its trends at the time that stars formed.\nDespite this discrepancy in the exact strength of the\nvertical gradient today, the general trend of a steepening\nvertical gradient with decreasing radius agrees with the\nMW. Because the vertical trends in FIRE-2 today are\nbroadly similar to those in the MW, and these trends in\nFIRE-2 were set mostly at birth, this suggests that the\nvertical trends of metallicity in the MW today were set\nmostly at birth as well.\n5.AZIMUTHAL SCATTER\n5.1. Results in FIRE-2\nWe measure the azimuthal scatter of [Fe/H], σ[Fe/H],\nas the standard deviation of [Fe/H] around an annulus\n(360 degrees) at a fixed R, with width ∆ R=±0.5 kpc.\nTo measure azimuthal scatter for stars at birth, we fol-\nlow Bellardini et al. (2022): we split each age bin of\n1 Gyr into 20 age bins, each spanning 50 Myr, and we\ncompute the azimuthal scatter within each one; then\nwe average across these 20 bins to compute the typi-\ncal azimuthal scatter within for a given 1 Gyr age bin.\nBellardini et al. (2022) (their Figure 9) examined how\nazimuthal scatter varies with azimuthal bin size (ar-\nclength) for stars at birth. They found a weak depen-\ndence on the arclength, such that the scatter across a lo-\ncal patch (azimuthal length ≲1 kpc) is only marginally\nsmaller ( ≲0.01 dex) than the scatter across an entire\n360-degree azimuth. Therefore, we measure the total\nazimuthal scatter across 360 degrees for simplicity.\nFigure 6 shows σ[Fe/H]versus cylindrical radius, R,\nboth today and at formation, in bins of stellar age. At a\ngiven age, σ[Fe/H]increases weakly with R, in agreement\nwith Bellardini et al. (2022) for stars at birth. This is\nbecause the outer gas disk is less well mixed, from a\ncombination of the longer orbital (and hence mixing)\ntimescales and the stronger influence of cosmic accre-\ntion. This radial dependence is slightly weaker for stars\ntoday, as expected from post-formation radial redistri-\nbution weakening any radial trends. For old stars, ra-\ndial redistribution does not change σ[Fe/H]much from\nformation to today, because there was no strong radial\ngradient at that time to add to this scatter. However,\nfor younger stars, radial redistribution acts to increase\nσ[Fe/H]from birth to today, because it adds additional\nscatter sourced from the radial gradient, as we explore\nin Section 5.3. However, this increase in σ[Fe/H]at a\ngiven Rfrom birth to today is generally smaller than\n0.030.060.090.12\n0-1 Gyrtoday\nformation\n0.030.060.090.12\n4-5 Gyr\n0.060.090.120.15\n8-9 Gyr\n0 2 4 6 8 10 12 14\nRadius [kpc]0.090.120.150.180.21\n11-12 Gyr[Fe/H]\nFigure 6. Azimuthal scatter in [Fe/H] across an annulus at\nfixed R,σ[Fe/H], versus cylindrical radius, R, for stars today\nand at formation (using their birth radii), in bins of stellar\nage. Each line and shaded region show the average and 1 σ\nscatter across the 11 galaxies. We show azimuthal scatter\nonly out to Rstar\n90at the time of formation. Azimuthal scatter\nincreases weakly with Rat all ages, more so at formation\nthan today. As a result of radial redistribution, azimuthal\nscatter is somewhat larger, with weaker radial dependence,\ntoday than at birth, but these differences are smaller than\nσ[Fe/H]in place at birth. Therefore, most of the azimuthal\nscatter in stars today was present at their birth.\ntheσ[Fe/H]already in place at birth. Thus, most of the\nazimuthal scatter was already in place at the time of\nstellar birth.\nFigure 7 shows σ[Fe/H]versus stellar age. The pur-\nple lines show the scatter across the Solar annulus\n(R= 8±0.5 kpc). There, the azimuthal scatter was\nfairly constant for ages ≲8 Gyr, at σ[Fe/H]≈0.087 dex\ntoday and ≈0.062 dex at formation. For stars older\nthan 8 Gyr, the azimuthal scatter increases rapidly with\nage, from increased patchiness of the gas at those times\n(Bellardini et al. 2021, 2022). Indeed, McCluskey et al.\n(2023) found that the disk began forming in these FIRE-\n2 galaxies ≈8 Gyr ago, on average. Before that time,12 Graf et al.\nazimuthal variations via patchiness were the primary\ncause of metallicity variations across the galaxy. After\nthat, the radial gradient dominated, as the azimuthal\nvariations smoothed away. As explained above, for this\nreason, the azimuthal scatter today versus at birth is\nnearly identical for stars formed in the “pre-disk” era\n≳8 Gyr ago, while radial redistribution causes the az-\nimuthal scatter today to be somewhat larger than at\nbirth during the “disk” era ≲8 Gyr ago. That said,\nas with radial and vertical profiles, the azimuthal varia-\ntions today primarily reflect those present at birth, with\nthe effects of radial redistribution generally adding only\nsecondarily; in the most extreme case they double the\npre-existing azimuthal scatter (a difference of 0 .042 dex\nfor 3 Gyr old stars).\nFor context, the orange line in Figure 7 also shows\nthe total scatter (standard deviation) in [Fe/H] across\neach entire galaxy ( R < 20 kpc). In this case, there is\nno distinction between stars today versus at formation,\ngiven our in-situ selection criteria from Section 2.4. The\ngalaxy-wide scatter is the same as the azimuthal scatter\natR= 8 kpc for old stars, because old stars had and\nhave negligible radial gradients. However, rather than\nremaining constant for stars ≲8 Gyr old, the galaxy-\nwide scatter rises toward younger ages, from the onset\nof the radial gradient, as Bellardini et al. (2022) showed.\n5.2. Comparison to the Milky Way\nMeasuring the azimuthal scatter of metallicity in the\nMW is challenging. Two recent observational analyses\nof the MW presented results against which we compare,\nbut we caution that, for azimuthal scatter in particular,\nmodeling/matching the selection function is particularly\nimportant, and we defer a more careful comparison for\nfuture work. For now, we compare with an emphasis on\ngeneral trends.\nRatcliffe et al. (2023) used APOGEE DR17 to study\nelemental abundances in the MW through the derivation\nof birth radii of 145,447 red giant stars in the disk. Their\nFigure 2 quantifies the standard deviation in [Fe/H] (az-\nimuthal scatter in the Solar neighborhood) versus stellar\nage, which we show in Figure 8 (left and right). Figure 8\n(left) also shows σ[Fe/H]versus stellar age from Figure 12\nof Anders et al. (2023), also based on APOGEE (see Sec-\ntion 3.2). One important difference between these obser-\nvational analyses is the effective radial range over which\nthey measured variations. Anders et al. (2023) sought\nto compute azimuthal scatter at a fixed R, that is, the\nscatter when subtracting out the effect of the radial gra-\ndient. By contrast, Ratcliffe et al. (2023) computed the\nscatter across their sample, which encompasses a much\nlarger radial range ( R≈5−12 kpc). Therefore, for\n0 2 4 6 8 10 12\nAge [Gyr]0.00.10.20.3[Fe/H]\ntoday\nformationR = 8 kpc\nR = 0-20 kpcFigure 7. Scatter in [Fe/H], σ[Fe/H], versus stellar age, for\nstars today and at formation (using their birth radii). Each\nline and shaded region show the average and 1 σscatter across\nthe 11 galaxies. The purple lines show the azimuthal scatter\nacross the Solar annulus, R= 8±0.5 kpc, which is nearly\nconstant with age over the last ≈8 Gyr and rises rapidly\nwith age before that. The azimuthal scatter today is only\nmoderately larger than at formation, from the effects of ra-\ndial redistribution, and only at ages ≲8 Gyr. Most of the\nazimuthal scatter in stars today was present already at their\nbirth. For comparison, the gold line shows the total scatter\nin [Fe/H] across the entire galaxy, R < 20 kpc, which shows\nthe same trend at old ages, but rises towards younger ages,\nfrom the onset of a radial gradient.\nyounger stars with ages ≲8 Gyr, the measured scatter\nis larger in Ratcliffe et al. (2023) than in Anders et al.\n(2023), a difference that diverges towards younger ages.\nAlso, as a result, the age dependence of the scatter in\nRatcliffe et al. (2023) is modest, varying from ≈0.16\nto≈0.23 dex across the entire age range. Both works\nagree, though, that the scatter in [Fe/H] was largest\n≈8−10 Gyr ago. Also, all observational works in\nSection 3.2 agree that the radial gradient in the MW\nis shallower for older stars, so if the radial gradient is a\nkey contributor to the measured scatter in Ratcliffe et al.\n(2023), this naively seems difficult to reconcile with the\ngradually rising trend with age in Ratcliffe et al. (2023).\nFigure 8 compares these observational analyses to our\nFIRE-2 galaxies, for which we measure σ[Fe/H]today\nversus stellar age. We show results from FIRE-2 using\ntwo radial selections, to bracket the radial selections of\nAnders et al. (2023) and Ratcliffe et al. (2023). Specif-\nically, the left panel shows stars at R= 8±0.5 kpc to-\nday, which in principle is most comparable to the anal-\nysis of Anders et al. (2023). Mimicking the selection\nfunction of Ratcliffe et al. (2023) is less straightforward,\nso we simply show the scatter across the entire galaxySpatial Variations of Metals in FIRE 13\n0 2 4 6 8 10 120.00.10.20.30.4[Fe/H]\nRadius = 8 kpc\nFIRE t/t=0\nFIRE t/t=10%\nFIRE t/t=25%\nFIRE t/t=50%\nFIRE t/t=10%: Romeo\nMW (Ratcliffe+2023)\nMW (Anders+2023)0 2 4 6 8 10 12Radius = 020 kpc\nFIRE t/t=0\nFIRE t/t=10%\nFIRE t/t=25%\nFIRE t/t=50%\nFIRE t/t=10%: Romeo\nMW (Ratcliffe+2023)\n Age [Gyr]\nFigure 8. Same as Figure 7, but assuming various fractional uncertainties in stellar ages, for stars today at R= 8±0.5 kpc\n(left) and across the entire galaxy, R= 0−20 kpc (right). Larger age uncertainty causes σ[Fe/H]to increase more rapidly with\nage, leading to larger scatter at a given age. For scatter across the entire galaxy (right), large age uncertainties can shift or erase\nthe intrinsic minimum σ[Fe/H]at ages ≈8 Gyr. We also show measurements of the scatter across the MW from Anders et al.\n(2023) and Ratcliffe et al. (2023). Using a larger radial range and larger age uncertainty leads to a normalization in σ[Fe/H]that\nis more similar with Ratcliffe et al. (2023), especially at younger ages, but it does not match the relative flatness with age that\nRatcliffe et al. (2023) finds. FIRE-2 galaxies also do not match the downturn at ages ≳9 Gyr that both Anders et al. (2023)\nand Ratcliffe et al. (2023) find.\n(R= 0−20 kpc) in the right panel, to conservatively\nbracket the extremes of radial selection.\nWe again assume various fractional uncertainties in\nstellar age, δt/t, in FIRE-2 when comparing with the\nMW. Again, the effects of these age uncertainties are\nmodest for δt/t≲10% and do not qualitatively change\nthe trends, except perhaps for the oldest stars. How-\never, δt/t≳25% affects the trend dramatically. For the\nazimuthal scatter at R= 8±0.5 kpc (left), larger age\nuncertainties cause the scatter to rise much more rapidly\nwith age at ages ≳3 Gyr. For the galaxy-wide scatter\n(right), larger age uncertainties cause the inferred mini-\nmum scatter to occur both at younger age and at larger\nscatter, thus “flattening” the relation with age. This\nhighlights that inferred trends of azimuthal or galaxy-\nwide scatter with age are sensitive to stellar age uncer-\ntainties.\nFor young stars, with ages ≲1 Gyr, the azimuthal\nscatter in FIRE-2 at fixed R(left) is comparable to\nthat measured in Anders et al. (2023). FIRE-2 alsoqualitatively agrees with Anders et al. (2023) that the\nazimuthal scatter at fixed Rincreases with age, if we\ninclude significant age uncertainties in FIRE-2. How-\never, the azimuthal scatter in Anders et al. (2023) ini-\ntially rises more rapidly with age than in FIRE-2, and it\nalso reaches a plateau at ≈0.22 dex, unlike the FIRE-2\ngalaxies which continue to rise with age to ≈0.35 dex\n(depending on age uncertainty).\nThe galaxy-wide scatter in FIRE-2 (right) is compara-\nble to Ratcliffe et al. (2023) at ages ≲4 Gyr. This likely\nreflects a trade-off that the radial selection in Ratcliffe\net al. (2023) is not quite as large as 20 kpc but also\nthat the MW has a stronger radial gradient than these\nFIRE-2 galaxies. The behavior of the scatter with age in\nFIRE-2 in the right panel depends sensitively on stellar\nage uncertainties. That said, none of the radial selec-\ntions or age uncertainties applied to FIRE-2 agree with\nthe relative flatness of the relation between scatter and\nage in Ratcliffe et al. (2023).14 Graf et al.\nWe conclude that, while the FIRE-2 simulations show\nqualitatively similar trends to the MW for the radial\nand vertical gradients, these trends of the (azimuthal)\nscatter with age are qualitatively discrepant with those\nmeasured in the MW, despite that they agree reasonably\nwell for just young stars. In general, FIRE-2 shows az-\nimuthal scatter that increases faster and monotonically\nwith stellar age, as measured today with non-trivial age\nuncertainties, and no level of assumed age uncertainty\nmatches the observed plateau at ≈9 Gyr. Again, we\ncaution that a more rigorous modeling of observational\nselection functions is essential for a more quantitative\ncomparison. We defer a more detailed analysis for fu-\nture work.\n5.3. Relation between the radial gradient and\nazimuthal scatter\nAs discussed in Section 5.1, the azimuthal scatter in\nthese FIRE-2 galaxies is relatively independent of Rand\nis nearly independent of age at ≲8 Gyr. Here, we seek\nto understand the origin of this azimuthal scatter, at\nleast for young stars today. In particular, we test for its\npossible relation to the radial gradient.\nWe extend the results of Orr et al. (2023), which stud-\nied the azimuthal variations of metals in gas in some of\nthese same FIRE-2 galaxies at z≈0. They found that\nradial transport of gas along spiral arms can be a signifi-\ncant driver for these azimuthal variations. In particular,\nspiral arms can act as “metal freeways”: their gravi-\ntational perturbations drive higher-metallicity gas from\nsmaller radii outward and lower-metallicity gas from\nlarger radii inward, thus contributing to the azimuthal\nscatter at a given Rtoday. Grand et al. (2016) pre-\nviously demonstrated a similar effect using the Auriga\ncosmological zoom-in simulations, and more recently, us-\ning a (non-cosmological) N-body baryonic simulation,\nKhoperskov et al. (2023) found a similar dynamical phe-\nnomenon and concluded that radial gradients are the key\ningredient for sourcing azimuthal variations across spiral\narms.\nAs a test of the importance of this effect across a pop-\nulation, we examine whether galaxies with stronger ra-\ndial gradients also have greater azimuthal scatter in their\nstellar metallicities, both at birth (as inherited from the\nstar-forming gas) and today.\nFigure 9 shows the azimuthal scatter in stars, σ[Fe/H],\natR= 8 ±0.5 kpc versus the radial gradient,\n∂[Fe/H]/∂R, within ±4 kpc of R= 8 kpc, for stars with\nage<1 Gyr both today and at birth. We use ±4 kpc\nas a conservative upper limit to the amount of redis-\ntribution from their birth radius that stars experience\nover≈1 Gyr in FIRE-2 (Bellardini et al. (in prep.).\n0.04\n 0.03\n 0.02\n[Fe/H]/R [dex kpc1]\n0.040.060.080.100.12[Fe/H]\ntoday\nformationFigure 9. Azimuthal scatter, σ[Fe/H], atR= 8±0.5 kpc\nversus radial gradient, ∂[Fe/H]/∂R across R= 4−12 kpc,\nfor stars younger than 1 Gyr today, measured using radii to-\nday and at birth, for each of our 11 galaxies. Galaxies with\nlarger azimuthal scatter have stronger radial gradients, with\nbest-fit slopes of −1.7 kpc today (navy dashed line) and −1.5\nkpc at formation (gold dashed line). This super-linear rela-\ntionship supports the idea that the azimuthal scatter at late\ncosmic times (once the disk has settled) is primarily sourced\nby the radial redistribution of gas before star formation (as\nthe trend at formation shows) and additionally by the radial\nredistribution of stars after formation (as the trend today\nshows), which mixes the radial gradient into azimuthal scat-\nter.\nEach point shows one of the 11 FIRE-2 galaxies, which\nwe measure both today and using the coordinates of the\nstars at birth. Galaxies with larger azimuthal scatter\nhave a stronger radial gradient. We test the strength\nof this correlation using a Pearson test, finding a coeffi-\ncient of −0.72 and p-value of 0 .012 for stars today, and\na somewhat weaker coefficient of −0.59 and p-value of\n0.054 for stars at formation. Therefore, both correla-\ntions are relatively strong and significant.\nFigure 9 also shows a fit linear relation between σ[Fe/H]\nand∂[Fe/H]/∂R, which has a slope of −1.7 kpc for stars\ntoday and −1.5 kpc for stars at formation. It is notewor-\nthy that both relations are steeper than linear. As pre-\nviously discussed, we necessarily measure the azimuthal\nscatter using a finite radial bin of ±0.5 kpc, so a galaxy\nwith a stronger radial gradient would have stronger (ra-\ndial) variation within this bin just from the radial gradi-\nent. However, as Figure 9 shows, the azimuthal scatter\nis generally much stronger than the strength of the ra-\ndial gradient multiplied by the radial bin width. This\nmeans that the azimuthal scatter is not simply dom-\ninated by radial variations within the bin, suggesting\nother contributions. To further test this, we reduced\nthe radial bin width in measuring the azimuthal scat-Spatial Variations of Metals in FIRE 15\nter from 8 ±0.5 kpc to 8 ±0.25 kpc: the scatter was\nessentially unaffected (∆ ≲0.001 dex/kpc). Thus, the\nscatter that we present represents the true azimuthal\nvariations along an annulus.\nThis relation between azimuthal scatter and the radial\ngradient supports the conclusions of Orr et al. (2023)\nand Khoperskov et al. (2023) that spiral arms act as\n“metal freeways”, and that strong radial gradients are a\nkey contributor to azimuthal variations in spiral arms.\nThis predicted correlation would be compelling to test\nusing a large sample of observed disk galaxies.\n6.SUMMARY AND DISCUSSION\n6.1. Summary\nWe analyzed 11 MW-mass galaxies from the FIRE-2\nsuite of cosmological zoom-in simulations. As several\nworks have shown, they reproduce many key observed\nproperties of the MW, M31, and similar-mass galaxies\nin the nearby universe. We analyzed the radial, verti-\ncal, and azimuthal variations of elemental abundances\nin stars. We focused on [Fe/H], but in Appendix B\nwe also show trends for [Mg/H] and [C/H], which are\nqualitatively similar. These simulations allow us to ex-\namine these trends both today and at the time that\nstars formed, in a realistic cosmological setting. The\nkey question that we explored is the extent to which\nspatial variations in elemental abundances today reflect\nthose at birth. We also compared with the MW, includ-\ning effects of uncertainties in stellar age in analyzing the\nsimulations. Our main results are as follows.\nAll spatial variations today (radial, vertical, az-\nimuthal), including their trends with age and location,\nwere already in place at the time that stars formed. The\nevolution of stars after birth, via radial redistribution\nand vertical heating, plays a subdominant role in de-\ntermining spatial patterns in elemental abundances to-\nday. That said, the combination of radial redistribu-\ntion and vertical heating of stars has somewhat weak-\nened (washed out) the radial and vertical gradients, and\nsomewhat increased the azimuthal scatter, since birth.\nRadial gradients : In FIRE-2 they are flat for the oldest\n(≳12 Gyr) stars and steepen systematically for younger\nstars. Importantly, this is true both today and at forma-\ntion, because the metallicity radial gradient in the ISM\nbecame steeper (more negative) over time. Radial gra-\ndients in the MW are generally stronger than in FIRE-2\ngalaxies at most ages today. We speculate that this is\nlargely because the MW’s disk settled earlier than the\ngalaxies in our FIRE-2 sample. That said, both FIRE-\n2 and the MW show the same key trend of a steeper\nradial gradient for younger stars, and both show nearly\nflat radial gradients when measuring all stars (regardlessof age) today. This implies that the radial gradient in\nthe MW today largely reflects what was in place in the\nMW at birth: the ISM had progressively steeper (more\nnegative) radial gradients in metallicity over time.\nVertical gradients : In FIRE-2 they have the opposite\nbehavior as the radial gradients. The vertical gradient\nfor all stars today is strong, while the vertical gradient\nfor mono-age stars is mostly negligible. The latter is\nlikely because the vertical extent of the star-forming gas\ndisk is set by turbulence, which also efficiently mixes\nmetals. Therefore, the strong vertical gradient we find\nwhen considering stars of all ages, present both today\nand at birth, primarily reflects the upside-down settling\nof the disk over time, while it enriched in metals. These\nvertical gradients are strongest at small radii, where the\nrange of stellar ages is largest, and they flatten with\nradius, being minimal in the outer disk where the age\nrange is minimal. Vertical gradients of all stars today\nare shallower in FIRE-2 galaxies than in the MW at\nmost radii, including at the solar annulus ( R= 8 kpc).\nWe speculate that the MW’s disk settled earlier and/or\nenriched more rapidly in metals as it settled than in\nthese FIRE-2 galaxies. However, the general trend of a\nsteeper vertical gradient at smaller radii agrees with the\nMW.\nAzimuthal scatter (at fixed radius) : In FIRE-2 it is\nnearly independent of radius today, and it increased\nweakly with radius at birth. At R= 8 kpc (Solar annu-\nlus), the azimuthal scatter is nearly constant with stellar\nage over the last ≈8 Gyr at σ[Fe/H]≈0.087 dex today\nand≈0.062 dex at birth after the galactic disk settled,\nand it rapidly rises with age before that from the increas-\ning “patchiness” of the gas at earlier times. The trend of\nazimuthal scatter versus age is different in FIRE-2 than\nin the MW, with the MW showing an increase with age\nat young ages, a peak at ≈9 Gyr, and then a decrease\nwith age toward older ages.\nEffects of stellar age uncertainties : Adding fractional\nuncertainties in stellar age of δt/t≳25% to these FIRE-\n2 galaxies can dramatically change (bias) the age depen-\ndence of these spatial variations.\nRelation between radial gradients and azimuthal scat-\nter: In FIRE-2, galaxies with larger azimuthal scatter in\nmetallicity for young stars today have a stronger radial\ngradient. This implies that azimuthal scatter today is\nlargely a result of radial redistribution of gas and stars,\nlikely driven by spiral arms (following Orr et al. 2023).\nConclusion : While the FIRE-2 simulations generally\nshow weaker gradients than the MW, the key trends\nwith stellar age and location today are similar. Because\nspatial variations in metallicity of stars today in FIRE-2\nwere largely in place when the stars formed, this there-16 Graf et al.\nfore implies that the spatial variations in metallicity of\nstars in the MW today were also largely in place when\nthe stars formed.\n6.2. Caveats\nWe describe some important caveats to our analysis.\nFirst, we analyzed a sample of 11 MW-mass galaxies.\nTheir cosmological selection function was simple: iso-\nlated or LG-like paired dark-matter halos at z= 0 with\nM200m≈1−2×1012M⊙, with no additional selection\nbased on galaxy properties or merger/formation history.\nOur results thus sample random formation histories of\nMW-mass galaxies. Indeed, as Bellardini et al. (2021)\nshowed, the radial gradients and azimuthal variations of\ngas metallicity in these simulations at z≈0 are com-\nparable to, and in some cases steeper than, those mea-\nsured in M31 and in surveys of nearby MW-mass galax-\nies. That said, as our comparisons have shown, these\nFIRE-2 simulations are not perfect analogs of the MW;\nin particular, they generally have shallower gradients.\nFurthermore, these FIRE-2 simulations do not model\npotentially important physical ingredients, including\nmagnetohydrodynamics or self-consistent cosmic rays\ninjection and transport (see discussions in Hopkins et al.\n2018; Wetzel et al. 2023). However, analyses of FIRE-2\nsimulations that included these processes indicate that\nthey are not likely to affect disk dynamics (and therefore\ndisk-wide metallicity variations) at fixed stellar mass\n(for example, Su et al. 2017; McCluskey et al. 2023).\nPotentially more important is that these FIRE-2 simu-\nlations do not include supermassive black holes and their\nAGN feedback. While the effects of AGN feedback are\nlikely most significant in the innermost regions of galax-\nies, and less likely to impact stellar dynamics and star-\nformation rates in the Solar annuli of MW-mass galaxies\n(Mercedes-Feliz et al. 2023; Wellons et al. 2023), some\nsimulations (for example ?) indicate that AGN feed-\nback can affect the kinematic and structural properties\nof MW-mass galaxies.\nFinally, in terms of stellar nucleosynthesis in FIRE-2,\nall results are based on a single set of models, which are\nthemselves uncertain. For example, Gandhi et al. (2022)\nshowed that the implementation of white-dwarf super-\nnovae in FIRE-2 likely underestimates their rates, lead-\ning to a modest underproduction of [Fe/H]. As we show\nin Appendix B, our key results are largely insensitive to\nwhich elements we examine, despite their contributions\nfrom different stellar evolution channels, which suggests\nthat the trends that we showed are likely robust to cur-\nrent uncertainties in stellar nucleosynthesis. Another\nkey limitation, especially for our results on azimuthal\nvariations, is that FIRE-2 models all core-collapse su-pernovae via the same (IMF)-averaged yields, but Mu-\nley et al. (2021) showed how different mass progenitors\nlead to different patterns in abundances, in particular,\nwith larger scatter.\n6.3. The strength of gradients compared with the Milky\nWay\nAs discussed in Section 3.2, and in detail in Bel-\nlardini et al. (2021), the radial gradients of metallic-\nity in these FIRE-2 MW-mass galaxies generally agree\nwell with measurements of M31, and they are gener-\nallysteeper than measurements of populations of nearby\nMW-mass galaxies. However, they also are generally\nshallower than in the MW: the strongest gradient in\nFIRE-2 agrees with the shallowest gradient measured\nin the MW, but otherwise observations of the MW are\nconsistently steeper. At face value, these comparisons\nwith nearby galaxies suggest that the MW has an un-\nusually steep gas-phase radial metallicity gradient (see\nalso Boardman et al. 2020).\nOne contribution to this could be distance uncertain-\nties in measurements of the MW. Donor et al. (2018)\nfound that the distance catalog used can change ob-\nserved radial gradients by up to 40%.\nAnother possibility is that the MW disk settled un-\nusually early compared to our simulations and compared\nwith nearby galaxies, as Bellardini et al. (2022) argued,\nfinding a weak, though non-zero, correlation between\nthe strength of the radial gradient and the time of disk\nsettling. McCluskey et al. (2023) found that the kine-\nmatics for young stars in FIRE-2 agree reasonably well\nwith the MW and nearby galaxies, but McCluskey et al.\n(in prep.) compares these kinematics as a function of\nstellar age against the MW, to understand better how\nthe dynamical evolution of these galaxies compares with\nthe MW (see also Belokurov & Kravtsov 2022).\n6.4. Comparison to previous works\nOur work builds on previous analyses of the FIRE\ncosmological zoom-in simulations (Ma et al. 2017a,b;\nBellardini et al. 2021, 2022), all of which agree with\nthe trends we presented. Various other works using dif-\nferent cosmological simulations also studied elemental\nabundance patterns.\nFont et al. (2011) analyzed the trends in the radial\nprofiles of [Fe/H] in stars today using the GIMIC suite\nof cosmological simulations of 412 galaxies. They found\na typical radial gradient for all stars measured across\nR < 20 kpc of ∂[Fe/H]/∂R≈ −0.025 dex kpc−1. More\nrecently, Font et al. (2020) used the ARTEMIS simula-\ntions to study the [Fe/H] profile in 42 MW-mass galaxies\nand found a gradient for all stars measured across theSpatial Variations of Metals in FIRE 17\nsame range of ≈ −0.015 dex kpc−1. While steeper than\nthe profiles we show in Figure 1, both works agree with\nour negative radial gradient in [Fe/H] for all stars at\nR≳6 kpc.\nSeveral works also studied how the radial profile of\nmetallicity changes over time in the (star-forming) ISM\nand/or in stars at birth. Some works found that ra-\ndial gradients were steeper in the past and became shal-\nlower (flatter) over time, opposite to our results. Vin-\ncenzo & Kobayashi (2018) analyzed [O/H] profiles in\nthe ISM for 10 MW-mass galaxies in their cosmolog-\nical simulations and found gradients measured across\nR≈0−15 kpc to be strongest at early times. Agertz\net al. (2021) studied the [Fe/H] profile in the ISM using\nthe VINTERGARTEN cosmological simulation of a sin-\ngle MW-mass galaxy and found similar results, though\nthey note that a relatively late-time merger may have af-\nfected the evolution of this particular galaxy. Buck et al.\n(2023) analyzed 4 galaxies from the NIHAO-UHD suite\nof cosmological simulations and found steeper gradients\nin gas at earlier times, and, in particular, that the gra-\ndients only steepened shortly after a gas-rich merger,\nwhich were more common in the past. However, they\nmeasured such gas-phase gradients across a fixed radial\nrange, R= 2.5−17.5 kpc, across cosmic time, which\ndiffers significantly from our measuring such gradients\nonly out to radii where stars form, ≈R90, which may\ncontribute to these different trends.\nOther analyses of cosmological simulations found re-\nsults similar to ours. Vincenzo & Kobayashi (2020) stud-\nied the [Fe/H] profile of stars at birth using a cosmologi-\ncal zoom-in simulation of a single MW-mass galaxy and\nfound a steepening gradient at later times. Similarly, Lu\net al. (2022) analyzed 4 simulations from the NIHAO-\nUHD suite and found only mild evolution in the [Fe/H]\nradial gradient of the ISM over time when measured over\nR= 0−10 kpc, with some galaxies showing mild steep-\nening and others showing mild shallowing over time.\nTherefore, expectations from cosmological simulations\nare mixed regarding the strength and even direction of\nthe evolution of radial gradients in the ISM across the\nhistories of MW-mass galaxies. The details of the mod-\neling of star formation and feedback can lead to differing\nexpectations. For example, Pilkington et al. (2012) and\nGibson et al. (2013) analyzed the MUGS and MAGICC\nsuites of cosmological simulations. The “weaker” feed-\nback model of MUGS results in stronger radial [O/H]\ngradients in gas at earlier times, which flatten over time.\nThe “enhanced” feedback model of MAGICC (otherwise\nidentical to MUGS) redistributes and mixes the ISM and\nwashes out any gradients early on, resulting in flat pro-files at early times that slightly steepen over time, as\nthe rotation of the disk becomes well-ordered.\nAnalyzing the larger-volume TNG50 simulation, Hem-\nler et al. (2021) studied 1595 galaxies with Mstar=\n109−1011M⊙, and found radial gradients in gas-phase\n[O/H] to be steeper in galaxies at higher redshift. On\nthe other hand, Tissera et al. (2022) studied 957 galaxies\nwith Mstar>109M⊙from the EAGLE simulation and\nfound radial gradients in gas-phase [O/H] to have essen-\ntially no evolution ( ≈0.001 dex kpc−1/δz) with red-\nshift. We note, however, that Hemler et al. (2021) and\nTissera et al. (2022) tracked galaxies of the same mass\nover cosmic time, rather than the histories of individual\ngalaxies as we did, which complicates the comparison,\ngiven expectations (at least for FIRE) that the strength\nof the radial gradient correlates with galaxy mass at a\ngiven redshift (for example, Ma et al. 2017a).\nFewer works have studied vertical profiles of metal-\nlicity using simulations or analytical models. Minchev\net al. (2013) used a cosmological simulation of a single\nMW-mass galaxy to study the vertical profile of stars\natR= 8±1 kpc. They found that [Fe/H] decreases\nwith|Z|when considering all stars, which agrees with\nour Figure 4, and that mono-age vertical gradients are\nessentially flat, which agrees with our Appendix B.\nRegarding azimuthal scatter, Lu et al. (2022) stud-\nied such variations in the ISM in 4 MW-mass galaxies\nfrom NIHAO-UHD and found that σ[Fe/H]slightly in-\ncreases with radius, in agreement with our Figure 6.\nFurthermore, Lu et al. (2022) found a generally increas-\ningσ[Fe/H]with increasing lookback time, also agree-\ning with our Figure 7. Khoperskov et al. (2023) used\n3 controlled (non-cosmological) N-body baryonic sim-\nulations to uncover the origin of abundance variations\nacross spiral arms in MW-like disk galaxies, and found\nthat radial gradients play a significant role in sourcing\nazimuthal variations across spiral arms, in agreement\nwith our Figure 9.\nAnalytic models also explored the expected evolution\nof spatial patterns in abundance. Minchev et al. (2018)\ndeveloped a method for determining stellar birth radii\nin the MW by extrapolating the evolution of the ISM\nmetallicity with radius and time. They argued for the\nexpectation that the radial gradient became shallower\nover time, from the “inside-out” nature of disk growth,\nthat is, increasing fractional stellar mass grows at in-\ncreasing radii over time. They also argued that radial\nredistribution plays a more significant role in flatten-\ning radial gradients for older stars, to become as weak\nas they are in the MW today. While we agree with\nMinchev et al. (2018) that radial redistribution acts to\nflatten radial gradients in mono-age stellar populations,18 Graf et al.\nwe disagree with the expectation that the radial gra-\ndient in the ISM was steeper in the past. The FIRE\nsimulations show the opposite trend, despite also expe-\nriencing radial inside-out formation, at least on average.\nIn earlier work, Chiappini et al. (2001) developed an ele-\nmental evolution model and argued that radial gradients\nin metallicity steepen over time, in agreement with our\nresults. They explained this as occurring from infalling\nmetal-poor gas in the outer galaxy at later times, which\ndiffers from the reasons for the steepening gradient over\ntime that we argue for. More recently, Sharda et al.\n(2021) developed an analytic framework that argued for\na steepening of radial gradients over time.\nObservationally, Cheng et al. (2024) studied [O/H]\ngas-phase radial gradients using a sample of 238 star-\nforming galaxies at z= 0.6−2.6 and found that galaxies\nhave generally flat, and sometimes positive radial gradi-\nents, at these early times, which tend to steepen in a\nnegative direction over time.\n6.5. Evolution of radial gradients\nThe most important source of spatial variations in\nmetallicity for a disk galaxy like the MW, at least to-\nday, is the radial gradient. And yet, as we described\nabove, theoretical works continue to debate the expected\nstrength and even simply the sign of the evolution of the\nradial gradient over a galaxy’s history. Determining the\nradial gradient of the ISM of the MW across its history,\nwhich set the birth conditions of stars, has been chal-\nlenging via galactic archeology, given additional effects\nlike radial redistribution of stars after birth that compli-\ncate interpreting their patterns today. However, as we\nshowed in Section 3.2, essentially all recent analyses of\nthe MW agree that the radial gradient today is system-\natically shallower (less negative) for older stars. The\nsimplest explanation for this observed trend is that it\nsimply reflects the conditions of the (star-forming) ISM\nacross the MW’s history, such that the radial gradient of\nISM metallicity in the MW became steeper (more neg-\native) over time, and that the effects of stellar radial\nredistribution were not so strong as to reverse or undo\nthis trend.\nWe showed that this “simple” behavior indeed occurs\nin the FIRE cosmological simulations. We reiterate ar-\nguments as to why this behavior should emerge, as pre-\nvious works articulated (for example, Ho et al. 2015; Ma\net al. 2017a; Bellardini et al. 2021).\nFirst, to source a metallicity radial gradient (in princi-\nple), smaller radii must have higher star-formation effi-\nciency, that is, mass in (young) stars that source most of\nthe metals, relative to mass in gas, over which the met-\nals are diluted. In other words, log(Σ star(R)/Σgas(R))∝[Fe/H](R) (at least for some age stellar population)\nshould decline with R. Gas generally distributes within\na galaxy with a declining (surface) density profile, often\nas Σ gas∼exp(−R/R s), where Rsis the scale radius.\nFurthermore, star formation density generally exhibits a\nsuper-linear relation to total gas density (Schmidt 1959;\nKennicutt 1998), such that Σ SFR∝Σn\ngas, often with\nn≈1.4. As long as n >1, this first condition should be\nmet, modulo the complications arising from metal mix-\ning and metal outflows. Indeed, Bellardini et al. (2022)\nshowed that in these FIRE-2 MW-mass simulations at\nz≈0, log(Σ star(<1 Gyr , R)/Σgas(R))∝[Fe/H](R) to\ngood approximation.\nSecond, to sustain a radial gradient in gas, the galaxy\nmust have sufficiently well-ordered rotation (in a disk) to\nprevent radial mixing of metals via turbulence. As these\nFIRE-2 MW-mass galaxies evolved, their ISM developed\nthinner disks with more ordered rotation (McCluskey\net al. 2023), thereby reducing radial mixing, which is a\nkey reason why the radial gradients became steeper over\ntime. Indeed, for young stars in these FIRE-2 galaxies,\nBellardini et al. (2022) showed a significant correlation\nbetween the strength of the radial metallicity gradients\nand the “diskiness” of the galaxy, defined by vϕ/σv.\nMore broadly, Ma et al. (2017a) analyzed the FIRE-1\nsimulations across a broad range of galaxy masses and\nredshifts and found that only galaxies with vϕ/σv≳1\nshowed significant (negative) radial gradients.\nThis idea of higher star formation efficiency at smaller\nradii in conjunction with more ordered rotation lead-\ning to a strengthening of radial gradients agrees with\nKhoperskov et al. (2023), which used controlled (non-\ncosmological) N-body baryonic simulations of disk\ngalaxies to show that their galaxy that experienced en-\nrichment over time developed a stronger radial gradient\nthan their (otherwise identical) galaxy that did not.\nWe emphasize that this steepening of radial metallic-\nity gradients in the FIRE cosmological simulations oc-\ncurs even though they (on average) experienced “inside-\nout” radial growth in stars, that is, higher fractional\ngrowth rate in stars at larger radii at later times. This\ncontrasts with arguments, such as in Minchev et al.\n(2018), that inside-out radial growth should lead to ISM\nradial gradients becoming shallower over time.\nOf course, we presented average/typical trends across\n11 MW-mass FIRE-2 galaxies. Individual galaxies, in-\ncluding the MW, could exhibit atypical behavior, and\nthe evolution of quantities like radial gradients can be\nnon-monotonic, especially from events like galaxy merg-\ners (for example, Ma et al. 2017a; Buck et al. 2023).\nMore measurements of stellar populations across the\nMW, with increased accuracy in determining stellar agesSpatial Variations of Metals in FIRE 19\nand birth radii, are necessary to address this question for\nthe MW’s history specifically. Nonetheless, we reiterate\nthat the trends in the FIRE-2 cosmological simulations\nare similar to the trends observed for the MW, which\nprovides evidence that they do reflect, to a significant\ndegree, the history of the MW.\n7.ACKNOWLEDGEMENTS\nWe thank Fiona McCluskey for detailed comments.\nRG, AW, and MB received support from: NSF via\nCAREER award AST-2045928 and grant AST-2107772;\nNASA ATP grant 80NSSC20K0513; HST grants AR-\n15809, GO-15902, GO-16273 from STScI.\nWe ran simulations using: XSEDE, supported by\nNSF grant ACI-1548562; Blue Waters, supported by theNSF; Frontera allocations AST21010 and AST20016,\nsupported by the NSF and TACC; Pleiades, via the\nNASA HEC program through the NAS Division at Ames\nResearch Center.\nThe data in these figures, and the Python code that\nwe used to generate them, are available at https://\ngithub.com/rlgraf/Graf-et-al.-2024.git. 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Figure A.1 shows\nthe vertical gradients, ∂[Fe/H]/∂|Z|, for mono-age stel-\nlar populations versus their age, both today and using\ntheir coordinates at birth, at three radii ( ±0.5 kpc):\nR= 4 kpc, R= 8 kpc, and R= 12 kpc. As in\nFigure 5 (left), we measure the vertical gradient across\n|Z|<1 kpc. For comparison, Figure A.1 shows the ver-\ntical gradient of all stars at that R, denoted with a star,\nthough in some cases the value extends well below the\nbottom axis, so we show an arrow. The vertical gradi-\nent of mono-age stars is fairly flat at all ages, at least\ncompared with the gradient for all stars. This is because\nthe vertical height of the disk is largely regulated by tur-\nbulence (the size of the largest turbulent Jeans length),\nand this same turbulence efficiently mixes metals verti-\ncally in the ISM, so stars forming at a given time reflect\nthis weak vertical gradient. This turbulent mixing is, in\npart, the same reason these galaxies do not have strong\nradial gradients for old stars.\nThat said, these FIRE-2 galaxies do show significant\nvertical gradients in the inner galaxy ( R≲4 kpc) for\nyoung stars ( ≲6 Gyr), as Bellardini et al. (2022) (their\nFigure 8) already noted for stars at the time of forma-\ntion. As Bellardini et al. (2022) argued, likely this re-\nflects the higher density and efficiency of star formation\nin the inner galaxy, such that in this regime the enrich-\nment timescale can be shorter than the mixing timescale.\nFurthermore, Figure A.1 shows that, unlike vertical gra-\ndients for all stars in Figure 5, these mono-age gradients\ntend to be steeper today than at formation. Vertical\nheating alone would tend to drive this change in the op-\nposite direction, so likely this arises from radial redistri-\nbution, potentially combined with disk flaring: higher-\nmetallicity stars from the inner galaxy redistribute out-\nward, enriching the galaxy at lower |Z|, and lower-\nmetallicity stars from outward redistribute inward, low-\nering the metallicity at higher |Z|.\nIn summary, Figure A.1 shows that the vertical gra-\ndient for mono-age stars is always significantly smaller\nthan that for all stars, so the vertical gradient for all\nstars primarily reflects disk vertical settling over time,\nas opposed to a vertical gradient in star-forming gas.\nThis is the opposite behavior to the radial gradient, as\nin Figure 1, for which the radial gradient can be strongfor star-forming gas, and thus for mono-age stars, but it\nis nearly flat when measuring all stars today.\nB.SPATIAL VARIATIONS OF DIFFERENT\nELEMENTS\nThroughout we used [Fe/H] as our representative mea-\nsure of metallicity, as is common observationally. How-\never, as we discussed in Section 2.2, FIRE-2 tracks 11 el-\nements. Here we present results for [C/H] and [Mg/H] as\nwell. These elements trace different stellar nucleosynthe-\nsis channels. In the FIRE-2 model, [Fe/H] is sourced pri-\nmarily from white-dwarf (Ia) supernovae (though with\na significant contribution from core-collapse supernovae\nas well), [C/H] primarily through stellar winds, and\n[Mg/H] primarily through core-collapse supernovae.\nFigure B.1 (top) shows the radial gradient,\n∂[X/H]/∂R, versus stellar age for [Fe/H], [C/H], and\n[Mg/H], both today and at birth, using the same selec-\ntion criteria as in Figure 2. Radial gradients for [Fe/H]\nare systematically stronger than for [Mg/H], because\nof the significant population of older stars in the inner\ngalaxy, which leads to an enhanced rate of white-dwarf\nsupernovae, and thus Fe enrichment, there. [C/H] shows\nthe steepest gradients, given that its yields from stellar\nwinds increase with progenitor metallicity in the FIRE-\n2 model. Despite these differences, the prevailing trends\nin the radial gradients of all three elements are similar,\nboth today and at formation, in agreement with Bellar-\ndini et al. (2021) for gas and Bellardini et al. (2022) for\nstars at formation.\nFigure B.1 (middle) compares the trends for azimuthal\nvariations, σ[X/H], versus stellar age, following the meth-\nods of Figure 7. As expected, given our results in Sec-\ntion 5.3, the element with the weakest radial gradient,\n[Mg/H], also has the lowest azimuthal scatter, whereas\nthe element with the strongest gradient, [C/H], has the\nhighest azimuthal scatter. The trends for the overall\nscatter across R= 0−20 kpc reflect those of the ra-\ndial gradient above. Again, the trends in azimuthal and\ntotal variations are similar across these elements.\nFinally, Figure B.1 (bottom) shows the vertical gra-\ndient, ∂[X/H]/∂|Z|across |Z|<1 kpc, of all stars re-\ngardless of age, versus R, for [Fe/H], [C/H], and [Mg/H],\nusing the same selection criteria as in Figure 5. Similar\nto the radial gradients for these elements, [Mg/H] has a\nshallower vertical gradient than [Fe/H], and [C/H] has\na steeper vertical gradient than [Fe/H], both today and\nusing coordinates at formation. But again, the trendsSpatial Variations of Metals in FIRE 23\n0 2 4 6 8 10 120.08\n0.04\n0.000.04[Fe/H]/Z [dex kpc1]\nRadius = 4 kpctoday\nformation\n0 2 4 6 8 10 12\nAge [Gyr]Radius = 8 kpc\n0 2 4 6 8 10 12Radius = 12 kpc\nFigure A.1. Vertical gradient of [Fe/H], ∂[Fe/H]/∂|Z|, of stars across |Z|<1 kpc, versus age, for stars both today and at\nformation (using birth |Z|andR), at different R(±0.5 kpc): 4 kpc (inner disk), 8 kpc (Solar annulus), and 12 kpc (outer disk).\nEach line and shaded region show the average and 1 σscatter across the 11 galaxies. The vertical gradient for mono-age stars\ntends to be steeper for younger stars, both today and at formation, and the vertical gradient is generally steeper today than\nat formation, likely from the effects radial redistribution. However, there is little-to-no vertical gradient for older stars or at\nlarger R. Most importantly, for comparison, the stars show the vertical gradient for all stars regardless of age from Figure 4,\nand arrows indicate values that extend below plot, which are: −0.32 dex kpc−1(today) and −0.46 dex kpc−1(formation) at\n4 kpc, and −0.15 dex kpc−1at formation at 8 kpc. Thus, these vertical gradients for mono-age stars are always much smaller\nthan the total gradient for all stars. The vertical gradient of all stars today primarily reflects the “upside-down” settling of the\ndisk over time, and not the gradient of mono-age stars.\nwith R, and the differences between today and forma-\ntion, are similar.\nOverall, while the exact slope of the gradients or the\nstrength of the azimuthal variations do depend on the\nmeasured element, the overall trends, both today and at\nbirth are similar, so we conclude that all of the trends\nthat we explored for [Fe/H] hold true for other elemental\nabundances in FIRE-2.24 Graf et al.\n0 2 4 6 8 10 12\nAge [Gyr]0.04\n0.02\n0.00[X/H]/R [dex kpc1]\n today\nformation\n [Fe/H]\n[C/H]\n[Mg/H]\n0 2 4 6 8 10 12\nAge [Gyr]0.00.10.20.3 today R = 8 kpc\nformation R = 8 kpc\nR = 0-20 kpc[Fe/H]\n[C/H]\n[Mg/H]\n0 2 4 6 810 12 14\nRadius [kpc]0.6\n0.4\n0.2\n0.0[X/H]/|Z| [dex kpc1]\ntoday\nformation\n [Fe/H]\n[C/H]\n[Mg/H][X/H]\nFigure B.1. Comparing the result for [Fe/H], [C/H], and\n[Mg/H], for the radial gradient, ∂[X/H]/∂R (top) and the\nazimuthal/overall scatter, σ[X/H], (middle), versus age, and\nthe vertical gradient, ∂[X/H]/∂|Z|, versus cylindrical radius,\nR, (bottom), of stars both today and at formation (using\ntheir birth radii). We use the same methods as in Fig-\nures 2, 7, and 5. Radial and vertical gradients are some-\nwhat steeper and the scatter is somewhat higher in [Fe/H]\nthan in [Mg/H], given the significant contribution to [Fe/H]\nfrom white-dwarf (Ia) supernovae in the inner galaxy from\nolder stars. The gradients are somewhat steeper and the\nscatter is somewhat higher for [C/H], given its significant\ncontribution from (progenitor-metallicity-dependent) stellar\nwinds in FIRE-2. However, the radial gradients, vertical gra-\ndients, and azimuthal scatter are all qualitatively similar for\nall 3 elements, both today and at birth."    },    {        "title": "2402.15663v1.Leveraging_ChatGPT_in_Pharmacovigilance_Event_Extraction__An_Empirical_Study.pdf",        "content": "Leveraging ChatGPT in Pharmacovigilance Event Extraction: An\nEmpirical Study\nZhaoyue Sun1, Gabriele Pergola1, Byron C. Wallace2andYulan He1,3,4\n1Department of Computer Science, University of Warwick\n2Khoury College of Computer Sciences, Northeastern University\n3Department of Informatics, King’s College London\n4The Alan Turing Institute\n{Zhaoyue.Sun, Gabriele.Pergola.1}@warwick.ac.uk\nb.wallace@northeastern.edu, yulan.he@kcl.ac.uk\nAbstract\nWith the advent of large language models\n(LLMs), there has been growing interest in\nexploring their potential for medical applica-\ntions. This research aims to investigate the\nability of LLMs, specifically ChatGPT, in the\ncontext of pharmacovigilance event extraction,\nof which the main goal is to identify and extract\nadverse events or potential therapeutic events\nfrom textual medical sources. We conduct ex-\ntensive experiments to assess the performance\nof ChatGPT in the pharmacovigilance event ex-\ntraction task, employing various prompts and\ndemonstration selection strategies. The find-\nings demonstrate that while ChatGPT demon-\nstrates reasonable performance with appropri-\nate demonstration selection strategies, it still\nfalls short compared to fully fine-tuned small\nmodels. Additionally, we explore the poten-\ntial of leveraging ChatGPT for data augmenta-\ntion. However, our investigation reveals that the\ninclusion of synthesized data into fine-tuning\nmay lead to a decrease in performance, possibly\nattributed to noise in the ChatGPT-generated\nlabels. To mitigate this, we explore differ-\nent filtering strategies and find that, with the\nproper approach, more stable performance can\nbe achieved, although constant improvement\nremains elusive1.\n1 Introduction\nPharmacovigilance stands as a pivotal discipline in\nhealthcare that encompasses a range of processes:\nidentifying, evaluating, understanding, and prevent-\ning adverse effects and other medicine-related is-\nsues (World Health Organization, 2004). Within\nthis domain, pharmacovigilance event extraction\nemerges as a crucial practice aimed at extracting\nstructured medication-related event data from med-\nical text sources, serving as valuable inputs for\nautomatic drug safety signal detection. With the\n1Related code for this paper is available at\ngithub.com/ZhaoyueSun/phee-with-chatgpt.rapid expansion of electronic health records (EHR),\nmedical case reports, and other textual resources,\nthe need for efficient and accurate pharmacovig-\nilance event extraction has become increasingly\npressing.\nStudies have been conducted to extract\npharmacovigilance-related information from text\ndata. However, previous research mainly focused\non simple tasks such as entity extraction (Wun-\nnava et al., 2017) or binary relation extraction (Gu-\nrulingappa et al., 2012; El-allaly et al., 2021). Re-\ncently, Sun et al. (2022) introduced a novel dataset\nfor pharmacovigilance event extraction, which in-\ncludes hierarchical annotations of adverse events\nand potential therapeutic events, capturing informa-\ntion about the subject, treatment, and effect. Addi-\ntionally, they investigate the performance of vari-\nous models, including sequence labelling and QA-\nbased approaches, for this task, providing a foun-\ndation for further advancements in extracting struc-\ntured event data for pharmacovigilance research.\nThe rise of large language models (LLMs), espe-\ncially ChatGPT (OpenAI, 2022), has sparked con-\nsiderable interest in their potential applications in\nthe medical field (Lu et al., 2023; Zhu et al., 2023;\nAgrawal et al., 2022; Kung et al., 2023). In this\nstudy, our focus is on exploring different ways to\nincorporate ChatGPT into the pharmacovigilance\nevent extraction task. Figure 1(a) presents an ex-\nample of this task.\nWe first explore various strategies for prompt-\ning and demonstration selection to assess Chat-\nGPT’s performance in zero-shot and few-shot sce-\nnarios, comparing it with smaller fine-tuned mod-\nels. Our findings indicate that, with suitable demon-\nstration selecting strategies, ChatGPT performs rea-\nsonably well but still falls short of the performance\nachieved by fully fine-tuned smaller models, as\ndemonstrated in Figure 1(b).\nFurthermore, we delve into the utilization of\nLLMs for data augmentation, which is suggestedarXiv:2402.15663v1  [cs.CL]  24 Feb 2024BACKGROUND :    from    in    under  therapy is rare .\nBACKGROUND : in a is rare . \n  by  therapy in      is uncommon .Ov ar ian cancer ar ising an endometr iotic cyst a postmenop ausal w oman tamo xifen\nOv ar ian cancer ar ising fr om an endometr iotic cyst postmenop ausal w oman under tamo xifen therapy\nOst eonecr osis caused dexamethasone a y oung male with asthmaA D E . e f f e c t A D E . t r i g g e r A D E . s u b j e c t _ d i s o r d e r\nA D E . t r e a t m e n t _ d i s o r d e rA D E . s u b j e c t\nA D E . a g e A D E . g e n d e rA D E . d r u g\nA D E . t r e a t m e n t\nA D E . e f f e c t A D E . s u b j e c t\nA D E . s u b j e c t _ d i s o r d e r A D E . g e n d e r A D E . d r u g\nA D E . t r e a t m e n t\nA D E . e f f e c t A D E . t r i g g e r A D E . d r u g\nA D E . t r e a t m e n tA D E . s u b j e c t\nA D E . a g e A D E . g e n d e r A D E . s u b j e c t _ d i s o r d e r\nA D E . t r e a t m e n t _ d i s o r d e r(a) Example with h uman annotation.\n(c) ChatGPT -synthesized case using the example in (a) for demonstration.(b) Example with the prediction of ChatGPT (BM25).Figure 1: Snippets from biomedical documents: a comparison of human annotations, ChatGPT predictions, and a\nChatGPT-synthesized case.\nto be beneficial in improving small model’s per-\nformance in recent work (Pergola et al., 2021; Lin\net al., 2022; Liu et al., 2022; Zhu et al., 2022; Tan\net al., 2023; Whitehouse et al., 2023). We em-\nploy ChatGPT to generate sentences structurally\nresembling demonstration samples, as illustrated in\nFigure 1(c). However, our experiments show that\nsimply combining these generated samples with\nthe training set leads to an overall performance\ndecrease. Considering the possible influence of\nsynthesized data noises, we further introduce a fil-\ntering strategy for augmented data quality control,\nwhich, though still does not outperform the fully\nfinetuned model, reduces the performance drop and\nbrings it closer to the levels achieved with the origi-\nnal training data, while reducing the variance. This\nindicates enhanced stability when working with\nample high-quality data.\nIn summary, we compare various regimes of\nleveraging ChatGPT to assist in pharmacovigilance\nevent extraction, providing practitioners with mean-\ningful references for choosing suitable strategies.\nAdditionally, we conduct a fine-grained qualita-\ntive analysis of ChatGPT synthesized instances and\ndata augmentation and explore reasons for their\nlack of positive effect, laying the groundwork for\nimprovements in subsequent work.\n2 Prompt-based Learning with ChatGPT\n2.1 Zero-shot Prompting\nFor zero-shot prompting, a manually designed in-\nstruction is employed to query ChatGPT for an-\nswers. In this study, we devise four approaches to\nprompt the model: a) Schema: providing instruc-\ntions alongside enumeration of event types and ar-\ngument types; b) Explanation: providing instruc-\ntions with a detailed explanation of the schema;\nc)Code: formulating instructions and output for-mat using a combination of text descriptions and\ncode snippets; d) Pipeline: querying the model in\na pipeline manner, which first prompts for the main\narguments and then follows up with type-related\nquestions for each sub-argument. Details of the\nprompts are presented in Appendix G.\n2.2 Few-shot In-context Learning\nFor few-shot in-context learning, several demon-\nstrations are provided together with the instruction.\nThe selection of different demonstration examples\ncan yield varying results. We explore different\nstrategies for choosing in-context examples based\non a given test instance, including: a) Random:\nrandomly selecting examples from the training set;\nb)SBERT: choosing examples based on the simi-\nlarity of their dense representations to the test sen-\ntence. We utilize Sentence-BERT (Reimers and\nGurevych, 2019) to obtain the sentence representa-\ntions; c) BM25: selecting examples based on the\nsimilarity of their lexical representations to the test\nsentence. We employ BM25 (Trotman et al., 2014)\nas the ranking function; d) TreeKernel: choosing\nexamples based on the structural similarity to the\ntest sentence. We implement the tree kernel by\ncomputing the Jaccard similarity of the subpaths\nwithin the dependency trees of the sentences.\n3 ChatGPT as Data Synthesizer\nWe explore the potential of leveraging ChatGPT for\ndata augmentation purposes. To achieve this, we\nincorporate an example from the training set, along\nwith its annotated events, as input to ChatGPT. We\nthen prompt ChatGPT to generate a sentence that\nexhibits a similar event structure to the given sen-\ntence and extract the events from the generated\nsentence. However, based on our initial study, we\nobserved that ChatGPT tends to miss specific men-\ntions of drugs or excessively use certain drugs, suchas ‘ibuprofen ’. We address this issue by restricting\nthe inclusion of drug names and their correspond-\ning effects sampled from the training data in gen-\nerated sentences. Details of the prompt for data\nsynthesizing are shown in Appendix G.\nRecognizing that directly incorporating gener-\nated samples into the training data can lead to per-\nformance decline, possibly due to issues related to\ndata quality, we have introduced filtering strategies.\nThe main rationale behind the filtering is to retain\nannotations for which the model exhibits a certain\nlevel of confidence, based on the assumption that\na finetuned model possesses some discriminatory\nability regarding the quality of annotations, and in-\ncorrect annotations may result in lower confidence\nscores from the model for the annotation sequence.\nSpecifically, we introduce: a) Train Filter: Fil-\ntering the training set with sgold<mean (sgold),\nwhere sgoldis the average token probability given\nby the fine-tuned model on the ground-truth event\nlabel sequence. This means we filter out train-\ning instances for which the annotation sequence\nhas less model certainty than the average level;\nb)Augment Filter: Filtering augmented data with\nz(sgold)<0orz(sgold)< z(spred), where spred\nis the average token probability for predicted event\nlabel sequences. z(s) = (s−mean (sv))/std(sv),\nandsvrepresents the values of sin the validation\nset. In this case, we filter out samples generated\nby ChatGPT if their generated annotation sequence\nprobability, as calculated by a fine-tuned model,\nfalls below the average level or is less certain than\nthe sequence predicted by the fine-tuned model\nitself. Considering models potentially assigning\nhigh scores to the sequences they predict, we use\nz-score instead of direct predictive probabilities for\ncomparison.\nWith these filtering rules, we compare the\nmodel’s performance on several data settings, in-\ncluding: training data (Tr.), training data combined\nwith augmented data (Tr.+Aug.), filtered training\ndata (Tr. Fil.), training data with filtered augmented\ndata (Tr.+Aug. Fil.) and filtered training data with\nfiltered augmented data (Tr. Fil.+Aug. Fil.).\n4 Experiments\n4.1 Experimental Settings\nDataset We conducted experiments on the PHEE\ndataset (Sun et al., 2022), an English event ex-\ntraction dataset sourced from publicly accessible\nmedical reports, encompassing annotations for twoevent categories: adverse events andpotential ther-\napeutic events . The annotations follow a hierar-\nchical structure, with main arguments providing\ninformation on the subject ,treatment , and effect ,\nwhile sub-arguments offer more detailed informa-\ntion pertaining to the main arguments. However,\nduring our analysis, we observed that certain argu-\nment types showed low consistency. To address\nthis issue, we performed automatic and manual re-\nvisions on the subject.disorder ,time_elapsed , and\nduration arguments. For further details, please re-\nfer to Appendix A. The dataset contains around\n5k sentences and we split the data into training,\nvalidation, and test sets by 6/2/2.\nBaselines We compare ChatGPT’s performance\nwith the best-performing Generative QA model\nproposed in (Sun et al., 2022) and two widely\nadopted seq-to-seq models: 1) UIE (Lu et al.,\n2022), a model that is specifically pre-trained on\nstructured information extraction data; and 2) Flan-\nT5(Chung et al., 2022), a model trained on a di-\nverse range of tasks using instructional prompts.\nFor more information, see Appendix B.\nEvaluation We follow Sun et al. (2022) to eval-\nuate both exact matching F1 score (EM_F1) and\ntoken-level matching F1 score (Token_F1) for ar-\ngument extraction. During our preliminary experi-\nments, we observed that ChatGPT struggled to gen-\nerate reasonable results for trigger extraction. Con-\nsidering that even humans find trigger identification\nchallenging, and that it doesn’t significantly con-\ntribute to understanding pharmacovigilance events,\nwe did not query ChatGPT for triggers, but we still\nask ChatGPT to generate the event structure, en-\nabling the differentiation of multiple events. For\nthe trigger extraction results obtained from finetun-\ning models, please check Appendix D.\nWe perform 5-fold cross-validation for fine-\ntuning and data augmentation experiments, while\nlimiting ChatGPT-based zero-shot and few-shot\nlearning to a single split due to cost-related rea-\nsons. For more details about the experimental setup,\nplease refer to Appendix C.\n4.2 Results and Discussion\nChatGPT with Different Prompting Strategies\nTable 1 presents the argument extraction results\nfor ChatGPT using different zero-shot prompting\nstrategies. Providing only instructions yields un-\nsatisfactory performance, but including a detailedMain-arguments Sub-arguments\nEM_F1 Token_F1 EM_F1 Token_F1\nSchema 30.31 47.41 22.50 26.51\nCode 25.94 40.42 25.67 29.70\nExplanation 34.80 52.99 36.70 39.33\nPipeline 32.57 49.41 27.79 33.80\nTable 1: Argument extraction results for ChatGPT zero-\nshot prompting with different prompting strategies.\nexplanation of the event schema leads to notice-\nable improvement, highlighting the importance of\ncomprehensive guidance. Further qualitative ex-\namination reveals that end-to-end generation tends\nto miss arguments, whereas the pipeline approach\ntends to generate numerous false positive cases. It\nis surprising that the model performs poorly on\nseemingly simple arguments such as ‘ population ’,\n‘route ’, and ‘ age’. While providing explanations\nimproves the performance of some arguments (e.g.,\n‘route ’ and ‘ age’), all approaches still struggle with\n‘population ’ extraction. This difficulty may due to\nthe gap between the lexical meaning of the label\n‘population ’ and the semantic meaning of the ar-\ngument. Additionally, while the pipeline method\nhas advantages in extracting certain argument types\n(e.g., ‘ gender ’ and ‘ frequency ’), the inference time\nis proportional to the number of argument types,\nmaking it approximately 10 times longer than the\nend-to-end methods.\nTable 2 displays the few-shot argument ex-\ntraction results for ChatGPT using various in-\ncontext selection strategies. Dense representation-\nbased demonstration retrieval with SBERT does\nnot demonstrate superiority in this task, possibly\ndue to limited domain knowledge captured by the\npre-trained sentence representation model. Incor-\nporating structured information improves perfor-\nmance, while the simplest lexical-based retrieval\nstrategy shows the most noticeable performance\ngains. Upon examining the samples retrieved by\ndifferent example selection strategies, we observed\nthat SBERT and TreeKernel tend to retrieve struc-\nturally similar sentences, while BM25 is more in-\nclined to retrieve sentences containing matching\nentities such as drugs (since entities usually serve\nas keywords in a sentence). This observation sug-\ngests that the superior performance of BM25 in\nargument extraction can be attributed to the fact\nthat this task is more sensitive to entities. When\nmore examples with similar entities are covered,\nChatGPT learns more effectively from them.Main-arguments Sub-arguments\nEM_F1 Token_F1 EM_F1 Token_F1\nrandom 58.31 72.74 60.32 63.74\nSBERT 56.90 71.65 62.29 64.25\nTreeKernel 60.54 73.68 63.36 64.69\nBM25 60.39 76.15 67.35 68.67\nTable 2: Argument extraction results for ChatGPT few-\nshot prompting with different in-context demonstration\nselection strategies (results for 5-shot are reported).\nFinetuning Models vs. ChatGPT Table 3 il-\nlustrates the argument extraction results for differ-\nent methods. The findings indicate that there is\nminimal variation among the fine-tuning methods.\nSpecifically, the Flan-T5 model, despite not being\nexplicitly pre-trained for the information extrac-\ntion task, demonstrates slightly better performance\nthan the UIE model. In contrast, ChatGPT without\ndemonstrations exhibits poor performance. How-\never, when demonstrations are provided, ChatGPT\nshows improved results, although there remains a\nnoticeable gap compared to the fine-tuning meth-\nods. For a detailed breakdown of the results for\neach argument type, refer to Appendix E.\nMain-arguments Sub-arguments\nEM_F1 Token_F1 EM_F1 Token_F1\nFully supervised\nGenerative QA 68.85 81.63 77.33 78.83\nUIE(Large) 69.46±.49 81.20±.40 77.12±1.378.83±1.4\nFlan-T5(Large) 70.78±1.482.34±1.577.63±.1.679.52±1.3\nZero-Shot\nChatGPT(Exp.) 34.80 52.99 36.70 39.33\nFew-Shot\nChatGPT(BM25) 60.39 76.15 67.35 68.67\nTable 3: Argument extraction results for various meth-\nods. For fine-tuning methods, we report the mean ±std\nvalue of 5-fold cross-validation. For ChatGPT(BM25),\nwe provide the results for 5-shot. We obtain Generative\nQA results directly from the original paper.\nData Augmentation with ChatGPT Table 4\npresents the performance of Flan-T5 when augmen-\ntation and various filtering strategies are employed.\nIt can be seen that simply extending the training\ndata with ChatGPT-synthesized cases could lead to\nan obvious performance drop. In contrast, with the\nfiltered training set, although retained only 65% of\ntraining data, surpasses results obtained from over\n5,000 augmented instances, which may indicate the\ncritical role of data quality in pharmacovigilance\nevent extraction. Furthermore, training with filtered\naugmented data effectively restores performanceto the original level. In particular, training with\nboth filtered training data and filtered augmented\ndata displays only slight deviations from training\nwith the original data, yet it reduces variance. The\np-values for the variance difference significance,\nassessed through the F-test, are 0.29 and 0.39 for\nEM_F1 and Token_F1, respectively.\nEM_F1 Token_F1 Avg. Cases\nTr. 74.45±1.46 81.30±1.27 2897\nTr.+Aug. 73.07±0.92 79.93±1.51 5446\nTr. Fil. 73.92±.1.28 80.71±1.60 1873\nTr.+Aug. Fil. 74.26±.1.27 80.98±2.06 3702\nTr. Fil.+Aug. Fil. 74.19±1.09 81.05±1.09 2678\nTable 4: Argument (including main and sub-arguments)\nextraction results for Flan-T5 (Large) with augmentation\nand filtering strategies. The Avg. Cases column displays\nthe average number of training cases over 5 folds.\nWe conduct a qualitative analysis to explore\npossible reasons for the performance degradation\ncaused by data augmentation. Through sampling\nanalysis of examples where the fine-tuned model\nmade correct predictions but the augmented model\nfailed, we find that for main arguments, the er-\nrors mainly stemmed from inconsistency in text\nspan boundaries, while failures due to semantic\nmisunderstandings are relatively rare and primarily\noccurred in the misidentification of abbreviations,\nsuch as the model incorrectly recognizing an ab-\nbreviation for a disease as a medication. As for\nsub-arguments, semantic misunderstandings and\nmissing arguments are the main reasons the aug-\nmented model makes mistakes. Additionally, some\nerrors resulted from inconsistent boundaries and\nannotation noise, which may influence evaluation\nscores but not necessarily harm model utility.\nCases where the arguments are missing some-\ntimes show a pattern, e.g., a ‘ population ’ argument\ntends to be missed when it’s in an expression like\n‘xx cases’, and a ‘ route ’ argument may be missed\nwhen overlapping with the ‘ dosage ’ arguments.\nHowever, in many cases, there’s no obvious reason\nwhy the argument is not extracted. For semantic\nissues, we observe that the ‘ subject.disorder ’ is eas-\nily confused with ‘ treatment.disorder ’, and ‘ time\nelapsed ’ is easily confused with ‘ duration ’. Ad-\nditionally, some ‘ age’ expressions (e.g., ‘adults’)\ntend to be predicted as ‘ gender ’, and some ‘ dura-\ntion’ expressions when describing a long term may\nbe identified as ‘ frequency ’.\nFurthermore, we conduct an unconditional sam-pling of instances synthesized by ChatGPT and\nanalyze the mislabels. The analysis indicates that,\nalthough category labelling errors are not common\nin ChatGPT-synthesized samples, there are still\ninstances of mislabelling for some relatively chal-\nlenging arguments, such as ‘ subject.disorder ’ and\n‘treatment.disorder ’. Moreover, we observe that,\nin comparison to the types and quantities of argu-\nments present in the given templates, ChatGPT-\nsynthesized examples have less coverage for rare\narguments. This might be one of the reasons con-\ntributing to performance degradation in argument\nextraction when using synthetic data for augmenta-\ntion as well.\nAccording to the qualitative analysis, we suspect\nthe model may struggle to capture the intricate\nannotation rules when only one example is used as\na demonstration. For future work, providing more\ndiverse examples when synthesizing data may be a\nworthwhile direction to explore. For more details\non qualitative analysis, please refer to Appendix F.\n5 Conclusion\nThis paper provides empirical practice in various\napproaches to leveraging ChatGPT for the pharma-\ncovigilance event extraction task. Overall, Chat-\nGPT exhibits impressive few-shot learning capabil-\nities in pharmacovigilance event extraction. Never-\ntheless, considering the sensitivity of the medical\nfield, fine-tuned models retain a clear edge in the\npresence of abundant data. In our experiments,\nthe introduction of ChatGPT-synthesized instances\nfor data augmentation does not improve the per-\nformance of small model fine-tuning. However,\nappropriate quality control may increase the stabil-\nity of performance. Qualitative analysis indicates\nthat errors may arise in ChatGPT-synthesized data\nwhen distinguishing semantically complex argu-\nments, and the coverage of rare arguments is insuffi-\ncient. We emphasize the structural complexity and\nfine granularity of arguments in event extraction,\nwhich may pose challenges in generating synthetic\ndata. Future work can conduct more in-depth data\naugmentation research addressing these aspects.\nLimitations\nIn our preliminary study, we encountered limita-\ntions in exploring alternative open-source LLMs,\nsuch as LLaMA 30B (Touvron et al., 2023) and\nFlan-T5 XXL (Chung et al., 2022), for zero-\nshot/few-shot prompting. These models exhib-ited significant differences in generation quality\ncompared to ChatGPT, and their slow inference\nspeeds hindered a comprehensive evaluation. De-\nspite these limitations, we highlight the importance\nof further research to investigate the potential of\nleveraging different open-source LLMs.\nSecondly, while Chain-of-Thought (CoT) rea-\nsoning has demonstrated enhanced performance\nin few-shot learning for biomedical NLP tasks,\nwe have not yet introduced it into our evaluation.\nThis omission is attributed to the intricate nature\nof constructing reasoning steps for each argument\nwithin the fine-grained event extraction task. In our\npreliminary experiments, simply asking ChatGPT\nto explain its extraction rationale didn’t enhance\nperformance; instead, it complicated the accurate\ncollection of extraction results. In our current ex-\nperiment, we explored ways to retrieve the most\nrelevant examples from the training set, but we\nlacked annotated reasoning steps for all samples,\nhindering a comprehensive evaluation of the CoT\nmethod in this context. Given these limitations, we\nleave the exploration of CoT to future work.\nMoreover, our investigation focused solely on\nunsupervised methods for in-context demonstration\nselection. Future research could explore the incor-\nporation of annotations in the selection process,\nwhich may yield valuable insights and improve\nthe performance of ChatGPT in pharmacovigilance\nevent extraction.\nEthics Statement\nThe approaches outlined in this article focus solely\non extracting information from the textual level\nand do not suggest a direct causal relationship be-\ntween drugs and their effects. The causality assess-\nment of ADEs requires expert evaluation, and the\nmethodologies presented in this paper are intended\nas supplementary tools to accelerate the process.\nThis paper explores common errors inherent in\nthe model’s extraction, and users should be aware\nof the practical consequences associated with dif-\nferent error types. Caution is particularly advised\nwhen employing statistical inferences based on the\ntools proposed in this paper, as the model may\nsometimes miss an argument (such as failure to rec-\nognize a patient’s race in the case of certain place\nnames due to insufficient generalization). Addi-\ntionally, in many instances, the extracted sentences\nthemselves may not mention certain pieces of infor-\nmation. In comparison to free-text sources, struc-tured data such as EHRs may offer a more reliable\nbasis for conducting statistical inferences.\nFurthermore, it is worth noting that the use of\nChatGPT-synthesized data may alter the data dis-\ntribution. For example, we observe that ChatGPT\nis more likely to generate the most common drug-\nADE pairs. Although this is reasonable, including\na substantial number of such \"correct\" examples in\nthe training data may lead to model bias, causing\nit to overlook rare but significant side effects men-\ntioned in the text. Data synthesized by ChatGPT\nmay also introduce incorrect knowledge, while the\nimpact of this on event extraction may be limited\nbecause the given sentence constrains the extrac-\ntion results. However, caution is needed when ap-\nplying the methods described in the text to other\napplication domains, such as ADE generation.\nAcknowledgements\nThis work was supported in part by the UK Engi-\nneering and Physical Sciences Research Council\nthrough a Turing AI Fellowship (EP/V020579/1,\nEP/V020579/2) and the National Science Founda-\ntion (NSF) grant 1750978.\nReferences\nMonica Agrawal, Stefan Hegselmann, Hunter Lang,\nYoon Kim, and David Sontag. 2022. Large language\nmodels are few-shot clinical information extractors.\nInProceedings of the 2022 Conference on Empiri-\ncal Methods in Natural Language Processing , pages\n1998–2022.\nHyung Won Chung, Le Hou, Shayne Longpre, Bar-\nret Zoph, Yi Tay, William Fedus, Eric Li, Xuezhi\nWang, Mostafa Dehghani, Siddhartha Brahma, et al.\n2022. Scaling instruction-finetuned language models.\narXiv preprint arXiv:2210.11416 .\nEd-drissiya El-allaly, Mourad Sarrouti, Noureddine En-\nNahnahi, and Said Ouatik El Alaoui. 2021. Mttlade:\nA multi-task transfer learning-based method for ad-\nverse drug events extraction. Information Processing\n& Management , 58(3):102473.\nHarsha Gurulingappa, Abdul Mateen Rajput, Angus\nRoberts, Juliane Fluck, Martin Hofmann-Apitius, and\nLuca Toldo. 2012. Development of a benchmark\ncorpus to support the automatic extraction of drug-\nrelated adverse effects from medical case reports.\nJournal of biomedical informatics , 45(5):885–892.\nTiffany Kung, Morgan Cheatham, Arielle Medenilla,\nCzarina Sillos, Lorie Leon, Camille Elepaño, Maria\nMadriaga, Rimel Aggabao, Giezel Diaz-Candido,James Maningo, and Victor Tseng. 2023. 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Sentence-bert:\nSentence embeddings using siamese bert-networks.\narXiv preprint arXiv:1908.10084 .\nZhaoyue Sun, Jiazheng Li, Gabriele Pergola, Byron\nWallace, Bino John, Nigel Greene, Joseph Kim, and\nYulan He. 2022. PHEE: A dataset for pharmacovigi-\nlance event extraction from text. In Proceedings of\nthe 2022 Conference on Empirical Methods in Nat-\nural Language Processing , pages 5571–5587, Abu\nDhabi, United Arab Emirates. Association for Com-\nputational Linguistics.\nXingwei Tan, Gabriele Pergola, and Yulan He. 2023.\nEvent temporal relation extraction with Bayesian\ntranslational model. In Proceedings of the 17th Con-\nference of the European Chapter of the Association\nfor Computational Linguistics , pages 1125–1138,\nDubrovnik, Croatia. Association for Computational\nLinguistics.Hugo Touvron, Thibaut Lavril, Gautier Izacard, Xavier\nMartinet, Marie-Anne Lachaux, Timothée Lacroix,\nBaptiste Rozière, Naman Goyal, Eric Hambro,\nFaisal Azhar, et al. 2023. Llama: Open and effi-\ncient foundation language models. arXiv preprint\narXiv:2302.13971 .\nAndrew Trotman, Antti Puurula, and Blake Burgess.\n2014. Improvements to bm25 and language models\nexamined. In Proceedings of the 2014 Australasian\nDocument Computing Symposium , pages 58–65.\nChenxi Whitehouse, Monojit Choudhury, and Al-\nham Fikri Aji. 2023. Llm-powered data augmen-\ntation for enhanced crosslingual performance. arXiv\npreprint arXiv:2305.14288 .\nWorld Health Organization. 2004. Pharmacovigilance:\nensuring the safe use of medicines. Technical report,\nWorld Health Organization.\nSusmitha Wunnava, Xiao Qin, Tabassum Kakar, Vimig\nSocrates, Amber Wallace, and Elke Rundensteiner.\n2017. Towards transforming fda adverse event nar-\nratives into actionable structured data for improved\npharmacovigilance. In Proceedings of the Sympo-\nsium on Applied Computing , pages 777–782.\nLixing Zhu, Zheng Fang, Gabriele Pergola, Robert Proc-\nter, and Yulan He. 2022. Disentangled learning of\nstance and aspect topics for vaccine attitude detec-\ntion in social media. In Proceedings of the 2022\nConference of the North American Chapter of the\nAssociation for Computational Linguistics: Human\nLanguage Technologies , pages 1566–1580, Seattle,\nUnited States. Association for Computational Lin-\nguistics.\nLixing Zhu, Runcong Zhao, Gabriele Pergola, and Yu-\nlan He. 2023. Disentangling aspect and stance via\na Siamese autoencoder for aspect clustering of vac-\ncination opinions. In Findings of the Association\nfor Computational Linguistics: ACL 2023 , pages\n1827–1842, Toronto, Canada. Association for Com-\nputational Linguistics.A Data Annotation Revision Details\nIn the original dataset, we observed particularly\nlow levels of annotation inconsistency for ‘ sub-\nject.disorder ’, ‘time_elapsed ’, and ‘ duration ’ ar-\nguments, as illustrated by the examples provided\nin Table A1. To address this, we conducted an\nautomatic revision for the ‘ subject.disorder ’ an-\nnotation and hired annotators to manually cor-\nrect the ‘ time_elapsed ’ and ‘ duration ’ annota-\ntions. For ‘ subject.disorder ’ correction, if a ‘ treat-\nment.disorder ’ was present in the ‘ subject ’ argu-\nment but not annotated as ‘ subject.disorder ’, we\nadded it to the ‘ subject.disorder ’ annotation. For\n‘time_elapsed ’ and ‘ duration ’ correction, detailed\nguidelines were provided to the annotators to en-\nsure consistent annotations. We employed three\nannotators and informed them about the purpose\nof the data. The annotators are all PhD students\nwho volunteered for this task, receiving compensa-\ntion through the university’s payment platform for\ntheir annotation work. Two of the annotators have a\nbackground in computer science, and one annotator\nhas a medical background. Two of the annotators\nare non-native English speakers, and one is a native\nEnglish speaker. Following the approach used in\n(Sun et al., 2022), we evaluated the consistency\namong the annotators using the EM_F1 score. The\naveraged EM_F1 scores for both ‘ time_elapsed ’\nand ‘ duration ’ annotations were 75.3%.\nB Details of Baseline Implementation\nFor the implementation of seq-to-seq baselines, we\nformulate pharmacovigilance event extraction as a\nconditional text generation task. Concretely, given\na sentence xand additional auxiliary information\na, the model is trained to generate a linearized\nsequence yrepresenting the output event structure.\nFor UIE, we refer to the methodology outlined\nin the original paper by utilizing the Structural\nSchema Instructor (SSI) as the auxiliary informa-\ntionaand constructing the target sequence ywith\nStructural Extraction Language (SEL). However,\nspecial tokens used in SSI and SEL in UIE can\nresult in a decrease in performance if no external\npre-training is applied. Thus for Flan-T5, we substi-\ntute the SSI with a concise instruction accompanied\nby a natural language enumeration of the schema.\nAdditionally, for the target sequence construction,\nwe utilize square brackets as the structural symbol.\nFor both UIE and FLan-T5, we use the large\nmodel which comprises 770M parameters. Train-ing an epoch typically takes around 2 minutes, and\nvalidation, which utilizes beam search, requires\napproximately 10 minutes with an NVIDIA A100\n(80G) GPU. The fine-tuning models generally con-\nverge within 10 epochs.\nC Details of Experimental Setup\nC.1 Few-shot Prompting Settings\nIn the context of event extraction, each shot in-\ncludes one example for each event type. In Sec-\ntion 4, we report the 5-shot results for in-context\ndemonstration selection strategies, which entails\nproviding a total of 10 examples for each instance.\nThe selection of the number of demonstration cases\nwas based on ChatGPT’s input length capacity.\nWe further evaluate the argument extraction per-\nformance of several in-context demonstration selec-\ntion strategies when different numbers of demon-\nstration examples are selected in Figure A1. No-\ntably, when the first example is added, all methods\nexperience a significant performance boost. How-\never, as the number of examples increases, the\nperformance gains become more minimal. Five-\nshot prompting (involving 5 ADE examples and\n5 PTE examples) has approached the maximum\ninput limit that ChatGPT can handle. Neverthe-\nless, we reasonably suspect that further increasing\nthe number of examples would not get significant\nperformance improvements.\n01020304050607080\nrandom SBERT TreeKernel BM25Performance with Different Demonstration Sizes\n1350\nFigure A1: Token_F1 scores for argument extraction\nwith different demonstration sizes. The blue line repre-\nsents the performance of zero-shot prompting with the\nexplained schema.\nC.2 Hyperparameter Details\nThe order and occurrence of events and arguments\nin the generated sequence can impact the learn-\ning effectiveness of the model. To tackle this, Lu\net al. (2022) introduced the ‘Rejection Mechanism’,\nwhich generates a null span when a specific type\nof event or argument is absent in the sentence. In\nour preliminary experiments, we determined thatSubject.Disorder: We report two patients with acne vulgaris with a fourth type of minocycline-induced cutaneous\npigmentation.\nWe observed that when a disorder span is included in a ‘subject’ argument and also as ‘treatment.disorder’, annotations in\nthe original dataset show inconsistency on whether to annotate this span as ‘subject.disorder’.\nTime_elapsed & Duration:\nIn this article, we describe another case of subcutaneous changes following repeated glatiramer acetate injection, presented\nas localized panniculitis in the area around the injection sites, in a 46-year-old female patient who was treated with\nglatiramer acetate for 18 months.\nAnnotation inconsistencies arise when a ‘time_elapsed’ argument can also be described as ‘duration’.\nTable A1: Inconsistent examples from the PHEE dataset.\nthe noise injection ratio has little impact on the per-\nformance but the order of the argument generation\nmatters. Therefore, we choose to set the noise in-\njection ratio to 0 and keep the arguments generated\nin order to reduce the fluctuation caused by random\ninsertion during model comparison.\nTo fine-tune the models, we establish a maxi-\nmum length of 512 tokens for both input and out-\nput. We utilize a total batch size of 32 for the large\nmodel, and 64 for the base model. The learning\nrates are configured as 3e-4 for the large model and\n5e-4 for the base model, with a warm-up ratio of\n0.06. We train the models for a maximum of 50\nepochs, early stopping if there is no improvement\nfor 5 epochs. During the generation process, we\nemploy beam search with a beam size of 3.\nWe employ the ‘gpt-3.5-turbo-0301’ version of\nChatGPT for prompting-based event extraction and\nsynthesized data generation. The temperature is set\nas 0 for zero-shot and few-shot prompting, and 0.2\nfor data generation.\nD Trigger Extraction Results for\nFinetuning Methods\nTable A2 displays the results of trigger extraction\nand event type classification for the fine-tuning\nmodels. In general, there is little difference in the\nperformance of trigger extraction and event type\nclassification between different models. Further-\nmore, training with filtered training and augmented\ndata still exhibits the smallest variance, which is\nconsistent with the observation for argument ex-\ntraction.\nE Argument Extraction Results for Each\nArgument Type\nTable A3 provides a detailed overview of argu-\nment extraction results for Flan-T5 with two aug-\nmentation strategies and ChatGPT. In comparison,\nChatGPT exhibits a specific vulnerability in accu-\nrately matching main arguments, likely attributedTrigger Event Type\nUIE(Large) 69.92 ±1.72 94.78±.72\nFlan-T5(Large) 69.60±1.87 95.04±.97\nw/ Tr.+Aug. 68.46±1.83 94.92±.60\nw/ Tr. Fil. 69.50±1.61 94.92±.88\nw/ Tr.+Aug. Fil. 69.68±1.36 95.00±.79\nw/ Tr. Fil.+Aug. Fil. 69.73±1.14 95.13±.48\nTable A2: Results for trigger extraction (EM_F1) and\nevent type classification (F1).\nto their greater length, which poses challenges in\nprecise boundary determination. When it comes\nto sub-arguments, ChatGPT demonstrates a perfor-\nmance distribution similar to fine-tuning models\nbut achieves lower overall scores. Notably, for cer-\ntain argument types of which ChatGPT performs\nnotably worse, such as ‘ frequency ’ and ‘ duration ’,\nthese shortcomings also negatively impact the per-\nformance when training with ChatGPT-generated\ndata. However, after filtering, the performance on\nthese argument types can be improved to the extent\nthat they may even outperform fine-tuning with\nannotated training data alone.\nF Supplementary material on qualitative\nanalysis for data augmentation\nTo elucidate the potential performance decline as-\nsociated with synthetic data, we sampled five in-\nstances for each argument type, where the fine-\ntuned model made correct predictions, while the\naugmented model (without filtering) made incor-\nrect predictions. We conducted a statistical analysis\nof the error types and Table A4 presents the distri-\nbution of error categories for each argument type.\nTo delve further into the origins of these errors,\nwe also sampled the data generated by ChatGPT,\nconducting a statistical analysis of its labelling\nerrors for comparison. Given the absence of a\ncomparable gold standard for the synthesized dataFlan-T5Flan-T5\n(Tr.+Aug.)Flan-T5\n(Tr. Fil.+Aug. Fil.)ChatGPT\nEM_F1 Token_F1 EM_F1 Token_F1 EM_F1 Token_F1 EM_F1 Token_F1\nSubject 73.11 82.37 70.93 80.90 72.39 82.15 57.96 75.20\nAge 88.12 92.07 87.21 92.55 87.50 92.82 86.62 90.18\nDisorder 69.80 77.13 63.81 72.76 69.73 77.45 53.90 61.08\nGender 86.73 86.51 86.03 85.78 87.15 87.00 84.29 85.07\nPopulation 74.83 75.72 72.30 73.94 75.90 76.69 49.30 42.11\nRace 93.20 93.35 93.29 91.20 92.02 91.52 87.5 77.78\nTreatment 66.35 79.82 66.27 79.00 65.90 79.68 57.67 73.49\nDrug 87.03 88.32 85.84 87.45 86.65 87.99 80.78 82.59\nDisorder 67.19 73.14 65.24 71.73 66.64 72.57 55.89 62.01\nRoute 67.76 69.34 63.55 65.55 66.37 70.39 56.66 63.73\nDosage 65.95 76.40 63.58 72.17 62.91 73.16 47.11 61.05\nTime elapsed 61.56 71.21 54.11 61.25 62.09 71.98 40.68 51.67\nDuration 60.40 64.91 56.12 60.42 61.47 58.77 47.56 56.58\nFrequency 51.26 54.37 43.43 46.19 53.25 52.10 36.36 33.09\nCombination.Drug 69.77 71.18 66.87 68.93 69.34 70.90 60.79 62.90\nEffect 74.33 84.73 74.68 83.94 74.75 84.65 64.60 79.19\nTable A3: Argument extraction results for each argument type. To accommodate space limitations, we showcase\nresults for Flan-T5 with two augmentation strategies and ChatGPT. The Flan-T5 results represent the average score\nacross 5-fold cross-validation, while the ChatGPT results showcase the performance of the 5-shot BM25 approach.\nby ChatGPT, we randomly sampled 30 generated\ncases and assessed errors across all argument types.\nThe resulting statistical findings are detailed in Ta-\nble A5.\nG Prompt Details\nTable A6 shows the instructions utilized for Chat-\nGPT’s zero-shot prompting. Through our prelim-\ninary experiments, we discovered that ChatGPT\nexhibits better performance when tasked with gen-\nerating structured output in JSON format rather\nthan textual output. Based on this finding, we ex-\nplore additional possibilities. For the end-to-end\ngeneration approach, we experiment with modify-\ning the instructions to a code style or providing a\ndetailed explanation of the schema. In the case of\npipeline prompting, we initially prompt ChatGPT\nto generate the skeleton of the output, encompass-\ning multiple events in a competent manner. Subse-\nquently, in the second stage, we provide the gener-\nation from the first stage and ask specific questions\nfor each sub-argument type.\nTable A7 presents the prompt employed to query\nChatGPT for the generation of synthesized in-\nstances for examples with adverse events. We em-\nploy a similar prompt for the data generation of\ncases with potential therapeutic events and mul-\ntiple events. Differently, we apply only the drug\nconstraint to instances related to potential thera-\npeutic events, as these typically do not involve arelevant effect. In addition, we refrain from impos-\ning such constraints on multi-event instances, as\ndoing so may complicate the preservation of event\nstructure in synthesized samples.\nH Licenses\nThe PHEE dataset employed in this study is sub-\nject to the MIT License. The UIE model is\ncovered by the Creative Commons Attribution-\nNonCommercial-ShareAlike 4.0 International Pub-\nlic License. The Flan-T5 model under the Apache\nLicense 2.0, and ChatGPT is a commercial service\nfor which we adhere to OpenAI’s terms of use. We\nuse the dataset and tools within the scope of their\nintended use.Argument\nMissingSemantic\nMisunderstandingBoundary\nProblemsAnnotation\nNoises\nSubject - - 2 3\nAge 3 2 - -\nDisorder - 4 1 -\nGender 1 3 - 1\nPopulation 3 - 2 -\nRace - - - -\nTreatment - 1 4 -\nDrug 1 1 1 2\nTreatment.Disorder 3 1 1 -\nRoute 3 - 1 1\nDosage 1 1 3 -\nTime elapsed 2 2 1 -\nDuration 1 2 1 1\nFrequency 1 3 1 -\nCombination.Drug 1 4 - -\nEffect - 2 3 -\nTotal 20 26 21 8\nTable A4: Statistics of error types for each argument type in qualitative analysis for data augmentation.\nSemantic\nMisunderstandingSemantic\nIncompletenessArgument\nMissingBoundary\nProblem#. In Synthesized\nSamples#. In Template\nSamples\nSubject - - - - 17 15\nAge - - - - 2 7\nDisorder 1 - 2 - 8 10\nGender - - - - 4 4\nPopulation - - - - - -\nRace - - - - - -\nTreatment - 1 - - 30 30\nDrug 1 - 5 2 32 36\nDisorder 1 - - - 8 11\nRoute - - - - 2 6\nDosage - - - - - -\nTime elapsed 1 - - - 1 1\nDuration - - - - 2 1\nFrequency - - - - - 1\nCombination.Drug 1 - 3 - 6 9\nEffect - 2 - 1 29 28\nTotal 5 3 5 3 141 159\nTable A5: Results of qualitative analysis for ChatGPT-synthesized data.Prompting Strategy Example\nSchema Extract event information from the following sentence and return events in json\nformat as this: [{\"event_type\": event type, \"arguments\":[{\"argument_type\":\nargument type, \"argument_span\":argument extraction}]}]. Event type: ad-\nverse event, potential therapeutic event. Argument type: subject, age, gender,\nrace, population, subject_disorder, treatment, drug, dosage, route, duration, fre-\nquency, time_elapsed, indication, combination_drug, effect. Sentence: <SEN-\nTENCE> Output:\nCode Argument = {\"argument_type\": str, #options: [subject, age, gender,race,\npopulation, subject_disorder, treatment, drug, dosage, route, duration,\nfrequency, time_elapsed, indication, combination_drug, effect]\n\"argument_span\": str,}\nEvent ={\"event_type\": str, #options: [adverse_event, poten-\ntial_therapeutic_event]\n\"arguments\": List[Argument],}\nevents: List[Event] = extract events in the sentence: <SENTENCE>\nprint(json.dumps(events))\nExplanation Extract event information from the following sentence and return events in json\nformat as this: [{\"event_type\": event type, \"arguments\":[{\"argument_type\":\nargument type, \"argument_span\":argument extraction}]}]. Event type: adverse\nevent (an event shows the use of a drug or combination of drugs cause a harmful\neffect on the human patient), potential therapeutic event (an event shows the\nuse of a drug or combination of drugs bring a potential beneficial effect on the\nhuman patient). Argument type: subject (overall description of the patients\ninvolved in the event), age (the concrete age or an age range of the subject),\ngender (the subject’s gender), race (the subject’s race or nationality), population\n(the number of patients receiving the treatment), subject_disorder (the subject’s\ndisorders), treatment (overall description of the therapy administered to the\npatients), drug (the drugs used as therapy in the event), dosage (the amount of\nthe drug is given), route (the route of the drug administration), duration (how\nlong the patient has been taking the medicine), frequency (the frequency of\ndrug use), time_elapsed (the time elapsed after the drug was administered to\nthe occurrence of the side effect), indication (the target disorder of the medicine\nadministration), combination_drug (the drugs used in combination), effect (the\nside effect in the adverse event or the beneficial effect in the potential therapeutic\nevent). Sentence: <SENTENCE> Output:Pipeline Stage 1 :\nExtract adverse events and potential therapeutic events in the sentence, as well\nas the information about the subject (the patient), the treatment and the effect\nof the treatment involved in the event. Return the output in json format as this:\n[{\"event_type\": event type, \"subject\": span of subject information, \"treatment\":\nspan of treatment information, \"effect\": span of effect information}]. Event\ntype: adverse event, potential therapeutic event. Sentence: <SENTENCE>\nOutput:\nStage 2 : Answer the question related to the given sentence and given event\ninformation. The answer should be a span exactly extracted from the sen-\ntence. If no answer can be found from the sentence, return N/A. Sentence:\n<SENTENCE> Event: Event type: <EVENT_TYPE> Subject: <SUBJECT>\nTreatment: <TREATMENT> Effect: <EFFECT>. <QUESTION>\nQuestions for each sub-argument type:\nage: What’s the age of the subject?\ngender : What’s the gender of the subject?\nrace: What’s the race or the nationality of the subject?\npopulation : How many subjects are involved in the event?\nsubject_disorder : What disorders do the subjects suffer from?\ndrug : What drugs are administered to the subject?\ndosage : What amount of the drug is administered to the subject?\nroute : What route is the drug given to the subject?\nduration : How long have the subject been taking the drug until the event oc-\ncurred?\nfrequency : How frequently does the subject take the drug?\ntime_elapsed : How long has elapsed since the patient started or ended dosing\nuntil the event occurred?\nindication : What’s the target disease of the treatment?\ncombination_drug : What drugs are used in combination in the event\nTable A6: Instructions for zero-shot prompting. <SENTENCE> is replaced with the query sentence. In the second\nstage of the pipeline prompting, <EVENT_TYPE>, <SUBJECT>, <TREATMENT>, <EFFECT> are replaced with\nthe generated results from the first stage, and <QUESTION> is replaced with manually crafted questions for each\nargument type. To enhance clarity, we substitute the ‘ treatment_disorder ’ in the dataset with ‘ indication ’ when\nquerying ChatGPT.Sentence: <SENTENCE> The events involved in the sentence are: <OUTPUT> Event type: adverse\nevent (an event shows the use of a drug or combination of drugs cause a harmful effect on the human\npatient), potential therapeutic event (an event shows the use of a drug or combination of drugs bring\na potential beneficial effect on the human patient). Argument type: subject (overall description of\nthe patients involved in the event), age (the concrete age or an age range of the subject), gender\n(the subject’s gender), race (the subject’s race or nationality), population (the number of patients\nreceiving the treatment), subject_disorder (the subject’s disorders), treatment (overall description\nof the therapy administered to the patients), drug (the drugs used as therapy in the event), dosage\n(the amount of the drug is given), route (the route of the drug administration), duration (how long\nthe patient has been taking the medicine), frequency (the frequency of drug use), time_elapsed (the\ntime elapsed after the drug was administered to the occurrence of the side effect), indication (the\ntarget disorder of the medicine administration), combination_drug (the drugs used in combination),\neffect (the side effect in the adverse event or the beneficial effect in the potential therapeutic event).\nGenerate a sentence with an adverse event which has a similar structure as the given sentence, and\nextract the events in the generated sentence. The drug <CONST_DRUG> must appear in the event,\nand the effect should be <CONST_EFFECT>. Return in the following json format: {\"sentence\":the\ngenerated sentence, \"output\": [{\"event_type\": event type, \"event_trigger\": the token indicating the\nexistence of the event, \"arguments\":[{\"argument_type\": argument type, \"argument_span\":argument\nextraction}]}]}. Return the json output only.\nTable A7: The prompt used to query ChatGPT for generating synthesized instances for ADE cases, with <SEN-\nTENCE> representing an example sentence from the training set, <OUTPUT> representing the annotation of the\nexample sentence, <CONST_DRUG> and <CONST_EFFECT> representing a pair of sampled drug and effect from\nthe training set."    },    {        "title": "2402.15668v1.Pruned_Pivot__Correlation_Clustering_Algorithm_for_Dynamic__Parallel__and_Local_Computation_Models.pdf",        "content": "Pruned Pivot: Correlation Clustering Algorithm for Dynamic,\nParallel, and Local Computation Models\nMina Dalirrooyfard\nMachine Learning Research\nMorgan StanleyKonstantin Makarychev\nDepartment of Computer Science\nNorthwestern University\nSlobodan Mitrovi´ c\nDepartment of Computer Science\nUC Davis\nAbstract\nGiven a graph with positive and negative edge labels, the correlation clustering problem aims to cluster\nthe nodes so to minimize the total number of between-cluster positive and within-cluster negative edges.\nThis problem has many applications in data mining, particularly in unsupervised learning. Inspired by\nthe prevalence of large graphs and constantly changing data in modern applications, we study correlation\nclustering in dynamic, parallel (MPC), and local computation (LCA) settings. We design an approach\nthat improves state-of-the-art runtime complexities in all these settings. In particular, we provide the first\nfully dynamic algorithm that runs in an expected amortized constant time, without any dependence on\nthe graph size. Moreover, our algorithm essentially matches the approximation guarantee of the celebrated\nPivot algorithm.\n1 Introduction\nWe study algorithms for the Correlation Clustering problem, which has many applications in Machine Learning\nand Data Mining Bansal et al. (2004); Becker (2005); Kalashnikov et al. (2008); Arasu et al. (2009); Firman\net al. (2013); Bonchi et al. (2013); Li et al. (2017). Among the most prominent applications is clustering\nproducts into categories or detecting communities based on product co-purchasing Wang et al. (2013); Veldt\net al. (2020); Shi et al. (2021). In this problem, we are given a set of objects with “similar” or “dissimilar”\nlabels between every pair of objects, and the goal is to cluster these objects such that similar objects are in\nthe same cluster and dissimilar objects are in different clusters. Formally, given a complete graph with edge\nweights in R, correlation clustering with minimum disagreement asks to cluster the nodes such that the sum\nof the weights of positive edges between clusters plus the sum of the weights of negative edges inside clusters\nis minimized.1This paper focuses on the unweighted setting, where weights are in {−1,+1}.\nCorrelation Clustering is APX hard Charikar et al. (2005). There has been a long line of work on\napproximation algorithms for correlation clustering (see e.g., Bansal et al. (2004); Charikar et al. (2005);\nDemaine et al. (2006); Chawla et al. (2015); Jafarov et al. (2021); Cohen-Addad et al. (2022); Behnezhad\net al. (2022); Chakrabarty & Makarychev (2023); Cohen-Addad et al. (2023)).The best known approximation\nfactor is 1 .73 + εdue to Cohen-Addad, Lee, Li, and Newman (2023). However, all known algorithms with\napproximation factor less than 3 use linear programming (LP), which makes most of them impractical for\ndealing with massive data.\nIn their seminal work, Ailon, Charikar, and Newman (2008) introduced an elegant 3-approximation\nalgorithm called Pivot , which runs in linear time (time proportional to the number of positive edges in the\n1Correlation clustering has also been studied on weighted graphs and with other objectives such as maximum agreement or\nminimum ℓpnorm .\n1arXiv:2402.15668v1  [cs.DS]  24 Feb 2024graph). This algorithm is the algorithm of choice in practice. The algorithm has been adapted for various\ncomputational models including semi-streaming Behnezhad et al. (2023); Cambus et al. (2022); Chakrabarty\n& Makarychev (2023), parallel algorithms (MPC) Cambus et al. (2022); Behnezhad et al. (2022), local\ncomputation algorithms (LCA) Behnezhad et al. (2022), and dynamic algorithms (Behnezhad et al. (2022);\nsee also Chechik & Zhang (2019)).\nWe study correlation clustering for massive and dynamic graphs. Such graphs are used to represent social\nnetworks Tantipathananandh & Berger-Wolf (2011); Hafiene et al. (2020), knowledge graphs Fang et al.\n(2020); Yan et al. (2021), and user-product interactions Ding et al. (2019). We design a Pivot -like algorithm\nthat can be easily implemented in the fully dynamic regime and LCA and MPC models. Our algorithm is\ninspired by the recent works by Behnezhad, Charikar, Ma, and Tan (2022) and Chakrabarty and Makarychev\n(2023). Behnezhad et al. (2022) presented a (3 + ε)-approximate algorithm, called R-pivot, that runs in O(1\nε)\nMPC rounds, for any ε >0. This algorithm can be implemented in LCA with ∆O(1/ε)-probe complexity,\nwhere ∆ is the maximum node degree in the graph (consisting of positive edges). Chakrabarty & Makarychev\n(2023) give a (3 + ε)-approximate semi-streaming algorithm that uses O(n/ε) words of memory. Behnezhad\net al. (2019) show how to maintain the lexicographically first maximal matching in a fully dynamic graph.\nTheir result can be used to implement Pivot in the fully dynamic setting. The expected update time for\nrelabelling edges is O(log2nlog2∆) per operation. Chechik & Zhang (2019) provide a similar result for\nmaximal matching with expected worst-case running time O(log4n) per update.\n1.1 Our Contributions\nIn this paper, we provide a new variant of Pivot , which we call Pruned Pivot , that gives a (3 + ε)-\napproximation for Correlation Clustering (see Theorem 4.1). Our algorithm is local and parallelizable by\ndesign: Given a node uand common randomness, it returns the cluster of uafter exploring only O(1/ε)\nnodes of the entire graph. This makes it easy to implement Pruned Pivot in various computational models,\nincluding dynamic algorithms, MPC, and LCA.\nOur first result is an efficient algorithm for dynamically maintaining a clustering. This is the first dynamic\nalgorithm for Correlation Clustering, whose expected running time does not depend on the graph size.\nTheorem 1.1 (Fully-dynamic correlation clustering) .For any ε >0, there is a data structure that maintains\na3 +εapproximation of correlation clustering in a fully-dynamic setting with an oblivious adversary. The\nexpected update time is O(1/ε)per operation.\nTheorem 1.1 gives an almost 3 approximation fully dynamic algorithm with update time O(1), which\nanswers an open question posed by Behnezhad et al. (2022).\nTheorem 1.2 (Correlation clustering in MPC) .For any ε >0, there is a randomized O(log1\nε)-round MPC\nalgorithm that achieves a 3 +εapproximation for Correlation Clustering. This bound holds even when each\nmachine has a memory sublinear in the node-set size.\nThe previously best-known MPC algorithm by Behnezhad et al. (2022) requires O(1/ε) rounds. Hence,\nour approach improves the dependence on 1 /εexponentially.\nTheorem 1.3 (Correlation clustering in LCA) .For any ε >0, there is a randomized O(∆/ε)-probe complexity\nlocal computation algorithm that achieves a 3 +εapproximation of correlation clustering on graphs with\nmaximum degree ∆.\nTheorem 1.3 gives an almost 3 approximation LCA in, essentially, O(∆) probes, thus answering another\nquestion posed by Behnezhad, Charikar, Ma, and Tan (2022).\nWe provide empirical evaluations on synthetic graphs in Section 8. They show that exploring only 4 nodes\nto obtain a node’s clustering suffices for the cost of Pruned Pivot to be within 1% of the cost of Pivot .\nWe finally note that our main technical result (Theorem 4.10) is of independent interest. It shows that\none can check whether a node is in a maximal independent set (MIS) in the expected time proportional to the\nnode’s degree. A seminal work by Yoshida, Yamamoto, and Ito (2009) shows how to do this in the average\n2degree time. Observe that both for our and the algorithm by Yoshida et al. (2009), the time it takes to find\nall nodes in MIS is linear in the number of edges.\n1.2 Comparison to prior work\nSeveral closely related works have introduced variants of the Pivot algorithm. In Pivot , nodes are processed\naccording to a predefined and random order, and each node queries its neighbors that have already been\nprocessed to determine its cluster (for a formal description of Pivot see Section 3). Hence, to determine\nthe cluster of each node, multiple “query paths” are made, and the collection of these query paths makes a\n“query tree”. (For a more formal definition of query paths and query trees, refer to Section 4.1.)\nFirst, given a parameter R, the R-Pivot algorithm by Behnezhad et al. (2022) runs the Pivot algorithm\nbut only considers “query paths” of depth at most R. If to obtain the cluster of a node one needs to consider\nquery paths of length more than R, then this node is put into a singleton cluster. Behnezhad et al. (2022)\nshow that this algorithm has approximation factor 3 + O(1\nR).\nSecond, the semi-streaming algorithm by Chakrabarty & Makarychev (2023) is another modified version\nofPivot . In this approach, given a parameter R, every node only queries at most its Rtop-ranked neighbors\nwhen deciding on their cluster. This algorithm yields an approximation factor of 3 + O(1\nR).\nInPruned Pivot , instead of only limiting the number of neighbors each node queries or the depth of the\nquery paths, our algorithm limits the total size of the query tree. Our analysis of the algorithm uses a new\napproach to counting query and expensive paths, which is very different from the approach of Behnezhad\net al. (2022).\nOur main technical contribution is a proof that by limiting the total size of the query tree, the existence/non-\nexistence of each edge only influences at most a constant number of other nodes. This crucial point allows us\nto achieve low probe complexity in LCA and MPC, and constant update time in the dynamic setting.\n2 Preliminaries\nAn instance of the correlation clustering problem receives an unweighted graph G= (V, E) on input. We\nconsider Erepresenting positive and ( V×V)\\Erepresenting negative labels between the nodes of V. This\nproblem aims to cluster Vto minimize the number of positive between-cluster and negative within-cluster\nlabels. The neighbors of a node u∈Vare denoted by N(u). We let u∈N(u), i.e., uis a neighbor of itself.\nNext, we formally define the MPC and LCA models.\nThe MPC model. Massively Parallel Computation (MPC) is a theoretical model of real-world parallel\ncomputation such as MapReduce Dean & Ghemawat (2008). It was introduced in a sequence of works by\nDean & Ghemawat (2008); Karloff et al. (2010); Goodrich et al. (2011). In MPC, computation is performed\nin synchronous rounds, where in each round every machine locally performs computation on the data that\nresides locally and then sends and receives messages to any other machine. Each machine has a memory\nof size S, and can send and receive messages of total size S. As the local computations frequently run in\nlinear or near-linear time, they are ignored in the analysis of the complexity of the MPC model, and so the\nefficiency of an algorithm in this model is measured by the number of rounds it takes for the algorithm to\nterminate where the memory Splays a key role. We focus on the sublinear memory regime, where S=nα\nfor some constant α∈(0,1).\nThe LCA model. Local Computation Algorithms (LCAs) were introduced by Rubinfeld et al. (2011)\nfor tasks where the input and output are too large to be stored in the memory. An LCA is not required to\noutput the entire solution but to answer queries about part of the output by examining only a small portion\nof the input. In Correlation Clustering, the query is a node v, and the output is the cluster ID of v. Formally,\nan LCA Ais given access to the adjacency list oracle for the input graph G, a tape of random bits, and local\nread-write computation memory. When given an input query x,Amust compute an answer for xdepending\nonly on x, Gand the random bits. The answers given by Ato all possible queries must be consistent, meaning\nthat they must constitute some valid solution to the computation problem.\n3We use probe to refer to accessing a node in an adjacency list. The LCA complexity of an algorithm is\nmeasured by the number of probes the algorithm makes per single query.\n3 Recursive and Pruned Pivot\nThis section describes our variant of the Pivot algorithm that we call Pruned Pivot . We remind the reader\nhow the standard Pivot algorithm works. First, it picks a random ordering π:V→ {1, . . . , n }. We say that\nπ(u) is the rank of node u. Ifπ(u)< π(v), then uhas a higher rank than v. Therefore, the node with rank 1\nis the highest-ranked, and the node with rank nis the lowest-ranked node. The algorithm maintains a list of\nnot yet clustered nodes. Initially, all nodes are not clustered. At every step, the algorithm picks the highest\nnot yet clustered node, marks it as a pivot, and assigns itself and all its not yet clustered neighbors to a new\ncluster. The algorithm labels all nodes in this new cluster as clustered and proceeds to the next step. Each\ncluster created by the Pivot algorithm contains a unique pivot node. We say that the cluster is represented\nby that pivot. If node ubelongs to the cluster represented by pivot v, we say that uis assigned to pivot v.\nTo describe our variant of the Pivot algorithm, we first rewrite the standard Pivot as a recursive or\ntop-down dynamic programming algorithm. The algorithm relies on the recursive function cluster (see\nAlgorithm 1). For a given node uand random permutation π, this function returns the pivot node to which\nuis assigned, along with a flag indicating if uis a pivot. Note that uis a pivot if and only if it is assigned to\nitself.\nAlgorithm 1 Recursive Pivot\n1:function cluster (u, π):\n2:Sort all neighbors of u(including uitself) by their rank π(v). Denote the sorted list by Nπ.\n3:for all vinNπ:\n4: ifv=u:\n5: return ubelongs to the cluster of u;uis a pivot.\n6: cluster (v, π)\n7: ifvis a pivot:\n8: return uis in the cluster of v;uis not a pivot.\nTo reduce the running time, we can cache or memoize the values returned by the function cluster . We\nwant to use this recursive function in our local computation algorithm (LCA). The problem is, however, that\nto cluster some nodes, the algorithm may need to make as many as Ω( n) calls to cluster (for instance,\nif node uis connected to all nodes in the left part of the complete bipartite graph Kn,n). That is why we\npropose a crucial change: execute only krecursive calls of Pivot . If the status of the node is not set by then,\nmark that node as unlucky and make it a singleton. The algorithm is given below.\nAlgorithm 2 Pruned Pivot\n1:Initialize a global variable rec-calls to 0.\n2:function Pruned-cluster (u,π):\n3:Ifrec-calls ≥k:\n4: terminate this recursion\n5:Sort all neighbors of u(including uitself) by their rank π(v). Denote the sorted list by Nπ.\n6:for all vinNπ:\n7: ifv=u:\n8: return ubelongs to the cluster of u;uis a pivot.\n9: rec-calls ←rec-calls + 1\n10: Pruned -cluster (v,π)\n11: ifvis a pivot:\n12: return uis in the cluster of v;uis not a pivot.\n4The recursion tree for the modified Pruned-cluster function contains at most kedges. Consequently, if\nkis a constant, the running time of function Pruned-cluster is also constant. We show how to implement\nthis algorithm as a Local Computation (LCA), Massively Parallel Computation (MPC), and Dynamic Graph\nAlgorithm. In the next section, we prove that the approximation factor of Pruned Pivot is 3 + O(1/k).\n4 Sequential Implementation\nIn the previous section, we described Pruned Pivot algorithm. We now examine a sequential algorithm\nthat produces the same clustering as the recursive algorithm above and, moreover, marks the same set of\nnodes as unlucky. First, we consider the standard Pivot implemented as a bottom-up dynamic programming\nalgorithm (see Algorithm 3).\nAlgorithm 3 Sequential Pivot\n1:Pick a random ordering π:V→ {1, . . . , n }.\n2:LetVπbe the list of all nodes u∈Vsorted by the rank π(u).\n3:for each uinVπ:\n4:Sort all neighbors of uby their rank π(v). Denote the sorted list by Nπ.\n5:while uis not assigned to a cluster:\n6: Pick the next neighbor v∈Nπ(u).\n7: ifvis a pivot: place uin the cluster of v.\n8: ifv=u: mark uas a pivot; create a new cluster for u; and place uin that cluster.\nIn the main loop (see the for each loop above), the algorithm iterates over all nodes in V. At iteration\ni∈ {1, . . . , n }, the algorithm processes node uwith rank i, i.e., u=π−1(i). It checks all neighbors vofuwith\nrank higher than that of v. If one of these neighbors is a pivot, the algorithm assigns uto the highest-ranked\npivot neighbor of u. If none of these neighbors are pivots, the algorithm marks uas a pivot and assigns uto\nitself.\nLet us set up some notation. Consider a neighbor vofu. It is processed at iteration i=π(v). Suppose\nthat no other neighbor of u(including uitself) is marked as a pivot before iteration i. Then, we know that u\nwill be assigned to the cluster of v, since it is the highest-ranked pivot neighbor of u. Thus, we will say that u\nissettled at step i. In other words, uis settled when the first neighbor of uis marked as a pivot. We denote\nthe iteration when uis settled by σ(u). Note that node uis assigned to the node processed at iteration\nσ(u), i.e., node π−1(σ(u)). In particular, if uis a pivot, then it is settled at the iteration i=π(u), the same\niteration as it is processed. We always have σ(u)≤π(u), because if uis not settled before iteration π(u),\nthen it is marked as a pivot and assigned to itself at iteration π(u); thus, if σ(u)≥π(u), then σ(u) =π(u).\nIf neighbor vofuis considered in the while -loop of the Sequential Pivot algorithm, then we say that\nuqueries v. We denote by Q(u) the set of all neighbors queried by u, except for uitself, and call this set the\nset of queried neighbors of u. Observe that Q(u) ={v∈N(u)\\ {u}:π(v)≤σ(u)}.That is, Q(u) is the set\nof all neighbors of u, excluding u, whose rank is higher than the rank of the pivot to which uis assigned.\nFinally, we formally define the recursion tree Tufor node u. The definition is recursive: If Q(u) is empty,\nthenTuonly contains node u. Otherwise, Tuis the tree with root uand|Q(u)|subtrees eTvattached to it –\none tree for every v∈Q(u). Each eTvis a copy of the recursive tree Tv. We stress that the recursive tree\nmay contain multiple copies of the same node v. One can think of nodes of Tuas being “stack traces” or\n“execution paths” for the recursive function cluster .\nSequential Pivot with Pruning. We now describe how to modify the bottom-up algorithm to make it\nequivalent to Pruned Pivot algorithm. First, we run the bottom-up algorithm as is and record its trace.\nWe then define the dependency size for every node u. The dependency size of uequals the number of edges\nin the recursive tree Tu. It can be computed using the following recurrence relation:\ndep-size( u) =X\nv∈Q(u)(1 + dep-size( v)). (1)\n5IfQ(u) is empty, then the dependency size of uequals 0, by definition. We mark node uasunlucky if its\ndependency size is at least k. Note that if one of the queried neighbors of uis unlucky, then uis also unlucky.\nAlgorithm 4 Pruning\n1:Compute the dependency size of every node uusing recurrence relation (1).\n2:Mark all nodes uwith dep-size( u)≥kas unlucky.\n3:Create a new cluster for each unlucky node u, remove ufrom its current cluster, and place uin the new\ncluster.\nThe pruning step puts all unlucky nodes into singleton clusters. We refer to the standard Pivot algorithm\nasPivot without pruning or simply Pivot . We refer to the Pivot algorithm that runs the pruning as\nPivot with Pruning . We show that the expected cost of the Pivot with Pruning is (3 + O(1/k))OPT .\nTheorem 4.1. The expected cost of the clustering produced by the Pruned Pivot is(3 +O(1/k))OPT .\nAilon, Charikar, and Newman (2008) showed that the approximation factor of Pivot is 3. By Lemma A.1,\nSequential Pivot is equivalent to Pivot . Hence, its approximation factor is also 3. The pruning step\nofSequential Pivot with Pruning removes some nodes (namely, unlucky nodes) from their original\nclusters and puts them into singleton clusters. This pruning step can increase the number of pairs of nodes\n(u, v) disagreeing with the clustering. Note, however, that if uandvare dissimilar (i.e., not connected with\nan edge), then the pruning step will never make them disagree with the clustering if they agreed with the\noriginal clustering. Thus, the pruning step can increase the objective function only by separating pairs of\nsimilar nodes ( u, v)∈E. In such case, we say that the pruning step cuts edge ( u, v). Specifically, edge ( u, v)\nis cut by the pruning step of Pivot with Pruning ifuandvare in the same cluster after the Pivot\nstep of the algorithm, but are separated by the pruning step, because u,v, or both uandvare unlucky\nnodes. We say that an edge ( u, v)∈Eis cut by Pivot (without pruning), if Pivot places uandvin\ndistinct clusters. In the next sections, we show Lemma 4.2 that states that the expected number of edges cut\nby the pruning step of Pivot with Pruning is upper bounded by the expected number of edges cut by\nPivot divided by ⌈(k−1)/2⌉/2. The “triangle-based” analysis of Pivot by Ailon, Charikar, and Newman\n(2008) shows that Pivot cuts at most 2 OPT edges in expectation. Thus, the pruning step cuts at most\n4OPT /⌈(k−1)/2⌉edges in expectation. We conclude that the expected cost of Pivot with Pruning is at\nmost (3 + 4 /⌈(k−1)/2⌉)OPT .\nLemma 4.2. The expected number of edges (u, v)cut by the pruning step of Sequential Pivot with\nPruning is upper bounded by the expected number of edges cut by Pivot divided by ⌈(k−1)/2⌉/2.\n4.1 Query Paths\nOur goal now is to prove Lemma 4.2. In this section, we define query paths ,extended query paths , and\nexpensive extended query paths . We then show that on the one hand, the number of edges cut by the pruning\nstep of the Sequential Pivot with Pruning algorithm is upper bounded by the number of expensive\nextended query paths divided by ⌈(k−1)/2⌉(see Corollary 4.7); and, on the other hand, the expected\nnumber of expensive extended query paths equals two times the expected number of edges cut by the Pivot\nalgorithm (see Theorem 4.10). This will imply Lemma 4.2.\nDefinition 4.3 (Query Paths) .A path (u0, u1, . . . , u L)is aquery path if each ui(i >0) queries ui−1.\nDefinition 4.4 (Extended Query Paths) .A path (u0, u1, . . . , u L)of length L≥2is an extended query path\n(EQ-path) if the following two conditions hold: (1) (u0, u1, . . . , u L−1)is a query path; and (2) π(uL−2)≤σ(uL).\nWe say that EQ-path (u0, u1, . . . , u L)is an extension of the query path (u0, u1, . . . , u L−1). We also call every\npath consisting of one edge (u0, u1)an extended query path.\nNote that a proper prefix of a query or extended query path is a query path.\n6w\nv\nb\na\nFigure 1: This figure shows an extended query path in the recursion tree Tvfor node v. The path starts with\nedge ( a, b) goes to the root of the tree, node v, and then proceeds to node w. The path from atillvis a\nquery path . The path from atowextends the path from atov. If edge ( a, b) is cut by Sequential Pivot\nwithout Pruning but edge ( v, w) is not cut, then this path is expensive . We call it expensive because if v\nis unlucky, then ( v, w) is cut by the pruning step of Pivot with Pruning and the cost of ( v, w) is partially\ncharged to this path.\nRecall, that for every u, we have σ(u)≤π(u). Also, a node uqueries its neighbor v(v̸=u) if and only if\nuis not settled before vis processed, i.e., σ(u)≥π(v). Thus, ( u0, u1, . . . , u L) is a query path if and only if\nσ(u0)≤π(u0)≤σ(u1)≤π(u1)≤. . .\n≤σ(uL−1)≤π(uL−1)≤σ(uL)≤π(uL). (2)\nSimilarly, a path ( u0, u1, . . . , u L) of length L≥2 is an EQ-path if and only if\nσ(u0)≤π(u0)≤σ(u1)≤π(u1)≤. . .\n≤σ(uL−2)≤π(uL−2)≤min(σ(uL−1), σ(uL)). (3)\nWe will charge all edges cut by the pruning step of Sequential Pivot with Pruning to Θ( k)expensive\nEQ-paths which are defined as follows.\nDefinition 4.5 (Expensive Extended Query Paths) .An extended query path (u0, u1, . . . , u L)is expensive if\nσ(u0)< σ(u1)butσ(uL−1) =σ(uL). We denote the set of all expensive query paths by X.\nNote that in every expensive EQ-path, the first edge is cut by Pivot (because σ(u0)< σ(u1)) but the last\nedge is not cut (because σ(uL−1) =σ(uL)). A path ( u0, u1, . . . , u L) is an EQ-path if and only if condition (3)\nholds, thus ( u0, u1, . . . , u L) is an expensive EQ-path if and only if\nσ(u0)< σ(u1) and σ(u0)≤π(u0)≤σ(u1)≤π(u1)≤\n··· ≤ π(uL−2)≤σ(uL−1) =σ(uL). (4)\nThe first condition σ(u0)< σ(u1) in (4) can be replaced with σ(u0)̸=σ(u1), because we always have\nσ(u0)≤σ(u1) ifu0, . . . , u L−1is a query path.\n4.1.1 Charging Cut Edges to Expensive Paths\nWe now prove a lemma that establishes a connection between edges cut by the pruning step of Sequential\nPruned Pivot and expensive EQ-paths.\n7Lemma 4.6. For every unlucky node vand every edge (v, w)with σ(v) =σ(w), there exist ⌈(k−1)/2⌉\nexpensive extended query paths that end with edge (v, w).\nThe immediate corollary of this lemma gives us a bound on the number of edges cut by the pruning step.\nCorollary 4.7. The number of edges cut by the pruning step of Sequential Pivot with Pruning is\nupper bounded by |X|/⌈(k−1)/2⌉.\nProof of Corollary 4.7. Every edge ( v, w)∈Ecut by the pruning step of Sequential Pivot with Pruning\nis not cut by Pivot . Hence, σ(v) =σ(w). Moreover, if ( u, v) is cut by the pruning step, then v,w, or both v\nandwmust be unlucky. Thus, by Lemma 4.6, there are at least ⌈(k−1)/2⌉expensive EQ-paths that end\nwith ( v, w) or ( w, v).\nProof of Lemma 4.6. LetTvbe the recursion tree for node v. We first show that Tvcontains at least\n⌈(k−1)/2⌉edges cut by Pivot (formally, Tvcontains copies of edges cut by Pivot ). Consider an edge\n(u′, u′′) inT. Since ( u′, u′′) is an edge in the recursion tree, u′queries u′′. Thus, edge ( u′, u′′) is not cut by\nPivot only if u′′is a pivot and u′is assigned to u′′. This means that u′′is the highest-ranked pivot neighbor\nofu′. Consequently, for every u′, there is at most one child node u′′such that ( u′, u′′) is not cut. Moreover,\nif one such u′′exists, then u′is not a pivot, and hence the edge from u′to its parent is cut (unless u′is the\nroot). We get the following claim.\nClaim 4.8. For every node uin the recursion tree Tv, at most one edge incident on uis not cut by Pivot .\nNode vis unlucky. Hence, the recursion tree Tvmust have at least kedges. Therefore, by Claim 4.8 and\nLemma 4.9 (see below) there are at least ⌈(k−1)/2⌉cut edges in Tv. In Lemma 4.9, red edges are cut edges,\nand blue edges are not cut edges.\nLemma 4.9. Consider a tree Twith kedges colored red or blue. Suppose that every node in Tis incident\nwith at most one blue edge. Then, Tcontains at least ⌈(k−1)/2⌉red edges.\nProof. Tree Thaskedges and k+ 1 nodes. Each node is incident to at most one blue edge. So, blue edges\nform a matching. The size of this matching is at most ⌊(k+ 1)/2⌋. The number of edges not in the matching\nis at least k− ⌊(k+ 1)/2⌋=⌈(k−1)/2⌉. All of them are red.\nNow, for every edge ( b, a) inTsuch that bqueries aand is not cut by Pivot , we construct an expensive\nEQ-path. This path starts with edge ( a, b), then goes to the root of tree T– node v– along the edges of\nT, and, finally, proceeds to node w(see Figure 1). Observe that the subpath from atovis a query path\nsince each node on the path queries the preceding node. Then, since ( v, w) is cut by the pruning step of\nSequential Pruned Pivot , it is not cut by Sequential Pivot . Therefore, σ(w) =σ(v). Hence, by (3),\nthe path ( a, b, . . . , v, w ) is an expensive query path.\n4.1.2 Expected Number of Expensive EQ-Paths\nWe now prove that the expected number of expensive EQ-paths is at most 4 OPT and the expected number\nof query paths that start with a fixed directed edge ( a, b) – we denote these paths by Q(a, b) – is at most 2.\nTheorem 4.10. For every ordered pair (a, b)with (a, b)∈E, we have Eπ|Q(a, b)| ≤2, and\nEπ|X| ≤ 2EhX\n(u,v)∈E1(σ(u)̸=σ(v))i\n.\nWe will refer to the time when iteration tofSequential Pivot occurs as time t. For the sake of analysis,\nwe shall assume that the ordering πis initially (at time 0) hidden from us and is revealed one node at a time.\nAt the beginning of iteration t, we learn the value of π−1(t), or, in other words, the identity of the node\nprocessed at time t. Note that the state of the algorithm after the first titerations is completely determined\nby the nodes π−1(1), . . . , π−1(t). In particular, at time t, for every node u, we can tell if uis settled by time t\n8and, if it is settled, then we know the value of σ(u); otherwise, we know that σ(u)> t. LetFtbe the filtration\ngenerated by π−1(1), . . . , π−1(t). We will use the standard notation Pr[· | F t] and E[· | F t] to denote the\nconditional probability and conditional expectation given the state of the algorithm after iteration t. Note\nthat each π(u) and σ(v) is a stopping time with respect to Ft.\nLetP(a, b) be the set of all paths that start with edge ( a, b). As we run the Sequential Pivot algorithm,\nwe add paths to sets Qt(a, b) and Xt(a, b). Loosely speaking, we add a path from P(a, b) toQt(a, b) if\nwe can verify that this path is a query path using condition (2) at time t; we add a path from P(a, b) to\nXt(a, b) if we can verify that this path is an expensive EQ-path using condition (4) at time t. Formally,\nwe add path ( u0=a, u1=b, . . . , u L) toQt(a, b) at time π(uL−1) if condition (2) holds; and we add path\n(u0=a, u1=b, . . . , u L) toXt(a, b) at time σ(uL−1) =σ(uL) if condition (4) holds. Thus, Qt(a, b) is the\nset of all query paths P∈ P(a, b) for which π(uL−1)≤t; andXt(a, b) is the set of all expensive EQ-paths\nP∈ P(a, b) for which σ(uL−1) =σ(uL)≤t. Note that at the times π(uL−1) and σ(uL−1) =σ(uL), we can\ncheck conditions (2) and (4), respectively. We also define a set of dangerous paths at time t, denoted by\nDt(a, b), as follows.\nDefinition 4.11 (Dangerous EQ-path) .An extended query path u0, . . . , u L(L≥1) isdangerous at iteration\ntifπ(uL−2)≤t,π(uL−1)> t, and σ(uL)> t. We omit the first condition ( π(uL−2)≤t) for paths of length\n1. Denote the set of all extended query paths that start with edge (a, b)and are dangerous at iteration tby\nDt(a, b).\nNote that a path P∈ P(a, b) may become dangerous at some iteration t, stay dangerous for some time,\nbut eventually it will become non-dangerous. After that, it will remain non-dangerous until the end of the\nalgorithm. The definition of dangerous paths is justified by the following lemma, which, loosely speaking,\nsays that every query path and every expensive EQ-path is created from a dangerous path.\nLemma 4.12. Consider a path P= (u0, u1, . . . , u L)∈ P(a, b). Let P′= (u0, u1, . . . , u L−1). Then, the\nfollowing claims hold for every t≥0:\n•IfP∈ Qt+1(a, b)\\ Qt(a, b), then P∈ Dt(a, b)butP /∈ Dt+1(a, b).\n•IfP∈ Dt+1(a, b)\\ Dt(a, b), then P′∈ Dt(a, b)butP′/∈ Dt+1(a, b).\n•IfP∈ Xt+1(a, b)\\ Xt(a, b), then P∈ Dt(a, b)orP′∈ Dt(a, b)butP /∈ Dt+1(a, b)andP′/∈ Dt+1(a, b).\nWe prove this lemma in Appendix 4.2.\nOur approach to bounding E|Qt(a, b)|andE|Xt(a, b)|is based on the following idea: At time t= 0,\nthe path ( a, b) is dangerous, and there are no query or expensive EQ-paths that start with ( a, b). If P\nis a dangerous EQ-path at time t, then at the next iteration, it may be extended to a longer dangerous\npath, replaced with a query path, and/or created one or more expensive EQ-paths. A dangerous path may\nalso disappear without producing any new dangerous, query, or expensive EQ-paths. For every EQ-path\nPdangerous at iteration t, we will compute the probabilities of creating new paths and derive the desired\nbounds on E|Qt(a, b)|andE|Xt(a, b)|. To make our argument formal, we define two random processes:\nΦt(a, b) = 2|Dt(a, b)|+|Qt(a, b)|;\nΨt(a, b) = 2|Dt(a, b)|+|Xt(a, b)|.\nWe claim that Φ t(a, b) and Ψ t(a, b) are supermartingales. That is, E[Φt+1(a, b)| Ft]≤Φt(a, b); and\nE[Ψt+1(a, b)| Ft]≤Ψt(a, b).\nLemma 4.13. Random processes Φt(a, b)andΨt(a, b)are supermartingales.\nWe prove this lemma in Appendix 4.3. We now use it to finish the proof of Theorem 4.10. We first\nupper-bound Eπ|Q(a, b)|. Fix a directed edge ( a, b). At time 0, Φ 0(a, b) = 2, since ( a, b) is a dangerous\nEQ-path at time 0 but ( a, b)/∈ Q 0(a, b). Process Φ 0(a, b) is a supermartingale. Hence, E[Φn(a, b)]≤2. Note\n9that at time n, there are no dangerous EQ-paths because by time nall nodes are processed and settled.\nHence,\nE[Q(a, b)] =E[Qn(a, b)] =E[Φn(a, b)]≤2.\nWe now upper-bound E|X|. Every expensive EQ-path P= (u0, . . . , u L) starts with a directed edge\n(u0, u1) and at some iteration tis added to the set Qt(u0, u1). Hence,\nE|X|=X\na,b:(a,b)∈EE|Xn(a, b)|\n=X\na,b:(a,b)∈EE\u0002\n|Xn(a, b)| ·1(σ(a)< σ(b))\u0003\n.\nHere, we used the definition of expensive PQ-paths: In every expensive path PinP(a, b),σ(a)< σ(b).\nObserve that |Xn(a, b)|= Ψ n(a, b) and E\u0002\nΨn(a, b)| Fσ(a)]≤Ψσ(a)(a, b). Moreover,\nE\u0002\nΨn(a, b)·1(σ(a)< σ(b))| Fσ(a)]≤\n≤Ψσ(a)(a, b)·1(σ(a)< σ(b)),\nbecause the event {σ(a)< σ(b)}is inFσ(a), or, in other words, at time σ(a), we already know the value of\n1(σ(a)< σ(b)). Thus,\nE[|X| | F σ(a)]≤X\na,b:(a,b)∈EΨσ(a)(a, b)·1(σ(a)< σ(b)).\nIt remains to compute Ψ σ(a)(a, b)·1(σ(a)< σ(b)). If σ(a)< σ(b), then ais not a pivot (otherwise, we would\nhave σ(a) =σ(b) =π(a)). Thus, π(a)> σ(a), and the only EQ-path in P(a, b) dangerous at time τ=σ(a) is\nthe path ( a, b). Hence, |Dσ(a)(a, b)|= 1. Similarly, there are no expensive PQ-paths in Xσ(a)(a, b), because\nσ(b)> τ=σ(a). Therefore, Ψ σ(a)(a, b)·1(σ(a)< σ(b)) = 2 ·1(σ(a)< σ(b)), and\nE|X|=EE[|X| | F σ(a)]≤2EhX\n(a,b)∈E1(σ(a)̸=σ(b))i\n.\n4.2 Proof of Lemma 4.12\nProof of Lemma 4.12. I. IfP∈ Q t+1(a, b)\\ Qt(a, b), then Pis a query path, and π(uL−1) =t+ 1. Using\ncondition (2), we get π(uL−2)< π(uL−1) =t+ 1; σ(uL)≥π(uL−1) =t+ 1. We have π(uL−2)≤tand\nσ(uL)≥π(uL−1)> t. Also, Pis an extended query path (as every query path). Thus, Pis a dangerous\nextended query path at time t. However, it is no longer dangerous at time t+ 1, because π(uL−1) =t+ 1.\nII. If P∈ D t+1(a, b)\\ Dt(a, b), then the following hold: Pis an extended query path; π(uL−2) =t+ 1;\nπ(uL−1)> t+ 1; and σ(uL)> t+ 1. To see how we derived π(uL−2) =t+ 1, observe that P∈ Dt+1(a, b)\nimplies π(uL−2)≤t+ 1, but the fact that P /∈ Dt(a, b), together with π(uL−1)> t+ 1 and σ(uL)> t+ 1,\nimplies π(uL−2)> t. Note that L≥2 because every path of length 1 is dangerous at time 1 but P /∈ Dt(a, b).\nSince P′is a query path by Definition 4.4, we have σ(uL−1)≥π(uL−2) =t+ 1 (see (2)). This shows that\nP′∈ Dt(a, b)\\ Dt+1(a, b) if the length of P′is 1, i.e., if L= 2. For L≥3, we need to additionally check that\nπ(uL−3)≤t. This is the case because π(uL−3)< π(uL−2) =t+ 1.\nIII. If P∈ Xt+1(a, b)\\ Xt(a, b), then Pis an expensive extended query path and σ(uL−1) =σ(uL) =t+ 1.\nNote that π(uL−1)≥σ(uL−1) =t+1. Since P′is a query path, we have π(uL−2)≤σ(uL−1) =t+1. Consider\ntwo cases. If π(uL−2)≤t, then Pis dangerous at time tbut not at time t+ 1 (because σ(uL) =t+ 1). If\nπ(uL−2) =t+ 1, then P′is dangerous at time t(because π(uL−3)< π(uL−2) =t+ 1). However, P′is no\nlonger dangerous at time t+ 1 (because σ(uL−1) =t+ 1).\n10a b ··· uL−1uL\nws···w2w1\nFigure 2: Path ( a, b, . . . , u L−1, uL) is a dangerous EQ-path at iteration t. At iteration t+ 1, it may\nbecome a query path and/or an expensive EQ path. It may also get extended to EQ-paths Pw, where\nw∈Wt\\ {uL−1, uL}. These extended paths Pwmay be dangerous or expensive at iteration t+ 1, but they\nalso may be non-dangerous and non-expensive at iteration t+ 1.\n4.3 Proof of Lemma 4.13\nProof of Lemma 4.13. We first analyze process Φ t(a, b). By Lemma 4.12, if path Pis added to set Qt+1(a, b)\nor path Pwbecomes dangerous at step t+ 1, then Pis dangerous at step t. Hence,\nΦt+1(a, b)−Φt(a, b) =−2|Dt(a, b)\\ Dt+1(a, b)|+ 2|Dt+1(a, b)\\ Dt(a, b)|+|Qt+1(a, b)\\ Qt(a, b)|=\n=X\nP∈Dt(a,b)−2·1{P /∈ Dt+1(a, b)}| {z }\nno longer dangerous paths+1{P∈ Qt+1(a, b)}| {z }\nnew query paths+2X\nw∈V1{Pw∈ Dt+1(a, b)}| {z }\nnew dangerous paths.\nWe show that the conditional expectation of every term in the sum above given Ftis non-positive. Consider\na path P= (u0, . . . , u L)∈ P(a, b). Let\n∆t+1(P) =−2·1{P /∈ Dt+1(a, b)}+1{P∈ Qt+1(a, b)}+ 2X\nw∈V1{Pw∈ Dt+1(a, b)}.\nWe need to show that\nEh\n∆t+1(P)| Ft, P∈ Dt(a, b)i\n≤0.\nNote that if P∈ Dt(a, b) and P∈ Dt+1(a, b), then ∆ t+1(P) = 0 by Lemma 4.12. Thus, it suffices to prove\nthatEh\n∆t+1(P)| Ft, P∈ D t(a, b), P /∈ D t+1(a, b)i\n≤0. Let Wt=\b\nw∈N(uL) :σ(w)> t\t\nbe the set of\nneighbors of uLthat are not settled by iterations t. Note that Pstops being dangerous at iteration t+ 1 only\nifuL−1is processed or uLis settled at iteration t+ 1. The latter event occurs if and only if one of the nodes\ninWtis processed at iteration t+ 1. Hence,\nEh\n∆t+1(P)| Ft, P∈ Dt(a, b), P /∈ Dt+1(a, b)i\n=Eh\n∆t+1(P)| Ft, P∈ Dt(a, b), π−1(t+1)∈Wt∪{uL−1}i\n.\nHere, π−1(t+ 1) is the node processed at iteration t+ 1. Observe that π−1(t+ 1) is uniformly distributed in\nWt∪ {uL−1}given that π−1(t+ 1)∈Wt∪ {uL−1}. Now, if π−1(t+ 1) = uL−1, then π(uL−1) =t+ 1, and\npath Pis added to Xt+1(a, b) (it is an expensive path because condition (2) is satisfied). If π−1(t+ 1) = uL−1,\nthen some or all paths Pw, where w∈Wt\\ {uL−1, uL}, are added to the set of dangerous paths Dt+1(a, b).\nNote that no path Pw′with w′/∈Wt\\ {uL−1, uL}is added to Dt+1(a, b) since for w′∈N(uL)\\Wt\nandw′/∈ {uL−1, uL}, we have σ(w′)≤t, but for PwinDt+1(a, b), we must have σ(w)> t. Thus, if\nπ−1(t+ 1) = uL−1, then ∆ t+1(P)≤ −2 + 1 + ( |Wt| −1) (here, we use that uLalways belongs to Wtif\nP∈ Dt(a, b)). If, however, π−1(t+ 1)∈Wt\\ {uL−1}, then ∆ t(a, b) =−2, because (1) Pdoes not become a\nquery path, and (2) Pis not extended to any dangerous paths at iteration t+ 1. We obtain the following\nbound\nEh\n∆t+1(P)| Ft, P∈ Dt(a, b), π−1(t+ 1)∈Wt∪ {uL−1}i\n≤ −2 +1 + 2(|Wt| −1)\n|Wt∪ {uL−1}|<0.\n11This proves that Φ tis a supermartingale. We now show that Ψ( t) is also a supermartingale. Using Lemma 4.12,\nwe get\nΨt+1(a, b)−Ψt(a, b) =X\nP∈Dt(a,b)∆′(P),\nwhere\n∆′\nt(P) =−2·1{P /∈ Dt+1(a, b)}| {z }\nno longer dangerous paths+2X\nw∈V1{Pw∈ Dt+1(a, b)}\n| {z }\nnew dangerous paths+1{P∈ Xt+1(a, b)}+X\nw∈V1{Pw∈ Xt+1(a, b)}\n| {z }\nnew expensive paths.\nAs before, it suffices to show that for every P∈ P(a, b), we have\nEh\n∆′\nt+1(P)| Ft, P∈ Dt(a, b), P /∈ Dt+1(a, b)i\n≤0.\nFrom the previous argument, we know that π−1(t+ 1) is uniformly distributed in Wt∪ {uL−1}given Ft,\nP∈ Dt(a, b), and P /∈ Dt+1(a, b). We now consider two cases: uL−1is already settled by iteration t(then\nuL−1/∈Wt) and uL−1is not yet settled (then uL−1∈Wt). In the former case, Pmay never become an\nexpensive EQ-path, since σ(uL−1)≤t, but σ(uL)> t. Moreover, if uL−1is processed at iteration t+ 1, then\nit will not be marked as a pivot, and, consequently, no nodes will be settled at iteration t+ 1. In particular,\nifPis extended to some EQ-path Pw, then this path will not be added to the set of expensive EQ-paths\nXt+1(a, b) (even though it may eventually be added to some set Xt′(a, b) with t′> t+ 1). In fact, every path\nPwwith w∈Wt\\ {uL}will be added to the set of dangerous EQ-path Dt+1ifuL−1is processed at iteration\nt+ 1. If another node in Wt– not uL−1– is processed at iteration t+ 1, then Pis not extended to any other\npaths. Denote the event {P∈ Dt(a, b)\\ Dt+1(a, b)}byEt. Then,\nEh\n∆′\nt+1(P)| Ft,Et, σ(uL−1≤t)i\n=−2 + 2|Wt| −1\n|Wt|+ 1<0.\nWe now consider the case when uL−1is settled by time t. To this end, we define two disjoint subsets\nofWt:W(1)\nt=Wt\\N(uL−1) and W(2)\nt=Wt∩N(uL−1)\\ {uL−1, uL}. IfuL−1is processed at iteration\nt+ 1, then uL−1is marked as a pivot at this iteration. In this case, all nodes in W(2)\ntas well as node uL\nare settled at iteration t+ 1. Consequently, path Pand all paths Pwwith w∈W(2)\nt\\ {uL}are added to\nthe set of expensive paths Qt+1(a, b). No other extensions of Pare added to this set. All paths Pwwith\nw∈W(1)\ntare added to the set of dangerous paths Dt+1(a, b). Hence, if uL−1is processed at iteration t+ 1,\nthen ∆′\nt+1(P) =−2 + 2|W(1)\nt|+ (|W(2)\nt|+ 1). If uLor any node in W(2)\ntis processed at time t, then path P\nbecomes expensive, but it is not extended to any other EQ-path. Hence, ∆′\nt+1=−2 + 1. Finally, if a node\ninW(1)\ntis processed, then Pis not extended to any other paths, and Pdoes not become expensive. Thus,\n∆′\nt+1=−2. Therefore,\nEh\n∆′\nt+1(P)| Ft,Et, σ(uL−1)> t)i\n=−2 +2|W(1)\nt|+|W(2)\nt|+ 1\n|Wt|+|W(2)\nt|+ 1\n|Wt|\n=−2 +2|Wt|+ 2\n|Wt|<0.\nThis completes the proof of Lemma 4.13.\n5 MPC Algorithm\nNow we discuss how to simulate Pruned Pivot in MPC. To that end, observe that Pruned Pivot has\ndepth at most k. Moreover, for each node v, there are at most kneighbors of v, which the computation\n12forvdepends on. To find those kneighbors of v, we sort all the neighbors of vand select the top kones.\nSorting can be done in O(1) MPC rounds Goodrich et al. (2011) and O(m) total memory for all the nodes\nsimultaneously.\nDefine a directed graph HonVsuch that Hcontains an edge ( u, v) if and only if vis among the top\nkneighbors of u. The k-hop neighborhood of uinHcontains all the edges and nodes needed to process u\nbyPruned Pivot . Given this, our MPC algorithm simultaneously gathers k-hop neighborhood for each\nu. This can be done in O(logk) MPC rounds and O(n·kk+1) total memory via graph exponentiation (see\nthe paper by Lenzen & Wattenhofer (2010)). The output for uofPruned Pivot is computed on a single\nmachine using its relevant k-hop neighborhood.\n6 LCA\nGiven a node v, our LCA algorithm simply simulates Pruned Pivot . The probe complexity of such an\napproach is almost direct. Namely, the algorithm visits at most knodes via recursive calls. Each call scans\nneighbors of the corresponding node to find the k-top ones. That scan takes O(∆) probes. Therefore, the\nprobe complexity is O(∆·k).\nIt remains to analyze the space complexity and, if any, its effect on the approximation.\nRandom ordering. Our algorithm assumes it has access to a random node-permutation π. However, it is\nunclear how to obtain πin LCA. So, instead, each node vdraws an integer rank rv∈[0, . . . , n10) uniformly\nat random. If values rvare drawn independently of each other, then, with probability at least 1 −n−5, for\neachu̸=vit holds ru̸=rv. Therefore, the values rvimplicitly define a random permutation, which is enough\nto simulate our algorithm. (For now, assume that ru̸=rv, and at the end of this section, we discuss how to\nhandle the case when ru=rvfor two nodes.)\nSource of randomness. If the algorithm has access to an arbitrary long tape of random bits with random\naccess, then whenever a node wants to learn rv, it reads 10 lognbits starting at position v·10logn. If such\ntape is not accessible, then the corresponding local computation algorithm has to store random bits in its\nmemory. It is obvious how to keep all the required random bits in O(nlogn) memory; a node vusesO(logn)\nbits independent of other nodes to obtain rv. However, we show that substantially fewer bits suffice for small\n∆. To that end, we first recall the definition of w-wise independence hash functions and a folklore result\nabout its construction.\nDefinition 6.1 (w-wise independent hash functions) .Letw, b, N ∈Nandsbe a seed of independent random\nbits. A function hs:{0,1}N→ {0,1}bis a called w-wise independent hash function if for any I≤w, all\ndistinct x1, . . . , x I∈ {0,1}Nand all distinct y1, . . . , y I∈ {0,1}bit holds\nPr I^\ni=1hs(xi) =yi!\n= 2−I·b.\nTheorem 6.2 (Folklore) .Letw, b, N ∈N. There exists a w-wise independent hash function hs:{0,1}N→\n{0,1}bwith a seed sof length w·max{N, b}. Moreover, hscan be stored using O(w·(N+b))bits of space.\nLeth:V→ {0,1,2, . . . , n10−1}be a w-wise independent hash function; we will set win the remainder\nof this section. Consider an LCA execution of our algorithm from vthat, instead of randomly pre-generated\nr-values, uses h; letALCA refer to that algorithm. That is, whenever ALCA needs π(u) it uses h(u) instead.\nSo, invoking ALCA onv, the algorithm has to learn the rank-ordering of the neighbors of v. To achieve that,\nALCA evaluates h(u) on each u∈N(v).\nIn the process, ALCA invokes hat most ∆ ·kmany times. Hence, if w≥∆·k, each invocation of\nhis entirely independent of the previous invocations of h. Therefore, from the point of view of v, the\nexecution ALCA(v) is equivalent to that of executing ALCA with rvalues pre-generated using O(nlogn) bits\nof randomness.\n13DoesALCA executed on all v∈Vresemble our algorithm? Very likely, no! Namely, for w < n , we should\nexpect lots of dependencies between execution trees for some, possibly far away, nodes. Nevertheless, we will\nargue that, in expectation, the cost of ALCA and our main algorithm are the same for a proper value of w.\nTo that end, in our main algorithm, let cu,vbe the expected cost that pair {u, v}incurs. That is, if {u, v}is\nan edge, then cu,vis the probability that our main algorithm cuts it. If {u, v}is not an edge, then cu,vis the\nprobability that our main algorithm does not cut. In particular, the expected cost of our main algorithm isP\nu,v∈V×Vcu,v.\nConsider {u, v} ∈V×V. Then, ALCA behaves the same with respect to {u, v}ifALCA(v) andALCA(u)\ntogether resemble the execution of our main algorithm for vandu. This can be achieved by ensuring that\nthe randomness used for ALCA(u) and ALCA(v) is independent, which is achieved for w≥2·∆·k.\nIn this setup, we also have that in the execution tree for v, with probability 1 −n−5at least, no two ranks\nare the same. Taking the union bound over all v∈V, with probability at least 1 −n−4, no execution tree\nhas two same ranks.\nThe case when ru=rvforu̸=vin an execution tree. As argued above, this case happens with\nprobability at most n−4. We distinguish two cases: when the input graph is a union of cliques and when it is\nnot. In the latter case, the optimum value is at least 1. Since ru=rvhappens with probability at most n−4\nand the maximum possible cost of a clustering is n2, we have that the expected cost in this case is at most\n(3 +ε)OPT + 1/n2≤(3 +ε+ 1/n2)OPT .\nNow, consider the case when the input graph is a union of cliques. In this case, it holds OPT = 0, and\nan additive cost of 1 /n2does not yield a (3 + ε) multiplicative one. To handle this case, we add a rule for\nbreaking ties in ranks to our algorithm. Namely, if a node xhas two neighbors uandvsuch that ru=rv,\nthen our algorithm ranks ubefore viffu < v ;u < v means that the label of uis smaller than the label of\nv. Since each node sees the same neighborhood in a clique – this neighborhood includes the node itself –\nthen each node chooses the same pivot even if ru=rvfor two distinct nodes. So, each clique is clustered\ncorrectly.\n7 Fully Dynamic Algorithm under Oblivious Adversary\nThis section explains how to implement Pruned Pivot in the fully dynamic setting with an oblivious\nadversary. On a high level, when an edge eis updated, our dynamic algorithm simply recomputes the\nclustering for each node that queries e. As Theorem 4.10 states, there are only O(1) such nodes in expectation.\nThis property is the key to enabling us to obtain only O(k) expected amortized update time.\nIn addition, our algorithm updates the neighbor list of the endpoints of e, which is needed to implement\nPruned Pivot . To obtain the desired running time, we use that Pruned Pivot visits at most the top k\nneighbors of a node. Therefore, instead of maintaining the entire neighborhood list of a node in a sorted\nmanner, we do it only for the top kneighbors of each node. We show how to dynamically maintain this list\nin expected amortized O(logk) time.\nWe now provide details. Our algorithm maintains the following information for each node u:\n•Nk(u): Top kneighbors of ukept in an ordered balanced binary tree.\n•Q−1\nP(u): The nodes that query u, except from uitself, kept in a double-linked list.\n•QP(u): The set of nodes queried by Pruned-cluster (π, u), which is maintained during the recursive\ncalls. Together with every node wqueried by u, inQP(u) is also stored the pointer to where uis in the\ndouble-linked list Q−1\nP(w). We use these pointers to efficiently update Q−1\nP(·).\nConsider an update of edge {a, b}, i.e., an edge insertion or removal. This update triggers several updates\nin the information we maintain for each node. Without loss of generality, assume that π(a)< π(b). Note that\namong all Nk(·), only Nk(b) changes. We show how to update Nk(b) inO(logk) expected time in Section 7.2.\nIfNk(b) is changed, but bqueries ( b, a) neither before nor after this update, then QPandQ−1\nPstructures\nremain the same as before the update. However, if whether bqueries ( b, a) changes after the update, then\n14all the nodes querying b, i.e., those in Q−1\nP(b) might change their structures. Moreover, only those nodes\nthat query bbefore the update might change their structures. To see that, assume that wdoes not query b\nbefore the edge update. First, if wqueries a, it will not query ( a, b) since π(a)< π(b). Second, and taking\ninto account that wnever queries ( a, b), ifwnever reaches bin the invocation of Pruned-cluster (w, π)\nbefore the update, then {a, b}cannot be queried by wregardless of the update. This yields the following\nobservation.\nObservation 7.1. Consider an edge update {a, b}withπ(a)< π(b). Then, only the nodes in Q−1\nP(b)∪ {b}\nare those whose maintained data structures can potentially change on the update.\nWe remind the reader that Pruned-cluster (w, π) does not use memoization. In Pruned Pivot-\nUpdate (·,·) (see Algorithm 6) we provide our update procedure, which uses Algorithm 5 as a subroutine.\nWe analyze its running time in Section 7.1.\nAlgorithm 5 A-Node-Update (w)\n1:forevery u∈QP(w):\n2: Remove wfrom Q−1\nP(u) by using the corresponding pointer from QP(w).\n3:Invoke Pruned-cluster (w,π) (Algorithm 2) by using Nk(·) as the neighborhood list. During the\nexecution, update QP(w).\n4:forevery u∈QP(w):\n5: Append wtoQ−1\nP(u) and record in QP(w) the pointer where wis appended to.\nAlgorithm 6 Pruned Pivot-Update (a, b)\n1:Update Nk(b) as described in Section 7.2.\n2:A-Node-Update (b)\n3:ifbqueries ( b, a):\n4:forevery w∈Q−1\nP(b):\n5: A-Node-Update (w)\n7.1 Running Time for our Dynamic Algorithm\nMaintaining Q−1\nP(u).For each u∈QP(w), updating double-linked list Q−1\nP(u) is done in O(1) time –\nappending takes O(1) time, while the removal also takes O(1) by using the pointers stored in QP(w).\nNow we upper-bound the expected running time of Pruned Pivot-Update (a, b). In that, we use the\nfollowing claim.\nLemma 7.1. The number of nodes in Q−1\nP(b)processed by Algorithm 6 within the ifcondition is O(1)in\nexpectation.\nProof. Ifbqueries ( b, a), then each node that queries balso queries ( b, a). Hence, the number of nodes in\nQ−1\nP(b) equals the number of nodes querying ( b, a). By Theorem 4.10, that number in expectation is at most\n2. Observe that Theorem 4.10 provides an upper-bound for Pivot . Hence, Pruned-cluster might query\n(b, a) only less frequently.\nLemma 7.2. Pruned Pivot-Update (a, b)takes O(k)amortized time in expectation per an edge update.\nProof. We show that maintaining Nk(b) takes O(logk) amortized time in expectation in Section 7.2.\nBy Lemma 7.1, in expectation, only O(1) nodes ware processed within the ifcondition. In addition to\nthem, bis also processed. For each such worbthe following is performed:\n15•Invocation of Pruned-cluster . This invocation visits O(k) nodes. To traverse neighbors of a node u,\nPruned-cluster usesNk(u). Even though Nk(u) is organized as a binary balanced tree, all its nodes\ncan be traversed in the rank-decreasing order in O(1) time per node.\n•Updating Q−1\nP(u) foru∈QP(w). As described above, this is done in O(1) per u. By design of Pruned\nPivot , we have |QP(w)| ≤k.\n7.2 Maintaining Nk(u)\nThis section describes how to dynamically maintain Nk(u) with O(logk) update time in expectation. To\nmaintain Nk(u), as the first step, the algorithm organizes the neighbors of uas we describe next. At each\nalgorithm step, we associate ˜duwith u. In particular, ˜duis such that the current degree of uis in the range\n(1/4·˜du,4·˜du). In other words, ˜duis a 4 approximation of the degree of u. Moreover, all the neighbors\nofuare placed in bu=⌈˜du/(80k)⌉buckets numbered 1 , . . . , b u. A neighbor wofuis placed in the bucket\njsuch that ( j−1)·n/bu< π(w)≤j·n/bu. Hence, a bucket corresponds to n/buconsecutive integers.\nThis is convenient as if uhasd(u) neighbors, then under randomly chosen πit holds that in expectation\nd(u)/n·n/bu=d(u)/buneighbors are in a given bucket. With at least two buckets, this ratio is in the range\n(10k,320k). When there is a single bucket only, it has at most 320 kelements by definition of buand˜du.\nEach bucket is organized as an ordered balanced binary tree. Therefore, if a bucket contains telements,\nthen insertion, deletion, and finding the i-th-rank node can be done in O(logt) time. We use Buto refer\nto these buckets for node v. We note that the number of buckets might change over time. We discuss that\ntowards the end of this section.\nEdge insertion. If a new edge {u, x}is inserted, then the algorithm adds xto the bucket in Bucorresponding\ntoπ(x). In expectation, that bucket has Θ( k) nodes. Hence, this operation is done in O(logk) time in\nexpectation. Our algorithm also checks whether xhas a higher rank than the lowest rank node in Nk(u). If\nthat is the case, it removes the smallest-rank node from Nk(u) and inserts x. This is done in O(logk) time.\nEdge deletion. If an edge {u, x}is deleted, then the algorithm first removes xfrom the bucket corresponding\ntoπ(x). In expectation, that bucket has Θ( k) nodes. Hence, by the concavity of the logfunction, this\noperation is done in O(logk) time in expectation. Second, if xdoes not belong to Nk(u), then the algorithm\ndoes nothing else. Otherwise, the algorithm removes xfrom Nk(u) and finds the k-th highest-ranked element\ninBu. It does so in the following way: it visits bucket by bucket in the decreasing order of ranks until it\nreaches a bucket containing the desired element. We now analyze the complexity of this search.\nLetYbe a random variable representing the cost of this search. Let YBbe the time spent searching\nbucket B; we count 1 even if Bis accessed but empty. Then, Y=P\nB∈BuYB. Let Bjdenote the j-th bucket\ninBu. Let Zibe the event that the buckets B1. . . B icontain less than kelements in total. We have\nE\u0002\nYBj\u0003\n=E\u0002\nYBj|Zj−1\u0003\n·Pr (Zj−1) +E\u0002\nYBj| ¬Zj−1\u0003\n·Pr (¬Zj−1).\nObserve that E\u0002\nYBj| ¬Zj−1\u0003\n= 0, as no search is performed on Bjif the buckets B1. . . B j−1contain at\nleast kelements. This effectively implies that\nE\u0002\nYBj\u0003\n=E\u0002\nYBj|Zj−1\u0003\n·Pr (Zj−1). (5)\nAlso, we have that\nE[|Bj| |Zj−1]∈O(˜du/(bu−(j−1))).\nHence,\nE\u0002\nYBj|Zj−1\u0003\n∈O(1 + log ( ˜du/(bu−(j−1)))). (6)\n16Next, we upper-bound Pr ( Zj−1). Definition of Ziimplies\nPr (Zi)≤Pr (|Bi|< k|Zi−1)·Pr (Zi−1) (7)\n=iY\nt=1Pr (|Bt|< k|Zt−1). (8)\nFori= 1, we have E[|B1| |Z0]≥d(u)/bu≥10k; the latter inequality follows by our discussion above. For\ni >1, we have\nE[|Bi| |Zi−1]≥d(u)−k\nbu−(i−1)≥E[|B1| |Z0].\nThe latter inequality can be easily verified algebraically, but also it is an easy observation that it holds as\nthe first i−1 buckets in expectation contain at least 10 k(i−1). Hence, d(u)−knodes distributed over\nbu−(i−1) buckets yield the bucket-average higher than 10 k.\nCondition on Zi−1. Then, let Xwbe a 0 /1 random variable that equals 1 if and only if the neighbor wof\nuis inBi; in particular, E[|Bi| |Zi−1]=EhP\nw∈N(u)Xw|Zi−1i\n. Observe that the random variables Xare\nnegatively correlated. Hence, we can apply Chernoff bound to upper-bound the probability that |Bi|< k.\nSince E[|Bi| |Zi−1]≥10k, we have that\nPr (|Bi|< k|Zi−1)≤e−k.\nPlugging this into Equation (7), we derive\nPr (Zi)≤e−ik. (9)\nWe now turn back to computing E[Y]. By plugging Equation (6) and Equation (9) into Equation (5), we\nobtain\nE[Y] =buX\nj=1E\u0002\nYBj\u0003\n(10)\n≤buX\nj=1e−k(j−1)·O\u0010\n1 + log ( ˜du/(bu−(j−1)))\u0011\n.\nTo upper bound Equation (10), we first let tu=˜du/bu. By definition, tu∈O(k). Observe that\nbu≤(bu−(j−1))·j\nwhen jranges in the interval [1 , bu]; the minimum is achieved for j= 1 and j=bu. This implies that\n˜du/(bu−(j−1)) = tu·bu/(bu−(j−1))≤tu·j.\nFrom Equation (10), this yields the upper-bound\nE[Y]≤O\nbuX\nj=1e−k(j−1)·(1 + log ( kj))\n\n=O(1) + O\nbuX\nj=1e−k(j−1)logk\n\n+O\nbuX\nj=1e−k(j−1)logj\n\n17≤O(1) + O(logk) +O\nbuX\nj=1e−jj\n\n≤O(1) + O(logk) +O(1)\n=O(logk),\nas desired. To upper-bound O\u0010Pbu\nj=1e−jj\u0011\nwe used that it holdsP∞\ni=1i/2i= 2.\nUpdating Buwhen du/˜du/∈(1/4,4).We first describe how to address this change in O(logk) amortized\nexpected time and then explain how to de-amortize this.\nWhen dubecomes ˜du/4 or 4 ˜du, then our algorithm updates the current ˜duto˜d′\nu, and re-creates Bufor\n˜d′\nu. Ifdu≤˜du/4, then ˜d′\nu←˜du/2. Similarly, if du≥4˜du, then ˜d′\nu←2˜du. This approach is standard and\ntypically illustrated through the example of dynamic arrays. For our problem, this technique yields amortized\nO(logk) expected update time.\nIf memory allocation takes O(1) time, this technique can be de-amortized. This de-amortization is\nstandard, but we provide a couple of sentences of explanation for the sake of completeness. Instead of creating\nall the buckets from scratch when du=˜du/4 ordu= 4˜du, a de-amortized algorithm does that gradually.\nThat is, as soon as it starts updating the buckets for the current ˜duvalue, it allocates the memory for buckets\nfor both ˜d′\nu=˜du/2 and ˜d′\nu= 2˜du; only an allocation is performed, without any initialization. The algorithm\nalso maintains a variable IDso-far, which, when ˜duis updated, is initialized to 0.\nOn a new neighbor wupdate, if πw≤IDso-far, the algorithm carries over that update for all three bucket\nstructures. If πw>IDso-far, the algorithm only updates the ˜du-buckets. In addition, the algorithm considers\n10 elements from the ˜du-buckets with smallest πbut greater than IDso-far values, and copies them to the\n(˜du/2)- and (2 ˜du)-buckets. The value of IDso-far is increased properly so to correspond to the last element\ncopied from ˜du-buckets.\n18Figure 3: Comparison of correlation clustering cost for Pivot ,R-Pivot , Narrow- Pivot andPruned Pivot .\nThe optimal clustering has an expected cost of less than 17970.\n8 Empirical Evaluation\nIn this section, we conduct a simple empirical assessment of our algorithm. We compare the correlation\nclustering cost of Pruned Pivot to that of Pivot , R-Pivot of Behnezhad et al. (2022), and Narrow- Pivot\nof Chakrabarty & Makarychev (2023) on synthetic graphs.\nSet-Up. We use stochastic block model graphs generated as follows: Each sample graph has three partitions,\neach with 200 nodes. The probability of the appearance of an edge inside each partition is 0 .9, and between\npartitions is 0 .1.\nWe generate 100 graphs and run Pivot ,R-Pivot , Narrow- Pivot , and Pruned Pivot on these graphs\nwith parameter Rranging from 2 to 30 (Here Ris the parameter kforPruned Pivot ). Note that Pivot\ndoes not depend on R. For each R, we take the mean error of these runs for each algorithm (see Figure 3). We\nremove the standard deviation of the error for figure readability; the std is very similar for all the algorithms\nforR >13 and is around 2300.\nResults. First note that the clustering that puts each partition in one cluster achieves an expected cost of\n17970, which is an upper bound on the average optimum value. We selected edge probabilities 0 .1 and 0 .9\npartly to approximate the optimal clustering cost, as computing the exact value is NP-hard. So, Pivot ’s\napproximation factor is at least 2 .35 in this case. Furthermore, the trivial clustering that puts each node in\none cluster results in 65730 average error which is significantly more than the error of the other algorithms.\nWe observe that all three Pivot variants converge fast to the cost of Pivot . 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In Proceedings of the\nAAAI conference on artificial intelligence , volume 35, pp. 4564–4572, 2021.\nYoshida, Y., Yamamoto, M., and Ito, H. An improved constant-time approximation algorithm for maximum˜\nmatchings. In Proceedings of the forty-first annual ACM symposium on Theory of computing , pp. 225–234,\n2009.\n21A Equivalence between Pivot and Sequential Pivot\nIn Algorithm 7 we recall the Pivot algorithm from Ailon et al. (2008).\nAlgorithm 7 Pivot\n1:function Pivot (G= (V, E))\n2:ifV=∅:\n3: terminate\n4: Pick a random pivot u∈V.\n5: Cluster together uand its neighbors.\n6: LetHbe obtained by removing uand its neighbors from G.\n7: Pivot (H)\nLemma A.1. Algorithm 7 and Algorithm 3 are equivalent\nProof. Instead of choosing a random pivot ueach time on Line 4 of Pivot (Algorithm 7), we assume that\nbefore the algorithm is invoked a random permutation πover the input nodes is chosen. Then, Line 4 is\nimplemented by choosing the first, with respect to π, node available in Vin that invocation of Pivot .\nWe prove the lemma by induction on π. First, consider the node with rank 1, i.e., let π(u) = 1. uis a\npivot in Pivot , and since uhas no higher ranked neighbors, it executes line 8 in Sequential Pivot and\nhence is a pivot there as well.\nSuppose that for some t≥1, both Pivot andSequential Pivot cluster all nodes with ranks 1 , . . . , t\nthe same way. Suppose π(u) =t+ 1. First, suppose that uis a pivot in Pivot . Then it must be that udoes\nnot have any pivot neighbor with a higher rank: if there is such node v, then when vis being clustered, uis\nput in the cluster of vin Line 5 of Algorithm 7. This means that in Sequential Pivot uis also a pivot.\nNow, suppose that uis not a pivot in Pivot . Let vbe the pivot of uinPivot . Then it must be that vis\nthe highest ranked pivot in the neighborhood of u: if there is a higher ranked pivot v′in the neighborhood of\nu, then when v′is being processed (before v),uis put in the cluster of v′in Line 5 of Algorithm 7. This\nmeans that in Sequential Pivot , when uis being processed, Line 7 is executed when the neighbor vofuis\npicked.\n22"    },    {        "title": "2402.15672v1.Chaotic_tides_as_a_solution_to_the_Hyperion_problem.pdf",        "content": "Chaotic tides as a solution to the Hyperion problem\nMax Goldberga,∗, Konstantin Batyginb\naDepartment of Astronomy, California Institute of Technology, 1200 E. California Blvd, Pasadena, CA 91125, USA\nbDivision of Geological and Planetary Sciences, California Institute of Technology, 1200 E. California Blvd, Pasadena, CA 91125, USA\nARTICLE INFO\nKeywords :\nSaturnian satellites\nNatural satellite evolution\nTides\nOrbital resonancesABSTRACT\nThe dynamics of the outer regular satellites of Saturn are driven primarily by the outward migration\nof Titan, but several independent constraints on Titan’s migration are difficult to reconcile with\nthe current resonant orbit of the small satellite Hyperion. We argue that Hyperion’s rapid irregular\ntumblinggreatlyincreasestidaldissipationwithasteepdependenceonorbitaleccentricity.Resonant\nexcitationfromamigratingTitanisthenbalancedbydampinginafeedbackmechanismthatmaintains\nHyperion’s eccentricity without fine-tuning. The inferred tidal parameters of Hyperion are most\nconsistent with rapid Titan migration enabled by a resonance lock with an internal mode of Saturn,\nbut a scenario with only equilibrium dissipation in Saturn is also possible.\n1. Introduction\nMuchlikeminiatureplanetarysystems,theregularsatel-\nlites of Saturn are expected to have originated on nearly\ncoplanarandcircularorbitswithinthecircumplanetarydisk\nor the planet’s rings. Since their formation, tidal dissipation\nwithin Saturn has caused the moons to migrate outwards\nand encounter mean motion resonances with each other. In\nsomecases,pairsofmoonscapturedintothesemeanmotion\nresonances and remain there today, while in others there is\nindirect evidence of the excitation caused by resonant en-\ncounters.Assuch,resonantdynamicsofferauniquewindow\nintothesystem’sevolutionarypast.Inthetightlypackedin-\nnersaturniansystem(i.e.interiortoTitan),acomplexwebof\nresonancessetsstrictconstraintsontherelativemigrationof\neachmoon(Ćuketal.,2016a;ĆukandElMoutamid,2023).\nIn contrast with the well-understood dynamical history of\nthe inner moons, the Titan–Hyperion system is strikingly\nenigmaticandoneofthemostremarkablemysteriesofSolar\nSystem dynamics.\nHyperion, the only satellite in the large gap between\nTitanandIapetus,istrappedinanexterior4:3mean-motion\nresonance with Titan. The origins of this orbital config-\nuration have historically been attributed to an outwardly\nevolving Titan capturing Hyperion into commensurability\n(Colombo et al., 1974). This scenario is accompanied by\nspecific consequences: preservation of the adiabatic invari-\nant (Henrard, 1982) implies that, assuming no dissipation\nwithinHyperion,Titanmustmigrate 4%insemi-majoraxis\npost-capture(Ćuketal.,2013)andthusthetidal 𝑄ofSaturn\nmust be𝑄Æ<1500.1\nThere are, however, contradicting constraints on Titan’s\nmigration from Iapetus, the outermost regular satellite of\n∗Corresponding author\nmg@astro.caltech.edu (M. Goldberg)\nORCID(s):0000-0003-3868-3663 (M. Goldberg); 0000-0002-7094-7908\n(K. Batygin)\n1However, too much migration of Titan is problematic: if Saturn has\nespecially strong dissipation ( 𝑄Æ<500), Titan and Hyperion would have\nstarted wide of the 3:2 resonance and captured into the wrong resonance.Saturn.Iapetusliesjust 0.4%insidethe5:1mean-motionres-\nonancewithTitan,implyingarecentbutsignificantdynam-\nicalinteractionbetweenthemoons.Duringthe5:1resonant\nencounter,theeccentricityandinclinationofIapetusevolve\nchaotically and Titan’s migration must be rapid enough to\navoid ejecting Iapetus (Ćuk et al., 2013; Polycarpe et al.,\n2018). Evidently, the preservation of Iapetus necessitates\nrapid migration of Titan, while Hyperion’s resonance de-\nmands slow, short-range migration.\nHyperion’srotationalpropertiesareequallyremarkable,\nand constitute a unique example of stochastic rotation in\nthe Solar System. Wisdom et al. (1984) predicted that it\nwould be in a chaotically tumbling state before its rotation\nwas directly observed. They argued that given its moderate\neccentricity ( 𝑒≈ 0.1) and highly elongated shape (seen\nin Voyager 2 images), regular rotation in the synchronous\n(1:1) or 3:2 spin-orbit state is impossible. Instead, a chaotic\nzone surrounds the 1:1 and 2:1 spin-orbit resonances and\nHyperion’s spin vector evolves over timescales of a few or-\nbitalperiods.Inaddition,Wisdometal.(1984)demonstrated\nthat much of the parameter space is attitude unstable, so\nthat an initial small obliquity is quickly amplified and rota-\ntion inevitably occurs on all three axes. Early ground-based\nlightcurveobservationsbyKlavetter(1989)confirmednon-\nperiodic rotation and suggested rotation at roughly the syn-\nchronous rate.\nDespite the remarkable predictive power of Wisdom\net al. (1984)’s calculations, images and light curves taken\nduring the Voyager 2 and Cassini visits to the saturnian\nsystemdemonstratedthatHyperionwasrotatingmuchfaster\nthan expected: roughly 4.2 times the synchronous rate\n(Thomasetal.,1995;Harbisonetal.,2011).Rotationmostly\noccurs around the longest axis and is quasi-regular. Never-\ntheless,thewobbleandprecessionareindeedclearlychaotic,\nwith typical Lyapunov times of several orbital periods, as\nmeasured by Black et al. (1995) and Harbison et al. (2011).\nUsingnumericalsimulations,Blacketal.(1995)showedthat\nthis state was not an unexpected outcome—initialized near\nthe synchronous state, Hyperion would irregularly alternate\nGoldberg & Batygin: Preprint submitted to Elsevier Page 1 of 11arXiv:2402.15672v1  [astro-ph.EP]  24 Feb 2024Hyperion rotation and orbit\nbetween slower chaotic tumbling and more rapid quasi-\nregular rotation, the latter being the state actually observed\nby Klavetter (1989) and Harbison et al. (2011).\nIn light of the discrepancies between the predicted rota-\ntional behavior of Hyperion and its observed state, as well\nas an unclear relationship between the Titan migration rate\nandtheTitan-Hyperionmean-motionresonance,acomplete\nunderstanding of the Hyperion problem remains elusive.\nPrevious work has generally considered the rotation to be\nsolely a consequence of the orbit and neglected the im-\npact of dissipation within Hyperion on its orbital evolution.\nWe argue that tidal dissipation within Hyperion is non-\nnegligible due to its rapid rotation, and in fact mediates\nitsorbitaleccentricitygrowthdespiteresonantforcingfrom\nTitan’s migration. As a result, several fine-tuning problems\nare avoided and the tidal quality factor of Hyperion can be\nestimated. We begin in Section 2 by studying the chaotic\nand quasi-regular rotation of Hyperion and calculating the\nresulting tidal dissipation. Then, in Section 3, we use this\nnewpictureofdissipationwithinHyperiontosetconstraints\non the range and rate of Titan’s outward migration. Finally,\nSection4discussestheimplicationsofthisproposeddynam-\nical history of Hyperion.\n2. Rotational dynamics of Hyperion\n2.1. Numerical procedure\nTo investigate its rotational dynamics, we numerically\nmodeled the spin and orientation of Hyperion under the\neffect of Saturn’s gravity. The satellite is assumed to be in\na fixed elliptical orbit around Saturn with eccentricity 𝑒and\ntrue anomaly 𝑓. The units are chosen such that the semi-\nmajoraxis𝑎isunity,theorbitalperiodis 2𝜋,and𝐺𝑀Æ=1,\nwhere𝑀ÆisthemassofSaturn.Hyperionismodeledasan\nellipsoid with principal moments of inertia 𝐴<𝐵<𝐶 and\nits spin is represented by 𝜔𝑎,𝜔𝑏, and𝜔𝑐, the projections of\nthe spin vector on the principal axes, so that the total spin\nrate is|𝜔|=√\n𝜔2\n𝑎+𝜔2\n𝑏+𝜔2\n𝑐. The spins evolve according\nto Euler’s equations,\ṅ 𝜔𝑎=𝐵−𝐶\n𝐴(\n𝜔𝑏𝜔𝑐−3\n𝑟3𝛽𝛾)\n(1)\ṅ 𝜔𝑏=𝐶−𝐴\n𝐵(\n𝜔𝑐𝜔𝑎−3\n𝑟3𝛾𝛼)\n(2)\ṅ 𝜔𝑐=𝐴−𝐵\n𝐶(\n𝜔𝑎𝜔𝑏−3\n𝑟3𝛼𝛽)\n(3)\nin which the external torque is provided by the gradient of\nthegravitationalfieldofSaturn(MurrayandDermott,1999).\nHere,𝑟istheinstantaneousHyperion–Saturndistanceand 𝛼,\n𝛽,and𝛾arethedirectioncosinesbetweentheprincipalaxes\nand the direction of Saturn.\nTo represent the orientation of Hyperion, we use the\nquaternion formalism, which avoids the coordinate singu-\nlarities that appear when using Euler angles (Mel’nikov,\n2020). The four quaternion components 𝜆0,𝜆1,𝜆2, and𝜆3arenormalizedandevolveaccordingto(Arribasetal.,2006)\ṅ𝜆0=1\n2(−𝜆1𝜔𝑎−𝜆2𝜔𝑏−𝜆3𝜔𝑐) (4)\ṅ𝜆1=1\n2(𝜆0𝜔𝑎−𝜆3𝜔𝑏+𝜆2𝜔𝑐) (5)\ṅ𝜆2=1\n2(𝜆3𝜔𝑎+𝜆0𝜔𝑏−𝜆1𝜔𝑐) (6)\ṅ𝜆3=1\n2(−𝜆2𝜔𝑎+𝜆1𝜔𝑏+𝜆0𝜔𝑐). (7)\nThe direction cosines are given by\n𝛼=(𝜆2\n0+𝜆2\n1−𝜆2\n2−𝜆2\n3)cos𝑓+2(𝜆0𝜆3+𝜆1𝜆2)sin𝑓\n(8)\n𝛽=2(𝜆1𝜆2−𝜆0𝜆3)cos𝑓+(𝜆2\n0−𝜆2\n1+𝜆2\n2−𝜆2\n3)sin𝑓\n(9)\n𝛾=2(𝜆0𝜆2+𝜆1𝜆3)cos𝑓+2(−𝜆0𝜆1+𝜆2𝜆3)sin𝑓.\n(10)\nWe use the moment of inertia parameters 𝐴= 0.314,𝐵=\n0.474,𝐶=0.542estimatedbyHarbisonetal.(2011)froma\nCassinishapemodel.WethennumericallyintegrateEqs.1–\n7 with a fifth-order Radau IIA method using relative and\nabsolute tolerances of 10−6and10−10, respectively. We ran\n10 integrations for 3×106orbits with orbital eccentricities\nranging uniformly in log-space from 0.01to0.631. Each\nwas started at synchronous rotation ( |𝜔|= 1) but with\nan obliquity of 1◦to induce tumbling (Black et al., 1995).\nAlthoughwobbledamping(BurnsandSafronov,1973;Wis-\ndom, 1987) may be relevant on such long timescales, the\npurposeofthesesimulationsistodeterminetherangeoftyp-\nical rotational dynamics; long integrations are more likely\nto capture rare behavior and less likely to be trapped in\n“smalltributariesofthechaoticzone,”asnoticedbyWisdom\n(1987). For comparison, we also integrated the rotation of\nHyperion from its observed state on 2005–06–10 for 103\norbits, using the orientation and spin vectors reported by\nHarbison et al. (2011).\n2.2. Rotational evolution of Hyperion\nOne example of the longer integrations is shown in\nFigure 1, where we have chosen 𝑒= 0.1. Hyperion begins\ninachaotictumblingstate,shadedinorangeontheplot,but\nintermittentlypassesthroughquasi-regularstates,shadedin\nblue. As noted by Black et al. (1995), quasi-regular states\nare typically associated with rotation primarily on axes 𝑎\nor𝑐(𝑏is not stable owing to the intermediate axis theo-\nrem). The right panel of Figure 1 shows the distribution of\nangular speeds in these two regimes. Chaotic tumbling is\nsmoothly distributed across all values of |𝜔|≲5. However,\nquasi-regular rotation is faster and dominated by peaks at\ndiscrete values of |𝜔|that correspond to half-integer spin-\norbitresonances.Athigher |��|,thepeakslieslightlywideof\nexactresonance.Inparticular,thestateofHyperionin2005\n(shadedingray)matchesthehighestpeak,whichappearsto\nbe associated with the 9∕2resonance.\nThe clustering behavior near spin-orbit resonance is\nprobably a consequence of the resonant “sticking” effect\nGoldberg & Batygin: Preprint submitted to Elsevier Page 2 of 11Hyperion rotation and orbit\n0246||\n0 2 4 6\n||\nRelative Frequencyp=13/225/237/249/2511/2\n1\n01a/||\n1\n01b/||\n0 25000 50000 75000 100000 125000 150000 175000\nt (years)1\n01c/||\nFigure 1: Left: A typical integration of the rotational equations of Hyperion for 3×106orbits, starting from a nearly synchronous\nstate and a realistic orbital eccentricity of 0.1. Chaotic tumbling (shaded orange) intermittently gives way to quasi-regular rotation\n(blue). Right: the distribution of |𝜔|in each state. The two distributions are shown to scale relative to each other. Dashed vertical\nlines mark spin-orbit resonances. The shaded gray region is the 2𝜎range of Hyperion’s rotation speed in its observed state in\n2005.\n(Karney, 1983; Meiss, 1992; Shevchenko, 1999). In the\nvicinityoftheseparatrixthatboundslargeresonantislands,\nthere are numerous small islands of secondary resonances.\nChaotic trajectories which wander near the separatrix may\nbe caught in one of these islands, which necessarily lie in\nproximity to the resonance. The trajectory will then evolve\nvery slowly through action space and the rotation will be in\na quasi-regular state for an extended duration.\nSimulations at other eccentricities were qualitatively\nsimilartothe 𝑒=0.1case.Alternationbetweenchaotictum-\nbling and quasi-regular rotation near spin-orbit resonances\nwasobservedatalltheeccentricitieswetested.Thetumbling\nstate accounted for 30–60% of the total duration, with no\nclear dependence of that fraction on eccentricity.\nHowever, the typical rotation speed in the long inte-\ngrations shows a strong dependence on orbital eccentricity\n(Wisdom, 1987; Quillen et al., 2020). Denoting the time\naverage of |𝜔|as⟨𝜔⟩, the typical ⟨𝜔⟩was much higher\nfor higher𝑒during both chaotic tumbling and quasi-regular\nrotation.Figure2showsthe ⟨𝜔⟩asafunctionofeccentricity\nfor the long simulations. We also computed the mean ⟨𝜔⟩\nduringthechaotictumblingonly,byremovingthetimeswith\nquasi-regular motion as in Figure 1. Both are fit well by\nan exponential dependence on 𝑒. Without removing quasi-\nregularmotion,wefind ⟨𝜔⟩≈2.59×1.42𝑒∕0.1.Considering\nchaotic tumbling only, we obtain ⟨𝜔⟩≈ 2.03×1.38𝑒∕0.1.\nFinally,forreasonsthatwillbecomeapparentinSection2.4,\nwefitthefourthrootofthetimeaverageof |𝜔|4asafunction\nof𝑒in the same way, finding ⟨𝜔4⟩1∕4≈ 2.88×1.37𝑒∕0.1\nand⟨𝜔4⟩1∕4≈ 2.27×1.38𝑒∕0.1for all rotation and chaotic\ntumbling, respectively.\n0.01 0.03 0.10 0.30\nOrbital eccentricity1.03.010.0 or pmax\n1D analytical estimate pmax(e)\nFit to simulations\nFit to simulations (tumbling only)Figure 2: Average rotation speed of Hyperion as a function of\nits orbital eccentricity. Black dots are the full long integrations\nand the black line is an fit with an exponential dependence\non eccentricity. Orange is the same but considering only the\nchaotic tumbling state, removing the quasi-regular rotation\n(see Figure 1). The magenta curve is the analytical estimate of\n𝑝𝑚𝑎𝑥from solving Eq. 14. The analytic solution, despite being\noffset from the numerical results by a factor of ∼2, captures\nthe qualitative behavior of ⟨𝜔⟩as a function of 𝑒.\n2.3. Analytical rotation model\nTo qualitatively understand the eccentricity dependence\nof⟨𝜔⟩seen in our simulations, it is instructive to consider\na simplified one-dimensional model of Hyperion’s spin,\neven though its rotation is fully three-dimensional. This\nGoldberg & Batygin: Preprint submitted to Elsevier Page 3 of 11Hyperion rotation and orbit\nsimplification ignores the obliquity and spin precession of\nHyperion, but can be studied analytically in much more\ndetail.FollowingWisdometal.(1984),assumethesatellite’s\nspinaxisisperpendiculartoitsorbitalplane.Theorientation\nof the satellite in the inertial frame is given by 𝜃. Then,𝜃\nevolves according to\n𝑑2𝜃\n𝑑𝑡2+𝜔2\n0\n2𝑟3sin2(𝜃−𝑓)=0 (11)\nwhere𝜔2\n0= 3(𝐵−𝐴)∕𝐶(Goldreich and Peale, 1966).\nEquation 11 is unwieldy because 𝑟and𝑓are complicated\nfunctionsoftime.However,itcanbeexpandedviaaFourier\nseries into\n𝑑2𝜃\n𝑑𝑡2+𝜔2\n0\n2∞∑\n𝑝=−∞𝐻(𝑝,𝑒)sin(2𝜃−2𝑝𝑡)=0 (12)\nwhere𝑝is a half-integer and the 𝐻(𝑝,𝑒)are coefficients\ngiven by\n𝐻(𝑝,𝑒)=1\n2𝜋∫2𝜋\n01\n𝑟3cos(2𝑝𝑡−2𝑓)𝑑𝑡, (13)\nwhich,for𝑒≪1and𝑝≲5isoforder𝐻(𝑝,𝑒)∼22𝑝−1𝑒2𝑝−2\n(Dobrovolskis, 1995). As is well-known (Goldreich and\nPeale,1966),Equation12pointsattheexistenceofadiscrete\nset of spin-orbit resonances in which 𝑑𝜃∕𝑑𝑡≈𝑝. For\nexample, the 3:2 spin-orbit state of Mercury corresponds to\n𝑝= 3∕2, or three rotations (in the inertial frame) for every\ntwo orbits (Goldreich and Peale, 1966). The half-width of\nthe spin-orbit resonance in frequency space is 𝜔0√\n𝐻(𝑝,𝑒),\nincreasing with 𝑒and decreasing with 𝑝. According to the\nresonanceoverlapcriterion(Chirikov,1979),chaoticbehav-\nior arises when neighboring resonances, (whose widths can\nbecalculatedtoleadingorderasiftheywereunperturbedby\neach other), would overlap. Thus, chaos will appear around\nthe𝑝and𝑝+1∕2resonances if\n𝜔0√\n𝐻(𝑝,𝑒)+𝜔0√\n𝐻(𝑝+1∕2,𝑒)≳1\n2. (14)\nWisdom et al. (1984) use Eq. 14 and the two widest reso-\nnances,𝑝=1and𝑝=3∕2, to generate a general condition\nfor the existence of a broad chaotic region and argue that\nHyperion must be in it.\nTurning this argument around, we can also ask, for a\ngiven𝑒and𝜔0,whatisthehighest 𝑝maxforwhichresonances\noverlapsuchthatthereisachaoticseasurroundingthe 𝑝max\nand𝑝max+1∕2resonances? Because of the dependence of\n𝐻(𝑝,𝑒)on𝑝, the chaotic sea will also extend for at least\n1≤𝑝≤𝑝max+ 1∕2, and a trajectory initialized near\n𝑝= 1will eventually explore up to 𝑝maxergodically. We\nsolveEq.14numericallyfor 𝑒byselectinga 𝑝maxandtaking\n𝜔0=0.94,correspondingtothevaluesof (𝐴,𝐵,𝐶)weused\nin the 3D simulations. The result is shown as the magenta\ncurve in Figure 2. The expression for 𝐻(𝑝,𝑒)ensures that\nthe size of the chaotic sea, and thus 𝑝max, grow steeply with\n𝑒(Wisdom, 1987). Our analytical model closely matchesthe steep dependence on 𝑒found in the numerical simula-\ntions, although the 1D model consistently underestimates\n⟨𝜔⟩. Evidently, the chaotic region is larger in 3D, and thus\nemerges at a smaller eccentricity for a given 𝑝. Indeed, the\nnotionthattheonsetofchaosoccursearlierinsystemswith\nmore degrees of freedom is qualitatively expected (see, e.g.\nMorbidelli, 2002).\nThe steep dependence of Hyperion’s rotation on its or-\nbitaleccentricityhasanimportantconsequence.Hyperion’s\neccentricityisresonantlyexcitedbyanoutwardlymigrating\nTitan, and in the absence of additional forces, increases\nmonotonically. Hence, we expect that the spin rate of Hy-\nperionhas grownovertime,incontrasttomostbodiesinthe\nSolar System.\n2.4. Tidal dissipation\nSome of the energy of the time-varying tidal torque is\ndissipated within Hyperion. The two main effects of the\ndissipation are despinning and eccentricity damping. The\ndespinning timescale at the present orbit is roughly of the\nordertheageoftheSolarSystem(Wisdometal.,1984).The\neccentricity damping timescale is usually much larger than\nthe despinningtimescale andhas thereforebeen ignored for\nHyperion. We will examine this in more detail.\nForasynchronouslyrotatingsatellitewithloweccentric-\nity, the eccentricity damping rate is given by\n𝜏−1\n𝑒,sync≡−1\n𝑒𝑑𝑒\n𝑑𝑡=21\n2𝑘2,H\n𝑄H𝑀Æ\n𝑀H(𝑅H\n𝑎H)5\n𝑛H(15)\nwhere𝑘2,His the tidal Love number of Hyperion, 𝑄His\nits tidal quality factor, 𝑅His the average radius of Hype-\nrion, and𝑀H,𝑎Hand𝑛Hare the mass, semi-major axis\nand mean motion of Hyperion, respectively (Goldreich and\nSoter, 1966). While the exact values of 𝑘2,Hand𝑄Hare\nunknown,Hyperionisbelievedtobearubblepilewithhigh\ninternal porosity (Thomas et al., 2007). Goldreich and Sari\n(2009) suggest that for such an object, 𝑘2,H≲1×10−3and\n𝑄H≲100.Assuming𝑘2,Hisattheupperbound, 𝑅H=150\nkm,𝑀Æ∕𝑀H=1.0×108,andHyperion’scurrentperiodof\n21.28d, we obtain\n𝜏𝑒,sync≈8×1013(𝑄H\n100)\nyr, (16)\nmuch longer than the age of the Solar System.\nHowever, Hyperion is manifestly notrotating synchro-\nnously. In non-synchronous rotation, the entirety of the tide\nisraisedandloweredduringonecycle,greatlyenhancingthe\ndissipationofenergy.BurnsandSafronov(1973)arguethat\ntheenergydissipatedinanon-synchronousrotatorperorbit\nis, to an order of magnitude,\nΔ𝐸∼|𝜔|4𝑅5\nH\n𝐺𝑘2,H\n𝑄H(17)\nwhere we have written the equation in terms of 𝑘2,Hrather\nthan the rigidity (Goldreich and Sari, 2009). Energy dissi-\npatedduringchaotictumblingshouldbeatleastcomparable\nGoldberg & Batygin: Preprint submitted to Elsevier Page 4 of 11Hyperion rotation and orbit\nto, if not much larger than this estimate (Wisdom, 1987;\nBrasser, 2020). Because angular momentum is conserved,\nthis dissipation must drive circularization of the orbit, and\nthus the eccentricity damping rate for Hyperion’s irregular\nrotation,𝜏−1\n𝑒,Hwillbeenhancedoverthesynchronousrateby\nroughly\n𝜏−1\n𝑒,H\n𝜏−1\n𝑒,sync∼1\n𝑒2\nH⟨𝜔4⟩\n𝑛4\nH. (18)\nForcurrentvaluesofHyperion, 𝑒≈0.1and⟨𝜔4⟩1∕4∕𝑛H≈\n4and the enhancement is ∼ 2×104. Quillen et al. (2020)\nperformedsimulationsofaviscoelasticmodeloftheMartian\nsatellitesPhobosandDeimosandconfirmedthattheenergy\ndissipationrateduringepisodesofrapidtumblingwaslarger\nthan the dissipation during synchronous rotation by 3 to 5\norders of magnitude. In their case, the process is naturally\nquenched as the eccentricity is damped, rotation slows, and\nthe satellites capture into a synchronous state. In contrast,\nHyperion’seccentricityiscontinuouslyexcitedbyaresonant\ninteractionwithTitan(Section3)andrapidrotationdoesnot\ncease.\nWiththeenhancementfromrapidrotation,theexpected\neccentricity damping timescale of Hyperion is now 𝜏𝑒,H∼\n4 × 109yr, of order the age of the Solar System. While\nthis estimate should not be taken to be exact (see Wisdom,\n1987) because of order-unity constants dropped by Burns\nand Safronov (1973), the result is that circularization of\nHyperion’s orbit can no longer be ignored despite its con-\nsiderable distance from Saturn. In addition, because of the\nexponential dependence of |𝜔|on𝑒, the damping timescale\nvaries considerably with 𝑒, in contrast to the synchronous\ntimescale,whichisindependentof 𝑒.Thisisthecriticalpiece\ncoupling Hyperion’s rotation to its orbital history.\n3. Orbital dynamics of the Titan–Hyperion\nsystem\nThepresenceofthe4:3mean-motionresonancebetween\nHyperion and Titan is usually interpreted as resulting from\nthe outward migration of Titan (Colombo et al., 1974; Ćuk\net al., 2013). The resonance has a libration amplitude of\n36◦, with a forced eccentricity of 0.1, and one possible\nexplanationforthisstateistheexpansionofTitan’sorbitby\n4%sincetheinitialencounterwiththeresonance(Ćuketal.,\n2013).\nInlightoftheorbitalcouplingdiscussedabove,however,\nit is critical to exmaine alternative scenarios. Much in the\nsame manner as the Moon recedes from the Earth due to\ntidesraisedontheocean,Titanmigratesoutwardbecauseit\nraises a tidal bulge on Saturn. As Saturn rotates faster than\nTitan orbits, the tidal bulge transfers angular momentum\nfromSaturn’srotationtoTitan’sorbit.Therateofexpansion\nof Titan’s orbit is given by\n𝜏−1\n𝑎,Ti=−1\n𝑎Ti𝑑𝑎Ti\n𝑑𝑡=−3𝑘2,Æ\n𝑄Æ(𝑛Ti)𝑀Ti\n𝑀Æ(𝑅Æ\n𝑎)5\n𝑛Ti(19)where𝑄Æ(𝑛)is the tidal quality factor of Saturn at forcing\nfrequency𝑛and𝑘2,ÆistheLovenumberofSaturn,whichwe\ntake to be0.382(Lainey et al., 2020; Jacobson, 2022). The\nsign convention, consistent with Equation 15, means that\n𝜏𝑎,Ti<0corresponds to outward migration. Quantitative\npredictions for 𝑄Æare challengingand highlydependent on\nSaturn’sinternalstructure(OgilvieandLin,2004).Observa-\ntionally,𝑄Æ(𝑛)canbemeasuredbyobservationsofoutward\nmigrationoftheinnersaturnianmoons(Laineyetal.,2012).\nInterestingly, recent works have shown that 𝑄Æis not the\nsame for each of Saturn’s moons, with Rhea especially\nhavinghighertidaldissipation(Laineyetal.,2017).Indeed,\nsome tidal theories predict that 𝑄Æshould depend on the\nforcingfrequency 𝑛,insomecasesquitesensitively(Ogilvie\nand Lin, 2004; Fuller et al., 2016; Terquem, 2021). There-\nfore, the tidal quality factor relevant for Titan’s migration\ncannot be assumed to be the same as the one measured for\nanother of Saturn’s moons.\nRecently, two groups have reported conflicting mea-\nsurements of outward migration of Titan. Lainey et al.\n(2020) used astrometry and Cassini radio tracking to obtain\n𝑄Æ(𝑛Ti) = 124+26\n−19(3𝜎uncertainties) corresponding to a\nmigration timescale of 𝜏𝑎,Ti≈ 10Gyr . They interpret this\nresult as consistent with the resonant locking model, in\nwhich satellites couple to inertial modes within Saturn and\nmigrate outwards as the interior of Saturn evolves over the\nlifetime of the Solar System. Long-range migration enabled\nby the resonant locking mechanism is also consistent with\ntheexpectationthatTitanformedatormigratedtotheinner\nedge of the circumplanetary disk, near a period of ∼ 3d\n(Batygin et al., 2023). However, such rapid migration is\ndisputed by Jacobson (2022), who uses a large corpus of\ntracking, astrometric, and other data to obtain 𝑄Æ(𝑛Ti) =\n1224±119 (1𝜎uncertainties), or 𝜏𝑎,Ti≈100Gyr , an order\nof magnitude slower than Lainey et al. (2020).\nThe capture and evolution of Hyperion in its mean-\nmotion resonance with Titan depends on the specifics of\nTitan’s outward migration. For completeness, we consider\ntwocasesbelow:oneinwhichTitan’smigrationisconsistent\nwith the results of Lainey et al. (2020), and one in which it\nis consistent with the results of Jacobson (2022).\n3.1. Analytical Results\nTo leading order, the mean motion and eccentricity of\nTitan and Hyperion near the 4:3mean-motion resonance\nevolve according to (e.g., Terquem and Papaloizou, 2019)\ṅ 𝑛Ti=3𝑛Ti\n2𝜏𝑎,Ti+3𝑛Ti𝑒2\nTi\n𝜏𝑒,Ti(20)\ṅ 𝑛H=12𝑛2\nH𝑀Ti\n𝑀Æ(𝑒Ti𝑓1sin𝜙1+𝑒H𝑓′\n2sin𝜙2)+3𝑛H𝑒2\nH\n𝜏𝑒,H\n(21)\ṅ 𝑒Ti=−𝑒Ti\n𝜏𝑒,Ti(22)\ṅ 𝑒H=−𝑛H𝑀Ti\n𝑀Æ𝑓′\n2sin𝜙2−𝑒H\n𝜏𝑒,H(23)\nGoldberg & Batygin: Preprint submitted to Elsevier Page 5 of 11Hyperion rotation and orbit\nwhere we have assumed that Hyperion’s mass and tidal\nmigration rate are negligible. Here, 𝜙1=4𝜆H−3𝜆Ti−𝜛Ti\nand𝜙2=4𝜆H−3𝜆Ti−𝜛Hare the critical resonant angles\nand𝑓1and𝑓′\n2are order unity constants. To reduce these\nequationsfurther,wenotethatthecaptureintoresonanceof\nHyperion implies that ̇ 𝑛Ti∕𝑛Ti=̇ 𝑛H∕𝑛H. Additionally, be-\ncauseTitan’spericenterprecessionisdominatedbySaturn’s\n𝐽2,𝜙1circulates and the time average of sin𝜙1is zero. We\nthus obtain\n4̇ 𝑒H𝑒H=−3𝑒2\nH𝜏−1\n𝑒,H−(1\n2𝜏−1\n𝑎,Ti+𝑒2\nTi𝜏−1\n𝑒,Ti)\n(24)\nwhich,despitebeingsimpler,muststillbeintegratednumer-\nically because of the distance and eccentricity dependence\nof the migration and damping timescales. Nevertheless, the\ncompeting effects of eccentricity excitation and damping\nsuggestthatwecancomputeanequilibriumeccentricitythat\nHyperion will tend towards. Setting ̇ 𝑒H=0, we find\n𝑒2\neq,H=−1\n3(1\n2𝜏−1\n𝑎,Ti+𝑒2\nTi𝜏−1\n𝑒,Ti)\n𝜏𝑒,H. (25)\nBecause𝑒Ti≪1andfortypicaltidalprocesses, |𝜏𝑒|∼|𝜏𝑎|,\nthe second term in the brackets can be neglected. Recalling\nEq.18,theequilibriumeccentricitycanbetranslatedintoan\nequation for the equilibrium rotation rate of Hyperion,\n⟨𝜔4⟩\n𝑛4\nH≈−1\n6𝜏𝑒,sync\n𝜏𝑎,Ti. (26)\nWith the expected damping of Hyperion (Section 2.4)\nand the current rotation rate of Hyperion (averaged over\ntheseculareccentricitycycleandthetworotationregimes),\n⟨𝜔4⟩1∕4∕𝑛H≈ 4, we find, assuming Hyperion is at its\nequilibrium eccentricity,\n𝑄H≈20(|𝜏𝑎,Ti|\n1010yr)\n. (27)\nThus, if Hyperion is at its equilibrium eccentricity, the\nLainey et al. (2020) Titan migration measurement implies\n𝑄H∼ 20, while the Jacobson (2022) value implies 𝑄H∼\n200. As we will see below, more accurate estimates of 𝑄H\nare larger because Hyperion does not typically reach the\nequilibrium eccentricity.\n3.2. Numerical results\nAs a means of testing our analytical theory, we ran a\nsuiteofN-bodysimulationsmodelingtheoutwardmigration\nof Titan and tidal dissipation of Hyperion resulting from its\nrapidchaotictumbling.Previousworkhascoupledrotational\nand orbital integrations to study the spin-orbit evolution of\nirregular satellites (Ćuk et al., 2016b; Quillen et al., 2017,\n2020; Agrusa et al., 2021; Quillen et al., 2022). However,\nour objective is to demonstrate the feasibility of resonant\ncapture under enhanced tidal damping. Accordingly, we do\nnot repeat the rotational simulations but instead model tidal\ndissipation with an eccentricity damping term estimated\nusing the results of Section 2. Our numerical integrationsuse the whfastsymplectic integrator in the reboundN-body\npackage (Rein and Tamayo, 2015). The integrator timestep\nwas chosen to be 1∕20the initial orbital period of Titan.\nAdditional forces for migration and eccentricity damping\nwere incorporated with reboundx(Tamayo et al., 2020). The\nintegrationincludestheSunandthe 𝐽2momentofSaturn,to\nwhichisaddedtheaveragedorbitsofthesatellitesinteriorto\nTitan.Ateachtimestep,wecomputetheeccentricitydamp-\ning of Hyperion using its instantaneous eccentricity, 𝑒H.\nTo accomplish this, we estimate the average rotation speed,\n⟨𝜔4⟩1∕4≈ 2.88×1.37𝑒H∕0.1(Section 2), and then use Eqs.\n15 and 18 to determine the enhanced eccentricity damping\ntimescale,𝜏𝑒,H. We also compute the migration rate and\neccentricitydampingrateofTitanateachtimestepaccording\ntotheprescriptionsspecifiedinthefollowingsections.Then,\nduringeach“kick”stepofthe whfastalgorithm,weapplyan\nadditional force to each satellite of\n𝒂damp,Ti=−2(𝒗Ti⋅𝒓Ti)𝑟Ti\n(𝒓Ti⋅𝒓Ti)𝜏𝑒,Ti−𝒗Ti\n2𝜏𝑎,Ti(28)\n𝒂damp,H=−2(𝒗H⋅𝒓H)𝑟H\n(𝒓H⋅𝒓H)𝜏𝑒,H(29)\nwhere 𝒓𝑖and𝒗𝑖aretheradiusandvelocityvectoroftheparti-\nclerelativetoSaturn,respectively(PapaloizouandLarwood,\n2000).\nThesimulationinitialconditionswerechosentobecom-\npatiblewithavailableconstraints.AlthoughtheageofHype-\nrionisnotknownprecisely,itsloworbitalinclinationimplies\nthat it, or its parent object, formed in the circumplanetary\ndisk.Highcraterdensitiesarealsoconsistentwiththenotion\nthat Hyperion is quite old (Plescia and Boyce, 1983; Bottke\netal.,2023).Wethusranthesimulationsoveratimespanof\n4.5Gyr.HyperionwasplacedexteriortoTitanwithaninitial\nperiodratioof1.35inordertoavoidcaptureintothewrong\nresonance. The initial eccentricity was varied between 0\nand 0.05 and the initial inclination was set to 0 relative to\nSaturn’s equator. The other orbital angles were randomized\nuniformly.\nIt is important to note that the measurements of Lainey\netal.(2020)andJacobson(2022)areonlyofTitan’scurrent\nmigrationrate(baselinesof ≈150yr)andarenotnecessarily\nrepresentative of the previous behavior of Titan. Accord-\ningly,weattempttoconstructareasonablemigrationhistory\nof Titan in each case and include that in the simulation as\ndescribedbelow.Toensurefeasiblecomputationaltimes,we\nsped up integrations by dividing the migration timescale of\nTitan and the eccentricity damping timescales of Titan and\nHyperionbyacommonfactorof 104.Wedonotexpectthis\nto impact our results because the accelerated migration and\ndampingtimescalesstillgreatlyexceedtheotherdynamical\ntimescales in this problem (which are ≲100yr), ensuring\nthatadiabaticityduringtheresonantencounterispreserved.\n3.2.1. Rapid Titan Migration\nFirst, we assume to be true the results of Lainey et al.\n(2020), who find 𝜏𝑎,Ti≈ −10Gyr and argue that Titan is\nin a ‘resonant lock’ with an internal mode of Saturn (Fuller\nGoldberg & Batygin: Preprint submitted to Elsevier Page 6 of 11Hyperion rotation and orbit\net al., 2016). In such a regime, the migration of Titan is set\nbytheinteriorevolutionofSaturn,whichunfoldsroughlyon\nthe timescale of its age. Following Lainey et al. (2020), we\nhypothesize\n𝜏−1\n𝑎,Ti≈−𝐵\n𝑡(30)\nwhere, to match the current 𝜏𝑎,𝑇𝑖measurement, 𝐵∼ 1∕3.\nThis equation has the solution\n𝑎lock(𝑡)=𝑎0(\n𝑡\n𝑡0)𝐵\n(31)\nwhere𝑎0isTitan’scurrentsemi-majoraxisand 𝑡0isSaturn’s\nage. Of course, Eq. 31 cannot be strictly true, because\n𝑎lock(0) = 0. Instead, a likely scenario is that Titan formed\nat an initial semi-major axis 𝑎𝑖and remained there until\nsometime𝑡lock,uponwhichpointitcaughtintotheresonant\nlock and Eq. 31 applies. Although 𝑡lock(equivalently 𝑎𝑖) is\nunknown,wefindthatourresultsdonotdependsignificantly\non its value.\nAs initial conditions, we choose 𝑡lock= 1,2or 3 Gyr,\nwhich correspond to an initial Titan semi-major axis of\n12.28,15.47, and17.71𝑅Ærespectively, and an initial Titan\neccentricity of 0.04. Titan migration occurs when 𝑡 > 𝑡lock,\nand we set the migration timescale to 𝜏𝑎,Ti=−3𝑡according\nto Eq. 30. The true eccentricity damping timescale of Titan\nis unknown, but is expected to be of the same order as the\nmigrationtimescale(Fulleretal.,2016).Accordingly,weset\n𝜏𝑒,Ti=𝜏𝑎,Ti, so that Titan’s final eccentricity is closed to its\nobserved value of 0.029.\nThe simulations consistently capture Hyperion into the\nobserved 4:3 resonant configuration with 𝑒𝐻≈0.1if𝑄H≈\n40. The final eccentricity does not depend strongly on 𝑡lock.\nFigure3showstwooftheseintegrationswherewehaveused\n𝑄H= 40and𝑡lock= 1Gyr(left panel) and 𝑡lock= 3Gyr\n(right panel). In both cases, Hyperion successfully captures\ninto the 4:3 mean motion resonance with Titan after Titan\nbegins migrating outward. Initially, Hyperion’s eccentricity\nis suppressed to Titan’s eccentricity, to which it is secularly\ncoupled,bythedissipationresultingfromchaotictumbling.\nOnce Hyperion reaches a sufficient semi-major axis, the\nresonant excitation from Titan becomes stronger than the\ntidaldampingandHyperionbecomesmoreeccentric.Bythe\nend ofthe simulation, Hyperionhas reached aneccentricity\nclose its present value of 0.1. The amplitude of libration of\nthe resonant angle is significant, even if somewhat smaller\nthan what is observed.\nCritically, this model of Hyperion’s capture into reso-\nnance is not compatible with long-range Titan migration\nif damping in Hyperion is ignored. In such an undamped\nscenario, to prevent Hyperion from having too large of an\neccentricity,Titanmustonlymigrate 4%insemi-majoraxis\n(i.e.𝑡lock≈4.0Gyr)afterthe4:3capture.If,however,Titan\nmigratedmorethan 11%fromitsinitiallocation(i.e. 𝑡lock≲\n3.2Gyr), Titan and Hyperion would have started wide of,\nand then encountered, the 3:2 resonance. The encounter\nis adiabatic (Batygin, 2015) and capture into the 3:2 is\n10203040Semi-major axis \n[Rsat]TitanHyperion\n0.000.050.100.15EccentricityTitan-Saturn\nresonant lock\n0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5\nTime [Gyr]0246Resonant angle\n10203040Semi-major axis \n[Rsat]TitanHyperion\n0.000.050.100.15EccentricityTitan-Saturn\nresonant lock\n0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5\nTime [Gyr]0246Resonant angleFigure 3: Capture of Hyperion into 4:3 resonance with Titan\nin the migration-by-resonant locking model. In both panels we\nhave assumed 𝑄H= 40, on the left 𝑡lock= 1Gyrand on the\nright𝑡lock=3Gyr. The resonant angle plotted in the bottom\npanel is𝜙2= 4𝜆H−3𝜆Ti−𝜛H. Dashed lines in the bottom\ntwo rows show the measured values of Hyperion’s (forced)\neccentricity and libration angle range.\nalmost guaranteed, unless the eccentricity of Hyperion is\nvery large.2capture into resonance is impossible regardless\nofmigrationspeed(Henrard,1982).Onceinthewrongreso-\nnance,Hyperionwillgrowineccentricityandeventuallybe\nejected,neverenteringthe4:3resonance.Thus,atfacevalue,\nany model in which Titan and Hyperion cross first-order\nresonances adiabatically requires that Hyperion and Titan\nmust have started interior to the 3:2 resonance. However,\nincorporatingtidaldissipationinHyperionremovesthefine-\ntuning restriction that the resonant lock-driven migration of\nTitan can only have begun recently.\n2Colombo et al. (1974) argue, using a backwards integration, that\nHyperion would have avoided capture into the 2:1 and 3:2 resonances, but\ntheir reasoning is flawed. Because resonant encounters in the backwards\nintegration are divergent, permanent\nGoldberg & Batygin: Preprint submitted to Elsevier Page 7 of 11Hyperion rotation and orbit\n3.2.2. Slower Titan Migration\nNow,weconsiderthemeasurementofJacobson(2022),\nwho find𝑄Æ(𝑛Ti) = 1224 ± 119 , or𝜏𝑎,Ti≈ −100Gyr .\nBefore proceeding, we remark that in the context of this\nmeasurement,itisnotobviouswhatthesourceofdissipation\nwith Saturn would be. Ćuk and El Moutamid (2023) argue\nthat𝑄Æ≈ 1200is the frequency-independent dissipation\nwithin Saturn and that the Jacobson (2022) measurement\nwould imply that Titan is experiencing equilibrium tides\noutside a resonant lock. However, the migration of Tethys\nimplies𝑄Æ≈ 7000(Lainey et al., 2020) and it is not clear\nhow one moon could experience tidal dissipation in Saturn\nweaker than equilibrium. In the absence of a clear guide,\nwe take𝑄Æ= 1200and assume 𝑄Æis constant over time\nand forcing frequency. Integrating Eq. 19 from 𝑡= 0to\n𝑡= 4.5Gyr, we find that with these assumptions, Titan’s\norbithasexpandedby 6.1%overtheageoftheSolarSystem,\nsoavoidingcaptureofHyperionintothe3:2resonanceisnot\na concern.\nWe ran another suite of simulations with this model\nof slower Titan migration. Titan was initialized with an\ninitial semi-major axis of 19.2𝑅Æand eccentricity of 0.04.\nMigration of Titan was computed with Eq. 19 and eccen-\ntricity damping was assumed, as in Section 3.2.1, to be\n𝜏𝑒,Ti=𝜏𝑎,Ti. The strength of dissipation in Saturn was set\nto𝑄Æ∕𝑘2,Æ=3000, so that after 4.5GyrTitan would reach\nits current semi-major axis. Hyperion was initialized with\nseveralperiodratiosbetweenthe4:3and3:2resonancewith\nTitan, and several eccentricities from 0 to 0.05.\nFigure4showstwosimulationsofcaptureinthismodel\nof Titan migration, with 𝑄H= 100(left panel) and 𝑄H=\n∞(right panel). The first value is generally expected for\nrocky bodies (Goldreich and Sari, 2009; Brasser, 2020)\nand puts Hyperion in the regime where tidal dissipation in\nHyperiondominatesoverresonantexcitationofeccentricity\nfrom Titan’s slower migration. Hyperion remains secularly\ncoupled to Titan and the resonant angle circulates. Con-\nversely,𝑄H=∞correspondstonodissipationinHyperion\nandisequivalenttothetidalcapturehypothesisofColombo\net al. (1974) and Ćuk et al. (2013). In this case, Hyperion\nreachesitspresenteccentricityandlibrationangleamplitude\nat the end of the simulation. Using the range of parameters\nin our simulation suite, we find that Hyperion only reaches\nits current eccentricity of 0.1if𝑄H≳1000.\nFigures3and4demonstratethatslowerTitanmigration\ndemands much weaker tidal dissipation in Hyperion. In the\nrapid migration scenario (Figure 3), eccentricity pumping\nfrom resonant excitation is roughly comparable to damping\nfrom tidal dissipation for 𝑄H∼ 100, allowing the eccen-\ntricity to gradually grow to its present value. In contrast,\nresonantexcitationfromaslowlymigratingTitan(Figure4)\nis dwarfed by tidal dissipation unless 𝑄His quite large.\nThis sort of weak dissipation in Hyperion is not physi-\ncallyimplausible.NimmoandMatsuyama(2019)arguethat\nin rocky rubble pile asteroids, energy losses occur only in a\nthin surface layer of regolith rather than the entire body. In\nthe context of that model, they find that 𝑄scales as𝑅2and\n10203040Semi-major axis \n[Rsat]TitanHyperion\n0.000.050.100.15Eccentricity\n0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5\nTime [Gyr]0246Resonant angle\n10203040Semi-major axis \n[Rsat]TitanHyperion\n0.000.050.100.15Eccentricity\n0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5\nTime [Gyr]0246Resonant angleFigure 4: Capture of Hyperion into 4:3 resonance with Titan\nin the equilibrium tides model where 𝑄Æ= 1200. On the left\npanel we have assumed 𝑄H=100, and on the right 𝑄H=∞,\ncorresponding to no damping in Hyperion. Dashed lines in\nthe bottom two rows show the measured values of Hyperion’s\n(forced) eccentricity and libration angle range.\n𝑄∕𝑘2as𝑅.Hyperionisicyandmuchlargerthanthetypical\nrubblepileasteroidstheyinvestigate,butextrapolatingtheir\nmodel would predict that Hyperion has a very large 𝑄∕𝑘2.\n4. Discussion and Conclusions\nDespite its apparent simplicity, the outer saturnian sys-\ntemhasconfoundedunderstandingfordecadesandarguably\nhasbecomemorechallengingwithnewdiscoveries.Atface\nvalue, the orbit of Hyperion implies that Titan migrated\nslowly, while the survival of Iapetus requires faster migra-\ntion. Both cases rely on tidal dissipation in Saturn much\nstrongerthanconventionallyexpected.Takentogether,these\nproperties are a challenge to reconcile, especially while\nremaining consistent with constraints provided by the inner\nsystem.\nGoldberg & Batygin: Preprint submitted to Elsevier Page 8 of 11Hyperion rotation and orbit\nWe have shown that Hyperion’s rotation is the key to\nresolving this discrepancy. Hyperion’s spin rate depends\nsteeplyonitsorbitaleccentricity,aconsequenceofachaotic\nseageneratedfromtheoverlapofspin-orbitresonancesthat\ngrows with eccentricity. Tidal dissipation is much stronger\nat higher spins and is thus a steep function of 𝑒Has well.\nAlthough the mean-motion resonant interaction between\nTitan and Hyperion grows the latter’s orbital eccentricity\nas Titan migrates outward, the enhanced tidal dissipation\nresulting from rapid tumbling damps the eccentricity. The\ndegree of dissipation within Hyperion can be probed by\ncomparingtherelativestrengthsofthesetwoeffectstomatch\ntheobserved 𝑒H=0.1.ThepreciserateofTitan’smigration\nisdisputedinrecentworks.Ifitisrapid,Hyperionmusthave\natidalqualityfactorof 𝑄H≈40,similartowhatistypically\nexpectedforrockybodies.Alternatively,ifmigrationisslow,\nHyperion must be only weakly dissipative ( 𝑄H≳1000),\nwhich is reasonable if dissipation occurs solely in a surface\nlayer, as suggested by Nimmo and Matsuyama (2019).\nThe dissipation itself could in principle be detected di-\nrectlythroughanexcessthermalsignature.Energyprovided\nby the orbit acts to heat up Hyperion and sublimate its\nwater ice. If Titan is migrating rapidly and 𝑄H= 40(e.g.\nright panel of Figure 3), the current energy dissipation rate\nis𝑑𝐸∕𝑑𝑡∼ 3MW . As a crude approximation, if this\ndissipation were constant over the lifetime of the Solar Sys-\ntem, and assuming Hyperion is made entirely of water ice,\napproximately3%ofthemassofHyperionwouldhavesub-\nlimated due to the tidal dissipation. While non-negligible,\nthis amount is insufficient to explain Hyperion’s high ( >\n40%) internal porosity (Thomas et al., 2007). However, it\nis possible that Hyperion experienced a transient phase of\nhigh eccentricity, perhaps due to a scattering event before\nthe resonant capture with Titan. An excitation to 𝑒H≈ 0.3\nfollowed by damping to a circular orbit could sublimate\n≈ 40%of Hyperion’s mass in <1Gyrand account for its\ncurrent porosity.\nSignificantsublimationcouldhaveotherimpacts.Selig-\nman and Laughlin (2020) demonstrate that uniform subli-\nmation across the surface of an ellipsoid acts to elongate it;\nthey use this effect to explain the extreme body axis ratio\nof ‘Oumuamua. If Hyperion underwent a similar process,\nits current shape could therefore be the consequence of a\nsmall irregularity in the shape of the primordial Hyperion\nthat grew as material preferentially sublimated from certain\nregions.\nThesetwocomplicationshighlightanimportantassump-\ntion we have made throughout this work. We have taken\nHyperion’s shape and material properties to be constant\noverthelifetimeoftheSolarSystem.However,sublimation,\ngravitational settling from tumbling, and impacts can vary\nthe mass, composition, porosity, strength, and shape of Hy-\nperion, all of which would affect the tidal dissipation rate.\nIncorporating all of these effects, while challenging, would\nprovide a more complete picture of Hyperion’s evolution.\nOurdetailedinvestigationsofHyperion’srotationshowed\nthatitismorecomplexthantheoriginalpredictionsderivedfromaone-dimensionalmodel.Evenso,generationofchaos\nvia overlap of non-linear spin-orbit resonances remains\nqualitatively useful. Our results suggest that Hyperion has\nhad a rich rotational history, alternating between tumbling\nand quasi-regular states that depend on its instantaneous\neccentricity. The orbit and rotation of Hyperion are inex-\ntricably coupled: orbital eccentricity sets the typical spin\nrate,andinturn,thenonsynchronousspindampstheorbital\neccentricity.Similarspin-to-orbitandorbit-to-spincoupling\nhas been suggested for asteroid binaries (Efroimsky, 2015;\nNimmo and Matsuyama, 2019; Quillen et al., 2022) and\nclose-in satellites (Dobrovolskis et al., 1997; Quillen et al.,\n2017, 2020). Hyperion, despite its small size and great dis-\ntance from Saturn, is subject to the same complex feedback\nbetween spin and orbital evolution.\nAlthoughwehaveidentifiedacompelling processtoex-\nplain the current state of Hyperion, we have not determined\nthe exact scenario that transpired. Of particular interest is\nHyperion’sinitialorbit.IfitsoriginalperiodratiowithTitan\nexceeded1.5,captureintothe3:2resonanceishighlylikely,\nunlessanothermechanismcanbreaktheresonanceoravoid\ncapture entirely. On the other hand, formation of Hyperion\nin such close proximity to the very massive Titan seems a\nprioriunlikely. We encourage further work on this topic.\nFinally, our rotation model of Hyperion does not include\nthe effects of tidal despinning. While it is clear that the\ncurrentrotationstateisnotinthe1:1or3:2island,itremains\npossible that it lies on a strange attractor with perpetual\nchaotic but quasi-regular motion (Melnikov, 2014). A self-\nconsistent rotation model incorporating wobble damping\nand tidal dissipation that is coupled to the orbital evolution\nwould thus be necessary to further constrain the history of\nHyperion and more precisely measure its tidal parameters.\nAcknowledgements\nWe thank the two referees for thorough reading of the\nmanuscript and helpful feedback. We also thank David\nNesvorný, Rogerio Deienno, Bill Bottke, and Jim Fuller\nfor insightful discussions. K.B. is grateful to Caltech, the\nDavidandLucilePackardFoundation,andNationalScience\nFoundation(grantnumber:AST2109276)fortheirgenerous\nsupport.\nReferences\nAgrusa, H.F., Gkolias, I., Tsiganis, K., Richardson, D.C., Meyer, A.J.,\nScheeres, D.J., Ćuk, M., Jacobson, S.A., Michel, P., Karatekin, Ö.,\nCheng,A.F.,Hirabayashi,M.,Zhang,Y.,Fahnestock,E.G.,Davis,A.B.,\n2021. The excited spin state of Dimorphos resulting from the DART\nimpact. 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Here, we demon-\nstrate the autonomous stabilization of remote entanglement between a pair of non-interacting super-\nconducting qubits connected by an open waveguide on a chip. In this setting, the interplay between\na classical continuous drive - supplied through the waveguide - and dissipation into the waveguide\nstabilizes the qubit pair in a dark state, which, asymptotically, takes the form of a Bell state. We\nuse field-quadrature measurements of the photons emitted to the waveguide to perform quantum\nstate tomography on the stabilized states, where we find a concurrence of 0 .504+0.007\n−0.029in the optimal\nsetting with a stabilization time constant of 56 ±4 ns. We examine the imperfections within our\nsystem and discuss avenues for enhancing fidelities and achieving scalability in future work. The\ndecoherence-protected, steady-state remote entanglement offered via dissipative stabilization may\nfind applications in distributed quantum computing, sensing, and communication.\nIntroduction\nEntanglement between distant physical systems is a\ncrucial resource for quantum information processing.\nOver long distances, entanglement can make communica-\ntion secure against eavesdropping and resilient to loss [1–\n3]. On shorter length scales, entanglement between dis-\ntant non-interacting modules can help realize non-local\ngate operations in a quantum computer [4–7]. Remote\nentanglement can be created by the deterministic ex-\nchange of photons between qubits located at remote sites\n(e.g., [8, 9]). Alternatively, quantum measurements can\nbe used to ‘herald’ entanglement in a probabilistic fashion\n(e.g., [10, 11]). Once established, entangled states have to\nbe protected from decoherence by the environment before\nthey can be used, a task that can be achieved via pas-\nsive storage in isolated quantum memories [3]. Current\nresearch actively pursues the development of physical sys-\ntems capable of high-bandwidth generation, distribution,\nand storage of remote entanglement.\nAn entirely different approach for the generation and\nprotection of entanglement is its stabilization . This\nmethod employs dissipation into a shared reservoir, in\ncombination with continuous drives, to establish entan-\nglement between two or more parties. Intriguingly, such\nan engineered dissipation can not only create entangle-\nment but also protect it - indefinitely - from environment-\ninduced errors [12, 13]. ‘Driven-dissipative’ processes\nthus provide an attractive route for the generation and\nautonomous preservation of entanglement [14–22]. Be-\nyond entanglement generation, dissipative processes have\nalso been studied for a variety of other tasks in quan-\ntum information processing as an alternative to unitary\ngate operations [13, 23–25]. Despite the wide interest in\n∗These authors contributed equally.\n†mohmir@caltech.com; http://qubit.caltech.eduthis area, however, stabilizing remote entanglement has\nremained elusive, primarily due to the challenge of engi-\nneering shared dissipation for remote sites.\nSpontaneous emission into a one-dimensional photonic\nbath can provide a shared dissipation channel for re-\nmote quantum emitters. Such a system can be realized\nwithin the paradigm of waveguide quantum electrody-\nnamics (QED), where two or multiple qubits - acting\nas quantum emitters - are strongly coupled to a shared\nwaveguide. In this setting, the interference of photons\nemitted by the qubits can give rise to the formation of\ncollective dark states that are protected from dissipation\nby their internal symmetries [26–30]. Theoretical work\nhas proposed a variety of methods for stabilizing collec-\ntive dark states [31–42]. However, experimental demon-\nstrations of these proposals have remained out of reach,\nowing to the need for components like the injection of\nnon-classical states into the waveguide and the require-\nments for either unidirectional or time-modulated qubit-\nphoton coupling – elements that present practical chal-\nlenges to implement.\nHere, we demonstrate the stabilization of remote en-\ntanglement by driven-dissipation, exploiting the emission\ninto a one-dimensional photonic bath. In our experiment,\na pair of superconducting qubits serve as remote quan-\ntum nodes, which are connected via an open microwave\nwaveguide on a chip. In following a previous theoretical\nproposal [43], our approach offers simplicity by relying\non classical drives and conventional bidirectional qubit-\nphoton couplings. The formation of a dark state in our\nexperiment is achieved via the precise tuning of the qubit\ntransition frequencies, ensuring that the wavelength of\nthe photons emitted by the qubits is precisely matched to\nthe inter-qubit physical distance. Supplying continuous\ndrives through the waveguide, we demonstrate the stabi-\nlization of a dark state in this system, which, asymptot-\nically, takes the form of a Bell state under strong drives.\nFollowing the stabilization, we conduct quantum state\ntomography to quantify the degree of entanglement inarXiv:2402.15701v1  [quant-ph]  24 Feb 20242\na\nb\nc\n1 mm \nQ1\n Q2\n100 /uni03BCm 10 /uni03BCm \nQ3\nInput Output\nWaveguide\nFIG. 1. Experimental setup. a) A pair of quantum emitters coupled to a shared waveguide. The qubit transition frequencies\nare offset in opposite directions with respect to a central frequency, and their separation is set equal to the wavelength of the\nradiation at their center frequency. A continuous Rabi drive is supplied through the channel. b) Left: The energy level diagram\nof the system. The triplet state |T⟩superradiantly decays into the waveguide. The singlet state |S⟩is coherently coupled to\nthe triplet but has no direct decay path into the waveguide. Right: The energy diagram in a basis including the dark state |D⟩.\nHere, the population is pumped from the triplet state into the dark state, which is protected from decay into the waveguide.\nc) Optical image of the fabricated device, where three transmon qubits are coupled to a shared coplanar waveguide on a chip.\nWe only utilize qubits 1 and 2 in this experiment. Each qubit can be addressed via flux and charge drives delivered via a pair\nof on-chip control lines.\nthe steady state of the system. Through repeated exper-\niments with different settings, we find a maximum con-\ncurrence of 0.504+0.007\n−0.029(95% confidence interval) with a\nstabilization time constant of 56 ±4 ns. Moreover, we\nstudy the trade-off between the stabilization time and the\nentanglement quality in the steady state and compare it\nwith a numerical model that considers the noise sources\nin our experiment. Finally, we discuss future improve-\nments in the experimental parameters, where we expect\nto reach fidelities exceeding 90% with improved thermal-\nization of the waveguide and coherence parameters within\nthe reach of state-of-the-art superconducting qubits. By\ndemonstrating driven-dissipative entanglement stabiliza-\ntion in a waveguide, our experiment marks an impor-\ntant step toward realizing a modular network architec-\nture where two- and multi-node remote entanglement is\naccessed on demand via an open radiation channel.\nTheoretical concept\nOur system includes two qubits coupled to a shared\nwaveguide with equal dissipation rates of Γ 1D(see\nFig. 1a). The qubit frequencies are offset symmetrically\nwith respect to a center frequency ( ω1,2=ω±δ). Fur-\nther, we choose the center frequency ωsuch that ℓ=mλ,\nwhere ℓis the physical distance between the qubits, λis\nthe wavelength of radiation at ω, and mis an integer.We assume the qubits are driven via a classical coherent\nfield at the frequency ω, supplied through the waveg-\nuide. In a frame rotating at the driving frequency, and\nafter applying the rotation wave approximation (RWA),\nthe Hamiltonian for this system can be written as\nˆH/ℏ=X\ni=1,2δi\n2ˆσz,i+1\n2\u0010\nΩˆσ†\ni+ Ω∗ˆσi\u0011\n. (1)\nHere, Ω denotes the Rabi frequency of the drive and\nδ1,2=±δ. While this Hamiltonian is separable, the inter-\nference of photons emitted by the two qubits in this set-\nting gives rise to suppression and enhancement of sponta-\nneous emission [44]. This can most easily be understood\nin the basis of the collective states corresponding to a pair\nof maximally entangled states, denoted by the triplet and\nsinglet states |T, S⟩= (|eg⟩±|ge⟩)/√\n2. The Hamiltonian\nin the basis of these states can be written as\nˆH/ℏ=Ω√\n2(|T⟩⟨gg|+|ee⟩⟨T|)−δ(|S⟩⟨T|) + H .c..(2)\nFigure 1b (left) shows the corresponding energy level\ndiagram, including the coupling and dissipation terms\n(see Appendix B 1 for derivation). As evident, the sin-\nglet state is sub-radiant and protected from direct dissi-\npation into the waveguide. However, in the presence of\nfrequency detuning, the singlet state can exchange pop-\nulation with the waveguide through coherent interaction\nwith the triplet state, which is super-radiant. In the3\nsteady state, the combination of this process and continu-\nous drive through the waveguide results in the formation\nof a superposition of the singlet and the ground states\nthat is dark to the waveguide. This stationary state can\nbe found by solving for an eigenstate of the Hamiltonian\nthat satisfies the condition (ˆ σ1+ ˆσ2)|D⟩= 0 (see Ap-\npendix B 2 and [35, 43]). The diagonalized level structure\nis shown in Fig. 1b (right), where the dark state is given\nby\n|D⟩=|gg⟩+α|S⟩q\n1 +|α|2. (3)\nHere, α= Ω/√\n2δis a drive-power dependent parameter\nthat sets the singlet fraction ( |α|2/(1 +|α|2)). As evi-\ndent, the stationary state is pure, and for strong drives\n(|α| ≫ 1) approaches the maximally-entangled singlet\nstate. As a result, by simply supplying the drive into the\nwaveguide, one can stabilize entanglement between the\nqubits starting from an arbitrary initial state.\nBy redrawing the energy level diagram to include the\ndark state, |D⟩, and a state normal to it ( |B⟩= (α|gg⟩−\n|S⟩)/q\n1 +|α|2), one can find the rate of pumping popu-\nlation into |D⟩. As shown in Fig. 1b (right), the popula-\ntion is dissipatively transferred from |T⟩to|D⟩, where it\nis trapped due to the decoupling of the dark state from\nthe remainder of the energy levels. The effective rate\nof this process can be found as (see Appendix B 2 for\nderivation)\nγeff,D=2Γ1D\n1 + Ω2/2δ2. (4)\nThe reciprocal of this ‘pumping’ rate sets the timescale\nfor stabilization tD= 1/γeff,D. We highlight the compe-\ntition between the singlet fraction |α|2/(1+|α|2) and the\nstabilization time tD= (1 + |α|2)/(2Γ1D), where achiev-\ning larger singlet fractions - corresponding to more entan-\nglement - requires higher drive powers and longer stabi-\nlization times. (see Appendix B 2 for further discussion).\nTo successfully implement the stabilization protocol\nunder consideration, a physical platform must fulfill sev-\neral requirements. Most importantly, it is necessary to\nhave precise control over the qubit transition frequency\nand to establish efficient interfaces between qubit and\npropagating photons. Precise control over the qubit fre-\nquency is needed to ensure that the interference of emit-\nted photons leads to destructive interference at both the\noutputs of the waveguide simultaneously, culminating in\na dark state. Formally, this condition can be articulated\nby defining the null spaces for the collective jump op-\nerators (see Appendix B 2) [45]. Efficient qubit-photon\ninterfaces are vital to minimize photon loss during the\nemission and re-absorption processes among the qubits,\nwhich can result in a reduced fidelity for the stabilized\nstate. A key metric in evaluating this effect is the Purcell\nfactor, defined as the ratio of an individual qubit’s decay\nrate to the waveguide to its intrinsic decoherence rate,P1D= Γ1D/Γ′, where Γ′= 2Γ 2−Γ1D= Γint+ 2Γ ϕ. (Γ 2\nis the total qubit decoherence, Γ intis loss to non-radiative\nchannels, and Γ ϕis pure dephasing). In addition to the\nfactors mentioned, another key ingredient is the charac-\nterization of the stabilized joint qubit state. This task\nis particularly challenging in the presence of dissipation\nfrom the waveguide. To overcome this challenge, charac-\nterization measurements need to happen either on very\nshort time scales, or, alternatively, temporary elimination\nof waveguide dissipation is needed during the character-\nization process. In the next section, we detail an exper-\nimental realization based on transmon superconducting\nqubits that satisfies these requirements.\nThe experiment\nThe fabricated superconducting circuit used to realize\nour experiment is shown in Fig. 1c. The circuit consists\nof three transmon qubits (1, 2, and 3), which are side-\ncoupled to the same coplanar waveguide (CPW). Each\nqubit has a weakly coupled charge control line (shown in\norange) and external flux bias port (shown in green) for\ntuning its transition frequency. Qubit 3 does not partici-\npate in any of our experiments and is decoupled from the\nrest of the system by tuning its frequency well away ( >1\nGHz) from the other two qubits. We list the details of\nour device fabrication in Appendix A.\nWe first tune the qubit transition frequencies to realize\nthe previously described level structure of Fig. 1b. We\nthen verify the required interference conditions by search-\ning for signatures of the sub-radiant and super-radiant\nstates. Figure 2a shows the transmission spectra through\nthe waveguide for weak microwave drives, measured as a\nfunction of the flux bias of Qubit 1. Meanwhile, Qubit\n2 has its transition frequency fixed at ωsuch that the\ninter-qubit separation along the waveguide equates the\ncorresponding wavelength ( ℓ=λ). As the two qubits\ncross, we note a broader resonant lineshape, which cor-\nresponds to the formation of a super-radiant state [44].\nFigure 2b shows the waveguide transmission spectrum\nfor the case where two qubits are precisely on resonance\natω(red), and for when Qubit 2 is at ωwhile Qubit 1\nis tuned out of the measurement window (blue). By fit-\nting Lorentzian lineshapes to these spectra, we find the\nradiative decay rate of the super-radiant (triplet) state,\nΓ1D,T/2π= 18.3 MHz, which is nearly twice the single-\nqubit decay rate [(Γ 1D,1,Γ1D,2)/2π= (10 .3,10.7) MHz],\npointing to the correct phase length between the qubits.\nWe characterize each individual qubit-photon interface\nusing fits to the single-qubit spectra at ω, extracting Pur-\ncell factors for each qubit [( P1D,1, P1D,2) = (11 .4,10.7)].\nAlthough the singlet is (ideally) protected from emission\nto the waveguide, in a realistic system with a finite Pur-\ncell factor, it has a finite lifetime. Being sub-radiant,\nthough, it is not visible in the waveguide transmission re-\nsponse. Instead, we measure the inelastic scattering from\nthe qubits using a spectrum analyzer (see Appendix A for4\nb a\nc d\n0 500 1000 1500 2000 2500 3000\nTime (ns)10-210-1100Excited state population\nSingle qubit\nSub-radiant\n0 20 40 60 80\n0.51\n0 500 1000 1500 2000 2500 3000\nTime (ns)0.51\n-25-20-15-10-50|t| (dB)\nSingle qubit\nSuper-radiant\n00.51|t|\n-0.378 -0.376 -0.374 -0.372-20-1001020Detuning (MHz)\n/0\nP = -131 dBmP = -113 dBm\n-6 -4 -2 0 2 4 6\nDetuning (MHz)\n-20 -10 0 10 20\nDetuning (MHz)\n0246PSD (W/Hz)00.511.510-23\nFIG. 2. Characterizing the super- and sub-radiant collective states. a) Transmission spectrum measured through the\nwaveguide as Qubit 1 is frequency-tuned across Qubit 2. The dashed line denotes the ℓ≈λpoint). b) Transmission spectra for\nthe single qubit (Qubit 2, blue) and two-qubit (red) settings, where a broader spectrum indicates the super-radiant (triplet)\nstate. Fitting Lorentzian lineshapes to the two data sets gives decay (intrinsic decoherence) rate rates of Γ 1D/2π= 10.7±0.4\nMHz (Γ′/2π= 1.0±0.5 MHz) and 18 .3±0.4 MHz (Γ′/2π= 1.3±0.5 MHz). The decay (intrinsic decoherence) rate for Qubit\n1, tuned to have the same transition frequency, is found as Γ 1D/2π= 10.3±0.3 MHz (Γ′/2π= 0.9±0.4 MHz, not shown). c)\nInelastic scattering spectrum. At higher drive powers (upper inset), the lineshape includes contributions from both the triplet\nand singlet states. Reducing drive power results in a lineshape predominantly set by the singlet state. Fitting using master\nequation simulations gives individual qubit dephasing of 174 ±24 kHz and correlated dephasing of 127 ±85 kHz. d) Measured\nrelaxation lifetimes for an individual qubit (Qubit 2) and the singlet state yielding T1= 16±1.9 ns and T1= 910 ±47 ns,\nrespectively. The insets show the linear scale plots of the same data.\nmeasurement details). While the resonance fluorescence\nspectrum at higher powers contains contributions from\nboth the super- and sub-radiant states, a measurement\ndone at a sufficiently low drive power is dominated by the\nresponse from the sub-radiant (singlet) state, as shown\nin Fig. 2c [26]. A master equation simulation fit to the\ninelastic scattering profile confirms the presence of the\nsub-radiant state (see Appendix C for a detailed discus-\nsion). Furthermore, we measure the singlet’s population\ndecay lifetime by resonantly exciting it via the individ-\nual charge lines. In this measurement, we read out the\npopulation of the ground state using the state-dependent\ntransmission coefficient through the waveguide (similar\nto [28], see Appendix D). Fig. 2d shows the measure-\nment results for the singlet state plotted next to the free\npopulation decay of an individual qubit (measured using\nresonant readout, Appendix D). We find a large contrast\n(factor of over 56) in the lifetimes of the single qubit and\nsub-radiant states, indicating that the qubit frequency\nconfiguration and Purcell factors are suitable for the re-alization of the stabilization protocol.\nHaving established the operation frequency from the\ncharacterization experiments detailed above, we proceed\nwith the stabilization and characterization of entangled\nstates. In this step, the qubits are detuned with respect\nto the target frequency (see Fig. 3a). Starting with a sys-\ntem at rest in this setting, we apply a narrow band drive\natωto initiate the stabilization process. After supplying\nthe drive for a finite duration of time, we turn it off and\nallow the qubits to freely decay into the waveguide. We\nnote here that strong engineered dissipation appears to\ndirectly preclude characterization of the qubit state; in\ntypical circuit QED settings, qubit state characterization\nfavors preserving the qubit state and minimizing qubit-\nenvironment coupling. A natural alternative is there-\nfore to recover the qubit state using the emitted photons.\nOur choice of frequency detuning between the two qubits\nresults in a low overlap between their spectral content.\nHence, the spontaneous emission from the two qubits into\nthe waveguide has spectrally distinguishable microwave5\na\n00.20.4Qubit 1\nRe(⟨σ 1⟩)\nIm(⟨σ 1⟩)\n|⟨σ1⟩|\n⟨σ†\n1σ1⟩\n00.20.4Qubit 2\nRe(⟨σ 2⟩)\nIm(⟨σ 2⟩)\n|⟨σ2⟩|\n⟨σ†\n2σ2⟩\n50 100 150 200 250 300 350\nTime(ns)00.10.20.3Cross Moments\n|⟨σ†\n1σ2⟩|\n|⟨σ1σ2⟩|b\nc−20 0 20\nDetuning from Drive (MHz)−20−15−10−50|t|(dB)δ/one.denominator\nδ/two.denominator\n⟨σ1⟩\n⟨σ2⟩\n⟨σ†1σ1⟩\n⟨σ†2σ2⟩\n⟨σ1σ2⟩\n⟨σ†1σ2⟩\n⟨σ1σ†2σ2⟩\n⟨σ†1σ1σ2⟩\n⟨σ†1σ1σ†2σ2⟩−0.20.00.20.4\nRe\nIm\nSim\nFIG. 3. Stabilization dynamics. a) The transmission spectrum measured through the waveguide when the two qubits are\ndetuned symmetrically ( δ/2π= 17 MHz). The black dashed line marks the target drive frequency ( ω/2π= 6.392 GHz, where\nℓ≈λ). b) The time dynamics of the first- and second-order moments after supplying a continuous drive with a power of −117\ndBm (corresponding to Ω 1/2π= 36 MHz and Ω 2/2π= 37 MHz) and frequency of ωto the waveguide. The moments are\nmeasured via the resonant readout of the two-qubit state as described in the main text. The ‘time’ axis on all plots shows the\nduration for which the drive has been applied to the waveguide. A delay following each measurement (200 ns, much longer\nthan the radiative lifetime of the individual qubits) ensures that the system is at ‘rest’ prior to the subsequent measurement.\nThe plots display average values derived from an experiment that has been repeated 150 million times. c) The measured\nmoments after a long drive (396 ns), corresponding to the system at the steady state. The plot includes all non-zero terms for\nan arbitrary state residing in a Hilbert space containing at most a single excitation in each qubit. The plots display average\nvalues derived from an experiment that has been repeated 3 billion times, and horizontal bars indicate standard deviations\n(see Appendix E 4). The values from a numerical simulation are displayed for comparison (boxed outlines). The model uses\nparameters based on the single-qubit characterization data. For further simulation details, see Appendix G.\nphotons with well-defined temporal wavepackets. Using\nquadrature amplitude detection and mode-matching to\nthe photonic time bins (following previous work, [46–49])\nwe successfully measure the self and cross-moments of\nthe emitted photons. Through input-output relations of\nmultiple qubits coupled to an open waveguide, we can\nrelate these photonic moments to those of the qubits\nthemselves. Additionally, since each photon is emitted\nby a qubit it may at most have only one excitation, we\ncan compute all the self and cross-moments of the form\n⟨(ˆσ†\n1)n1ˆσm1\n1(ˆσ†\n2)n2ˆσm2\n2⟩ ∀n1, n2, m1, m2∈ {0,1}, which\nsuffice to perform a joint state tomography of the two-\nqubit state (see Appendix E).\nFigure 3b shows the measured self- and cross-moments\nof the qubits as the state evolves in time for a Rabi drive\nof Ω/2π= 37 MHz and a detuning of δ/2π= 17 MHzbetween the qubits and the drive. As evident, the system\nstarts with the qubit pair in the ground state, where all\nthe moments are zero. With the continuous drive turned\non, we observe an initial increase in the average popula-\ntion of each qubit with time, followed by settling into a\nsteady-state value. The stabilization of a dark state with\nentanglement can be identified via the emergence and sta-\nbilization of a non-zero cross-qubit moment ⟨ˆσ†\n1ˆσ2⟩. We\nalso note the nonzero expectation values of the single-\nqubit first moments ⟨ˆσi⟩, which can be attributed to the\nfinite population of the ground state. Figure 3c plots the\nsummary of the measurement results of the steady state\nand includes all relevant moments for any joint state of\ntwo qubits. The small magnitudes of the third and fourth\nmoments indicate the low population of the doubly ex-\ncited state |ee⟩. Using the computed moments we recon-6\na\nb0 100 200 300 400\nTime (ns)0.000.050.100.150.200.250.300.35|⟨σ†\n1σ2⟩|\n/uni03A9/2π = 22 MHz (shifted by +0.1)\n/uni03A9/2π = 37 MHz\n/uni03A9/2π = 51 MHz\n25 30 35 40 45 50\nRabi Drive on Qubit 2 (MHz)0.20.30.40.50.6\nFidelity (to singlet)\nConcurrence\n012345\nc\nEntanglement Metric γeff/2π (MHz)\nFIG. 4. Interplay between drive power, stabilization\nrate, and entanglement a) Stabilization dynamics for dif-\nferent drive powers. The graph shows the ⟨ˆσ†\n1ˆσ2⟩cross mo-\nment, which is related to the amount of entanglement in the\nsystem. We fit exponential curves (solid lines) to the mea-\nsured data (dots) and extract the stabilization rates, given by\nγeff, for different drive powers. b) Variation in stabilization\nrates with power. Error bars represent ±standard deviations.\nWe can see a clear decrease in the stabilization rate with in-\ncreasing drive power, as expected from the theory (Eq. (4)).\nc) Fidelities (to a singlet state) and concurrence against vs\nthe drive power. Error bars represent 95% confidence inter-\nval. The maximum concurrence achieved is 0.504+0.007\n−0.029(95%\nconfidence interval). The corresponding time constant is 56\n±4 ns. The decrease in concurrence after a point is explained\nby a finite Purcell factor. Shaded regions in panels (b) and\n(c) represent simulations results with varying Purcell factors\n(from 10 to 30) for the qubits. See Appendix G for further\ndiscussion on simulations.struct the density matrix for the two qubit system using\nmaximum-likelihood estimation (Appendix E 4). The fi-\ndelity of the extracted density matrix to the singlet state\nat this drive power and detuning is 56.5% (95% confi-\ndence interval from 55.2% to 58.7%), which is already\nabove the 50% threshold that confirms entanglement.\nBeyond the quality of entanglement, the stabilization\nrate is another important metric for practical applica-\ntions. In the protocol being examined, the stabiliza-\ntion rate is anticipated to decrease as the drive power\nincreases, as indicated by Eq. (4). At the same time, the\nlevel of entanglement, quantified by the singlet fraction\n(see Eq. (3)), is expected to rise with increasing drive\npower. Figure 4a shows the time evolution of the cross\nmoment ⟨ˆσ†\n1ˆσ2⟩across three different power levels. The\nstabilization rates, derived from exponential fits to this\ndata (and experiments at a few different drive powers),\nexhibit a consistent decline with increased power, align-\ning with our expectations (see Fig. 4b). We also plot\nthe fidelity relative to the singlet state for each power\nsetting (see Fig. 4c), revealing that fidelities initially rise\nwith power before reaching a plateau.\nTo more accurately assess entanglement, we calculate\nand graph the concurrence [50] for each power setting\n(see Fig. 4c). This graph presents a more distinct trend,\nshowing concurrence peaking at intermediate drive pow-\ners and diminishing at both lower and higher powers.\nThe reduction of entanglement at low powers aligns with\nexpectations, mirroring the calculated fidelity and our\nmodel’s predictions. The reduction in entanglement at\nhigh power can be qualitatively understood by consid-\nering the role of parasitic (i.e., non-radiative) damping\nand dephasing, which our analytical model does not cap-\nture. Intuitively, these effects drive the qubits towards\na mixed state, with the decoherence rate Γ′competing\nagainst the effective pumping rate γeff(see [34]). This\nexplanation is supported by calculating the purity of the\nstabilized states, which drop from 90% at Ω /2π= 30\nMHz to 60% at Ω /2π= 51 MHz. Since our analytical\nmodel does not account for parasitic damping and de-\nphasing, we utilize master equation simulations to model\nour experimental setup more accurately. A complicat-\ning factor for these simulations is power-dependent de-\nphasing in driven qubits, as documented in previous re-\nsearch [51, 52], which effectively results in a variable Pur-\ncell factor at different drive powers (see Appendix F).\nTo capture this phenomenon, we conduct master equa-\ntion simulations of our experiment for a range of Pur-\ncell factors, setting the lower limit based on our qubit\nspectrum measurements ( P1D= 10) and selecting an up-\nper bound that qualitatively aligns with our experimental\ndata ( P1D= 30). The simulations, depicted in Fig. 4b\nand c, exhibit trends that closely mirror the observed\npower-dependent variations in our experimental results.\nIn our system, we can achieve a maximum concurrence of\n0.504+0.007\n−0.029(95% confidence interval) at an optimal drive\npower of Ω /2π= 30 MHz.7\nConclusions and outlook\nRemote entanglement is a crucial resource for quan-\ntum networks with a wide range of applications in dis-\ntributed quantum computing, communication, and sens-\ning. In platforms equipped with low-loss communication\nchannels, like superconducting qubits, remote entangle-\nment has been produced through deterministic processes\ninvolving direct photon exchange [8, 53–55] or virtual in-\nteractions via cavity relays [9, 56–58]. Although these\nmethods have achieved high fidelities, they typically re-\nquire pulse shaping or precise timing of qubit operations.\nRelying on driven-dissipative stabilization, as showcased\nin our experiment, offers a simple alternative that re-\nlaxes these requirements. Additionally, stabilizing the\ntarget state allows for maintaining the entanglement in-\ndefinitely, until it is needed for quantum networking oper-\nations, thereby providing an on-demand resource without\nthe latency caused by the propagation delay between the\nnodes.\nIn terms of physical dimensions, the inter-qubit dis-\ntance in our experiment is not particularly long (all com-\nponents are confined within a 1 cm by 1 cm chip) and is\ncomparable to previous work in cavity systems [18, 20–\n22]. Distinctly, however, relying on an open waveg-\nuide makes the concept insensitive to the distance be-\ntween the qubits (within integer multiples of λ/2), allow-\ning for the stabilization of long-distance entanglement.\nThis increased range of entanglement is not limitless, as\nthe breakdown of the Markov approximation (inherently\nassumed in our analysis) sets an upper bound on the\ndistance. Nonetheless, with the dissipation and drive\nrates attainable with superconducting qubits, this limit is\nrather long (meters), allowing for the method to be prac-\ntically feasible for a range of applications. Going beyond\nthis limit remains relatively unexplored, though propos-\nals based on injecting squeezed light into waveguides may\nprovide a route towards this [39, 41]. Additionally, the\nwaveguide’s broad-band nature enables frequency mul-\ntiplexing, permitting the simultaneous entanglement of\nmultiple qubit pairs at different frequencies through the\nsame channel. Finally, using chiral qubit-photon cou-\npling [59–63] can relax the requirement of precise phase\ntuning between the qubits and also lead to the formation\nof multi-partite entanglement [35, 43].In terms of entanglement fidelity, we achieve only mod-\nest results, which are limited by the qubit coherence (see\nAppendix H). However, with improvements to qubit co-\nherence and reduced waveguide temperature, we expect\nfidelities of over 90% (concurrence of 0.88) to be within\nthe reach of this method. In terms of bandwidth, the sta-\nbilization rate of 2.83 ( ±0.19) MHz (settling time of 56\nns) is comparable to previous demonstrations with super-\nconducting qubits [9, 53, 56, 64]. We note that predicted\nfidelity improvements are expected to be accompanied by\na reduction in stabilization times (a trend that goes be-\nyond our experiment and is typical of driven-dissipative\nprocesses). Nevertheless, for fidelities over 90% we expect\na stabilization rate of about 250 kHz to be within reach.\nUltimately, going beyond these limits requires further in-\nvestigations of protocols with better hardware efficiency\nand improved bandwidths [65].\nIn conclusion, we have demonstrated the generation\nand stabilization of entanglement between a pair of dis-\ntant qubits coupled to a shared waveguide via a driven-\ndissipative protocol. The presented scheme offers sim-\nplicity, steady state operation, and decoherence protec-\ntion, making it an attractive avenue for remote entangle-\nment generation. We have also identified the limitations\nof our experiment, finding high-fidelity remote entangle-\nment to be achievable using qubits with improved deco-\nherence characteristics well within reach of existing su-\nperconducting technologies. With future improvements\nin fidelities, we envision that entanglement stabilization\nprotocols may find applications in distributed quantum\ncomputing and quantum communication. Beyond appli-\ncations, extending the range of entanglement stabiliza-\ntion may prompt studies of non-local noise in photonic\nreservoirs, which may lead to potentially relevant insights\nfor quantum error correction.\nACKNOWLEDGMENTS\nThis work was supported by startup funds from Cal-\ntech’s EAS division, National Science Foundation (award\nnumber: 1733907), and Office of Naval Research (award\nnumber: N00014-24-1-2052). P.S.S. gratefully acknowl-\nedges support from the S2I-Gupta Fellowship. F.Y.\ngratefully acknowledges support from the NSF Gradu-\nate Research Fellowship. C.J. gratefully acknowledges\nsupport from the IQIM/AWS Postdoctoral Fellowship.\n[1] H. J. Briegel, W. Dr, J. I. Cirac, and P. 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Chaya-\nnun, C. Kriˇ zan, P. Malmberg, M. Rommel, C. War-\nren, P. Delsing, A. Yurgens, J. Bylander, and\nA. F. Roudsari, Mitigation of interfacial dielectric loss\nin aluminum-on-silicon superconducting qubits, arXiv\n10.48550/arXiv.2310.06797 (2023), 2310.06797.11\nAppendix A: Methods\n1. Fabrication\nOur device is fabricated on a 1 cm ×1 cm high-\nresistivity (10 kΩ-cm) silicon substrate. Electron-beam\nlithography is used to pattern the structures in separate\nmetal layers on the chip. Each lithography step is fol-\nlowed by electron-beam evaporation of metal and liftoff in\nN-methyl-2-pyrrolidone at 150◦C for 1.5 hours. Device\nlayers are as follows. (i) 150 nm thick niobium markers,\ndeposited at 3 ˚A/s. (ii) 120 nm thick aluminum ground\nplane, waveguide, flux lines, and qubit capacitors, de-\nposited at 5 ˚A/s. (iii) Josephson junctions evaporated\n(at 5 ˚A/s) using double angle evaporation and consisting\nof 60 nm and 120 nm layers of aluminum, with 15 min-\nutes of static oxidation between layers. We use asymmet-\nric Josephson junctions in the SQUID loop of each qubit\nto mitigate the effects of dephasing [66]. For further de-\ntails, see device parameters given in Table I. (iv) 150\nnm thick aluminum band-aids and air-bridges, deposited\nat 5 ˚A/s. Band-aids ensure electrical contact between\nJosephson junctions and qubit capacitors. Air bridges\nare used to ensure the suppression of the slot-line modes\nin the waveguide [67]. Air-bridges are patterned using\ngrey-scale electron-beam lithography and developed in a\nmixture of isopropyl alcohol and de-ionized water, fol-\nlowed by 10 minutes of reflow at 105◦C [68]. Electron\nbeam evaporation of the band-aid/bridge layer is pre-\nceded by 7 minutes of Ar ion milling.\n2. Measurement setup\nMeasurements are performed in a3He/4He dilution\nrefrigerator. A schematic of our measurement setup is\nshown in Fig. 5. The fabricated chip is wire-bonded to\na PCB and placed in an copper box. The box is then\nmounted to the mixing plate which is cooled to a base\ntemperature of 10 mK.\nThe waveguide input line (WG IN) is used to drive\nqubits and is attenuated at each temperature stage to\nminimize thermal noise; the total attenuation is 70 dB.\nAttenuators (not shown) are added to the input line at\nroom temperature to control input power. Four isolators\nare used to reduce thermal noise in the waveguide out-\nput line (WG OUT). The output is amplified by a high\nelectron mobility transistor (HEMT) amplifier at the 4\nK stage and a room temperature amplifier (not shown)\noutside of the fridge.\nA low noise, multi-channel DC source provides current\nbiases to flux tune qubit frequencies. Low-pass RF filters\n(Aivon Therma-uD25-GL RC filter with 15 kHz cutoff\nfrequency, Mini-Circuits VLFG490+ with 490 MHz cut-\noff frequency) suppress high-frequency thermal noise in\nthe DC lines, which are not attenuated. Qubits are also\nequipped with XY drive lines; RF inputs to the drive lines\nare attenuated (50 dB total) to reduce thermal noise.\nZ2WGINWGOUTXY2\nStillCold pla te10 dB 20 dB 10 dB 10 dB 20 dBMXC pla te4 K pla te50 K pla te300 K\nHEM T10 dB 20 dB 10 dBAiv.\nDC block attenuation circulator50/uni03A9 \ntermination\n10 dBoccEZ3\nAiv.XY3 10 dB 20 dB 10 dB 10 dB\nMiniZ1\nAiv.XY1 10 dB 20 dB 10 dB 10 dB\nMini Mini occEoccEFIG. 5. Measurement Setup schematic of dilution fridge\nwiring for measurement.\nSpectral measurements Elastic scattering measure-\nments (Fig. 2a,b; Fig. 3a) are performed using a vector\nnetwork analyzer (VNA, Agilent N5242A). VNA drives\nare attenuated to sub-single-photon power levels in order\nto prevent qubit saturation. The non-linear least squares\nmethod or circle fit method is used to fit transmission\ntraces. Inelastic scattering measurements (Fig. 2c) are\nperformed with an RF spectrum analyzer (Rohde and\nSchwarz FSV3013); the microwave excitation tone is pro-\nvided by the VNA in zero-span mode. The spectrum an-\nalyzer acquisition is performed with a resolution band-\nwidth of 20 kHz. Measured resonance fluorescence of the\nsub-radiant state is fitted to master equation simulations,\nas discussed in Appendix C.\nState tomography Qubit state tomography is per-\nformed using the Quantum Machines OPX+ (QM) mod-\nule, which is capable of arbitrary waveform generation\nand heterodyne detection. To generate the drive, MHz\nfrequency intermediate frequency (IF) signals from the\nQM are combined with a local oscillator (LO) supplied\nby an RF signal generator (Rohde and Schwarz SMB100)\nusing IQ mixers (Marki Microwave MMIQ-0520LS). For\nreadout, the signal from the output line is downconverted12\nusing an IQ mixer, and the resulting IF signal is digitally\ndemodulated. Details of state tomography are discussed\nin Appendix E 1. Data presented in Fig. 2d; Fig. 3b,\nc; and Fig. 4a, b are recorded using state tomography\ntechniques.\n3. Device parameters\nRelevant device parameters are given in Table I.\nDescription Symbol Value\nQubit 1\nMinimum frequency ω1,min/2π 6.409 GHz\nMaximum frequency ω1,max/2π 7.505 GHz\nTotal Josephson energy EJ(=EJ,a+EJ,b) 25.4 GHz\nJunction asymmetry αJ(=EJ,a/EJ,b) 6.6\nAnharmonicity Ec(=−α) 300 MHz\nQubit 2\nMinimum frequency ω2,min/2π 6.343 GHz\nMaximum frequency ω2,max/2π 7.457 GHz\nTotal Josephson energy EJ(=EJ,a+EJ,b) 25.1 GHz\nJunction asymmetry αJ(=EJ,a/EJ,b) 6.5\nAnharmonicity Ec(=−α) 300 MHz\nQubit 3\nMinimum frequency ω3,min/2π 6.565 GHz\nMaximum frequency ω3,max/2π 7.640 GHz\nTotal Josephson energy EJ(=EJ,a+EJ,b) 26.3 GHz\nJunction asymmetry αJ(=EJ,a/EJ,b) 6.9\nAnharmonicity Ec(=−α) 300 MHz\nSuper (sub)-radiant state (Fig. 2)\nWaveguide distance ℓ(=λ) 18.2 mm\nQubit 1 waveguide decay Γ 1D,1/2π 10.3±0.3 MHz\nQubit 1 intrinsic decoherence Γ′\n1/2π 0.9±0.4 MHz\nQubit 1 Purcell factor P1D,1(= Γ 1D,1/Γ′\n1) 11.4\nQubit 2 waveguide decay Γ 1D,2/2π 10.7±0.4 MHz\nQubit 2 intrinsic decoherence Γ′\n2/2π 1.0±0.5 MHz\nQubit 2 Purcell factor P1D,2(= Γ 1D,2/Γ′\n2) 10.7\nSuper-radiant state waveguide decay Γ 1D,T/2π 18.3±0.4 MHz\nSuper-radiant state intrinsic decoherence Γ′\nT/2π 1.3±0.5 MHz\nQubit 1 lifetime T 1,1 16.6±0.6 ns\nQubit 2 lifetime T 1,2 16.0±1.9 ns\nSub-radiant state lifetime T 1,S 910±47 ns\nStabilization experiment (Fig. 3)\nDrive frequency ω/2π 6.392 GHz\nQubit 1 waveguide decay Γ 1D,1/2π 8.7±0.1 MHz\nQubit 1 intrinsic decoherence Γ′\n1/2π 1.5±0.2 MHz\nQubit 1 Purcell factor P1D,1(= Γ 1D,1/Γ′\n1) 5.8\nQubit 2 waveguide decay Γ 1D,2/2π 10.5±0.1 MHz\nQubit 2 intrinsic decoherence Γ′\n2/2π 1.3±0.2 MHz\nQubit 2 Purcell factor P1D,2(= Γ 1D,2/Γ′\n2) 8.1\nTABLE I. Summary of experimental parameters.\nAppendix B: Theoretical modeling\nHere, we present the theoretical framework used to\nmodel two driven qubits coupled to a waveguide and the\nstabilization of the entangled dark state.\n1. Master equation\nThe most general master equation for this system is\ngiven below [43, 44, 69].˙ρ=−i\nℏ[ˆH+ˆHint, ρ] +Lρ (B1)\nwhere\nˆH/ℏ=X\ni=1,2δi\n2ˆσz,i+1\n2\u0010\nΩiˆσ†\ni+ Ω∗\niˆσi\u0011\n. (B2)\nis the system Hamiltonian written in the drive frame af-\nter applying the rotating wave approximation (RWA).\nBoth qubits are driven at the same frequency ωsuch that\nδi=ωi−ωis the qubit-drive detuning ( ωiis the qubit i\nfrequency). Ω idenotes the Rabi frequency on each qubit.\nIn the case of a drive applied via the waveguide (corre-\nsponding to our experiment), Ω i=|Ωi|eikxiincludes the\npropagation phase accumulated by the drive. Here, xiis\nthe position of qubit iandk=nω\ncis the wavevector of\nthe drive ( ndenotes refractive index, cdenotes speed of\nlight). ˆHintdenotes the photon-mediated interaction.\nˆHint/ℏ=Jˆσ†\n1ˆσ2+J∗ˆσ†\n2ˆσ1. (B3)\nHere, J=p\nΓ1D,2Γ1D,2(e+ik2d−e−ik1d)/4i. For the case\nof two qubits at the same frequency, this expression sim-\nplifies to the more familiar J= (p\nΓ1D,2Γ1D,2/2) sin( kd)\n[44]. Here, Γ 1D,iis the dissipation rate of qubit iinto the\nwaveguide, kiis the wavevector at ωi, and d=|x2−x1|\nis the qubit separation. The Liouvillian is given by\nLρ= Γ1D,1D[ˆσ1]ρ+ Γ1D,2D[ˆσ2]ρ\n+ Γ12D[ˆσ1,ˆσ2]ρ+ Γ21D[ˆσ2,ˆσ1]ρ.(B4)\nHere, Γ 12= Γ∗\n21=p\nΓ1D,1Γ1D,2(e+ik2d+e−ik1d)/2,\nwhich simplifies to Γ 12=p\nΓ1D,1Γ1D,2cos(kd) for res-\nonant qubits. This term denotes the correlated decay\nbetween qubits. The dissipator terms in the equation\nabove are given by\nD[A]ρ=AρA†−1\n2A†Aρ−1\n2ρA†A. (B5)\nD[A, B]ρ=BρA†−1\n2A†Bρ−1\n2ρA†B. (B6)\nIn our experiment, we drive both qubits through the\nwaveguide at a frequency ωcorresponding to qubit sep-\naration of λ. We then have Ω 1= Ω 2. Additionally,\neach qubit is equally detuned from the drive frequency\nso that ω1,2=ω±δ. At this detuning setting, the\nphoton-mediated interaction ( J) disappears exactly, and\nthe Hamiltonian is fully described by Eq. (1), reproduced\nbelow.\nˆH/ℏ=X\ni=1,2δi\n2ˆσz,i+1\n2\u0010\nΩˆσ†\ni+ Ω∗ˆσi\u0011\n. (B7)13\nUsing the triplet and singlet states |T, S⟩= (|eg⟩ ±\n|ge⟩)/√\n2, we can then rewrite the Hamiltonian as in\nEq. (2), repeated below.\nˆH/ℏ=Ω√\n2(|T⟩⟨gg|+|ee⟩⟨T|)−δ(|S⟩⟨T|) + H .c.(B8)\nFor this detuning setting, the correlated decay becomes\nΓ12=p\nΓ1D,1Γ1D,2exp (−i∆kd), where ∆ k=nδ/c. The\nfirst order correction to correlated decay (∆ kd) is ap-\nproximately 0 .02 in our experiment and may be safely\nignored so that Γ 12≈p\nΓ1D,1Γ1D,2. (For our experi-\nment, n= 2.6, δ= 2π×17 MHz , c= 3×108m/s,\nd= 18 .2 mm.) If we additionally take the assumption\nthat Γ 1D,1= Γ1D,2, the Liouvillian in Eq. (B4) simplifies\nto\nLρ= Γ1DD[ˆσ1+ ˆσ2]ρ (B9)\nwhich may be rewritten straightforwardly as\nLρ= 2Γ 1DD[|gg⟩⟨T|+|T⟩⟨ee|]ρ (B10)\nThe simplified Hamiltonian (Eq. (B8)) and Liouvil-\nlian (Eq. (B10)) are illustrated in the left sub-panel of\nFig. 1b. The triplet state |T⟩is super-radiant and decays\nat rate 2Γ 1Dto the ground state |gg⟩. The singlet |S⟩has\nno decay but exchanges population with the triplet at\nrate−δ. We note that we have considered here the case\nwhere qubit separation d=mλ. For qubit separations of\nd=λ/2±mλ, the singlet and triplet are exchanged ( |S⟩\ndecays directly to |gg⟩).\n2. Dark state formation\nIn our experiment, the interplay between drive, de-\ntuning, and decay results in the stabilization of a pure\nentangled state that is dark to the waveguide. In this sec-\ntion, we detail sufficient conditions for the formation of\nthe dark state and discuss their physical meaning, closely\nfollowing the references in [34, 35, 43].\nThe conditions for the existence of a pure, dark, sta-\ntionary state |Ψ⟩are as follows.\n1.|Ψ⟩is annihilated by the collective jump operators\n(or equivalently, |Ψ⟩exists in the nullspace of the\ncollective jump operators). cR,L|Ψ⟩= 0. This con-\ndition ensures that |Ψ⟩does not decay to the waveg-\nuide output ports and is therefore “dark.”\n2.|Ψ⟩is an eigenstate of the Hamiltonian. ˆH|Ψ⟩=\nE|Ψ⟩. This condition ensures stationarity.\nThe collective jump operators for qubits coupled to\na waveguide are given by cR=P\nieikxiˆσiandcL=P\nie−ikxiˆσi, corresponding to collective decay to the\nright and left output ports. Here, kis the wavevector\nat the drive frequency ( k=nω/c ) and xiis the posi-\ntion of qubit i. We note that these jump operators arevalid in the regime that ω≈ω1≈ω2. If waveguide\ndissipation rates Γ 1Dare small compared to qubit fre-\nquencies ( ω1, ω2), and the inter-qubit separation ddoes\nnot greatly exceed a wavelength, variations in phase shift\ndue to different frequencies is negligible [70]. Hence, the\ninterference effects discussed in the following section re-\nmain valid.\nWe first consider condition (1) that cL|Ψ⟩=cR|Ψ⟩= 0.\nFor the case of two qubits coupled to a waveguide, the\njump operators may be written as cR= ˆσ1+eikdˆσ2and\ncL= ˆσ1+e−ikdˆσ2, where d=|x2−x1|is the qubit\nseparation. The dissipators corresponding to these op-\nerators areΓ1D\n2D[cR,L]ρ. We note that for qubit separa-\ntion of d=λ, these dissipators sum to the Liouvillian of\nEq. (B9). In general, the nullspace of cR,Lconsists of |gg⟩\nand|ΨR,L⟩= (|ge⟩ −e±ikd|eg⟩)/√\n2. The state |ΨR,L⟩\nmay be interpreted as the collective qubit state that re-\nsults in destructive interference of waveguide emission in\nthe right (left) direction. For arbitrary qubit separation\nd, no state other |gg⟩than satisfies the destructive inter-\nference condition in both waveguide directions. However,\nwhen d=mλ(ord=λ/2+mλ) for integer m, the singlet\n(triplet) |S, T⟩= (|ge⟩ ∓ |eg⟩)/√\n2 will be annihilated by\nboth jump operators simultaneously, satisfying (1). We\nemphasize here the importance of precise control of phase\ndelay between qubits to achieve destructive interference\nto both waveguide ports. For our experiment, flux-tuning\nthe qubit frequencies allows us to tailor the phase delay\nin situ.\nWhile |gg⟩and|S⟩annihilate both jump operators\nford=mλ, neither are eigenstates of the Hamiltonian\n(Eq. (2), Eq. (B8)). We note here that for arbitrary qubit\ndetunings ( δi) and Rabi drive frequencies (Ω i), it is not\npossible to find an eigenstate of the Hamiltonian within\nthe considered subspace. However, letting Ω 1= Ω 2and\nδ1=−δ2is sufficient to obtain a “dark” eigenstate by\ncombining |gg⟩and|S⟩. This state is given below (and\nin Eq. (3)).\n|D⟩=|gg⟩+α|S⟩√\n1 +α2. (B11)\nHere, the coherent drive, waveguide dissipation, and\ndetunings all destructively interfere to remove coherent\ninteractions between |D⟩and the rest of the state space.\nThe parameter α= Ω/√\n2δsets the the singlet fraction\n(defined here as α2/(1 +α2)). For large Rabi drives, |D⟩\nasymptotically approaches the Bell state |S⟩. By express-\ning the orthogonal state to |D⟩in the dark subspace, we\nmay obtain intuition for the population dynamics. This\northogonal state is denoted as |B⟩, and is given below.\n|B⟩=α|gg⟩ − |S⟩√\n1 +α2. (B12)\nWe observe that the ground state |gg⟩may be re-\nexpressed as |gg⟩= (|D⟩+α|B⟩)/√\n1 +α2, allowing us\nto re-express the Liouvillian in Eq. (B10) as14\nLρ= 2Γ 1DD\u00141√\n1 +α2|D⟩⟨T|\n+α√\n1 +α2|B⟩⟨T|+|T⟩⟨ee|\u0015\nρ(B13)\nThis yields the effective decay (pumping) rates into |D⟩\nand|B⟩from|T⟩.\nγeff,D=1\n1 +α2×2Γ1D=2Γ1D\n1 + Ω2/2δ2(B14)\nγeff,B=α\n1 +α2×2Γ1D=2Γ1DΩ\n(1 + Ω2/2δ2)δ√\n2.(B15)\nEq. (B14) is given in Eq. (4) of the main text. Simi-\nlarly re-expressing the Hamiltonian in Eq. (B8) yields an\neffective Rabi drive between |B⟩and|gg⟩given below.\nΩeff,B=1q\n1 +Ω2\n2δ2\u0012Ω2+ 2δ2\n2δ\u0013\n(B16)\nThese decay and Rabi rates are illustrated in the right\nsub-panel of Fig. 1b. Here, we note the trade-off be-\ntween fidelity of the stationary state to |S⟩(increases\nwith α/(1 +α2)) and the decay rate into the dark state\n(increases with 1 /(1 + α2)). This trade-off is typical of\ndriven-dissipative stabilization schemes [22].\nAppendix C: Inelastic scattering of sub-radiant state\nResonance fluorescence measurements of the sub-\nradiant state ( |S⟩) are fit to master equation simulations,\nfollowing Eq. (B1) with slight modifications. First, the\nqubit-drive detuning is set to zero ( δi= 0) in Eq. (B2).\nNext, internal loss and dephasing terms are added to the\nLiouvillian, given below.\nLintρ=X\ni=1,2Γint,iD[ˆσi]ρ (C1)\nLϕρ=X\ni,j=1,2Γϕ,ij\n2D[ˆσz,i,ˆσz,j]ρ (C2)\nHere, Γ ϕ,iiis the dephasing of qubit iand Γ ϕ,ij=\nΓϕ,ji= Γϕ,corris the correlated dephasing. Correlations\nbetween qubit dephasing can arise when multiple qubits\nare coupled to a single noise source, such as a global mag-\nnetic field [27]. The total simulated Liouvillian accounts\nfor waveguide decay (Eq. (B4)), internal losses, and de-\nphasing ( Ltotρ=Lρ+Lintρ+Lϕρ). The power spectrumof the output radiation field is then given by the two-time\ncorrelation function\nS(ω) = ReZ∞\n0dτ\nπeiωt⟨c†\nR(t)cR(t+τ)⟩ (C3)\nwhere cRis the collapse operator ( cR= ˆσ1+eikdˆσ2)\nas defined in Appendix B. To reduce the number of free\nparameters in fitted simulations, we set Γ int,1= Γ int,2\nand Γ ϕ,11= Γϕ,22. Fitted parameters are Γ int= 0,Γϕ=\n174±24 kHz, Γ ϕ,corr= 127 ±85 kHz, as quoted in the\nmain text and shown in the bottom sub-panel of Fig. 2c.\nWe note that, following the treatment given in [27], the\nsub-radiant state’s decay lifetime (T 1) may be approx-\nimated as T 1= 1/(Γint+ Γϕ−Γϕ,corr) in the case of\nlarge Purcell factor. This relation yields an estimate for\nT1= 3.4µs, which greatly exceeds single qubit lifetimes\nof≈16 ns (see Appendix D).\nAppendix D: Lifetime measurement of an individual\nqubit and the sub-radiant state\nIndividual qubit and resonant sub-radiant state life-\ntimes are presented in Fig. 2d. To measure single qubit\nlifetimes, individual qubits are excited via the waveguide\nwith a constant πpulse. Qubit emission is demodulated\n(see Appendix E 1) using a 80 ns time windows after a\nvariable wait time τ, and⟨ˆσ†ˆσ⟩is calculated. Individual\nqubit T 1lifetimes are T 1= 16.6±0.6 ns for qubit 1 and\nT1= 16.0±1.9 ns for qubit 2.\nSub-radiant state lifetimes are measured by exciting\nthe singlet state ( |S⟩=|eg⟩ − |ge⟩) with a constant π\npulse. For inter-qubit separation of λ(ℓ=λ), driv-\ning through the waveguide can only excite the super-\nradiant triplet state ( |T⟩=|eg⟩+|ge⟩). To excite the\nsinglet state, XY lines are used to drive qubits out of\nphase simultaneously. After a variable wait time τ, the\nground state ( |gg⟩) population is then measured by state-\ndependent scattering [28]. In this scheme, if the ground\nstate is populated, photons are scattered between |gg⟩\nand|T⟩, reducing the transmission amplitude. Scattering\nresults in unit transmission if the system is fully excited\nto|S⟩. Fits to exponential decay profiles are used to ex-\ntract lifetimes. The extracted T 1lifetime for the singlet\nstate is T 1= 910 ±47 ns. We note here the discrep-\nancy between estimated T 1lifetime from inelastic scat-\ntering measurements (T 1= 3.4µs). We attribute this\ndiscrepancy to frequency shifts of the individual qubits\nover multiple days. Measured lifetimes of the sub-radiant\nstate are crucially dependent on the individual qubit fre-\nquencies due to the interference required to protect the\nstate from decay; frequency jitter can therefore strongly\naffect lifetimes. Despite these discrepancies, measured\nand estimated lifetimes of the sub-radiant state greatly\nexceed those of individual qubits, which are limited to\n≈16 ns.15\nAppendix E: State Tomography\n1. Measurement of qubit moments\nMost superconducting qubits use a dedicated resonator\nfor readout using single shot dispersive readout [71]. Due\nto the nature of our experiment, however, the qubits\nare strongly coupled to a separate waveguide. Hence,\nwithout the use of a tunable coupler between the qubits\nand the waveguide, the qubit would mostly be decaying\ninto this common waveguide, limiting the fidelities of the\nreadout that we can achieve. Additionally, unless care-\nfully designed, a readout resonator can act as another\nsource of dissipation for the qubits, thus inhibiting the\nformation of a truly dark state. We instead use the strong\ncoupling of the qubits to the common waveguide to per-\nform our readout using field emission tomography using\ntechniques similar to those presented in [72]. As shown\nthere, a direct mapping exists between the field emitted\nby a qubit and its state. This mapping is used to perform\nquantum state tomography on the two qubits. Following\n[72], the input-output relation for two qubits coupled to\nthe same waveguide is given by:\nˆaout(t) = ˆain(t) (E1)\n+e−iω1t1r\nΓ1D,1\n2ˆσ1(t) +e−iω2t2r\nΓ1D,2\n2ˆσ2(t)\nwhere ˆ σ1,2are the qubit annihilation operators, Γ 1D,1,2\nare the decay rates of the respective qubits to the waveg-\nuide, and t1,2represent the time-of-flight to the qubits.\nˆaoutand ˆainrepresent the output and input propagating\nmodes in the waveguide (this is for only one propoga-\ntion direction; a similar equation exists for the opposite\ndirection). Clearly, measuring the output field of the\nwaveguide can also be used to gain information of the\nqubits’ state. In fact, it can be used to determine the\nexact state of the qubits just before they begin emitting\ninto the waveguide.\nTo see this, consider the case of a single qubit at a fre-\nquency ωqwhose time-dependent annihilation operator is\ngiven by ˆ σ(t). The output field in this case, in the absence\nof any drive, is given by ˆ aout(t) =p\nΓ1D/2ˆσ(t) + ˆain(t),\nwhere ˆ ain(t) is the vacuum. For a qubit naturally decay-\ning into a waveguide, ˆ σ(t) evolves as e−Γ1Dt/2ˆσ(0)e−iωqt,\nwhere Γ 1Dis the coupling to the waveguide. Now con-\nsider the integral,\nˆσ=Z\ndtf(t)ˆaout(t)eiωqt(E2)\nwhere f(t) represents a weighting function to give a\nhigher weight to times where ˆ aout(t) is larger. This is\nknown as temporal mode matching, and Eq. (E2) is a\ncontinuous version of what actually happens during de-\nmodulation of the output field during the experiment (see\nAppendix E 2). By choosing f(t) =√2Γ1De−Γ1Dt\n2Θ(t),where Θ( t) is the Heaviside step function, one can show\nthat the integral ˆ σreduces to simply ˆ σ(0), thus recovering\nthe state of the qubit just before the emission began (our\nchoice of f(t) contains an additional√\n2 factor as com-\npared to previous work because qubits are side-coupled\nrather than end-coupled to the waveguide) [73]. Mode\nmatching is done to maximize the detection efficiency.\nThus, any choice of f(t) still retains all the statistical\nproperties of ˆ σand can be used for the measurement at\nthe cost of possibly decreased SNR. We henceforth use ˆ σ\nto denote the incoming mode, but it is equivalent to the\nstate of the qubit at the start of the emission.\nPrior to being detected, the field ˆ aout(t) passes through\na linear amplification chain with a net gain of G, usu-\nally on the order of many 10’s of dB. This is because\nthe actual detection is done using analog-to-digital con-\nverters that have a minimum voltage threshold, which is\ngenerally much higher than that of a single microwave\nphoton. Because of the amplification, there is some in-\nevitable noise added as well [46, 74], and hence what is\nactually detected is given by (in the limit of G ≫1),\nˆS=√\nG(ˆσ+ˆh†), with ˆhrepresenting the added noise.\nWe discuss how we calculate Gin Appendix E 3. For\nnow, consider the measurement of ˆS= ˆσ+ˆh†. We are\nprimarily interested in finding the moments of ˆ σ, and\nhence need to properly account for the noise ˆh. This can\nbe done by considering the binomial expansion of any\nmoment of ˆS:\n⟨ˆS†nˆSm⟩=nX\ni=0mX\nj=0(n\ni)(m\nj)⟨ˆσ†iˆσj⟩⟨ˆhn−iˆh†m−j⟩(E3)\nHere, the underlying assumption is that the noise is un-\ncorrelated with the signal. This equation can then be\nused to generate a system of equations that can be used\nto recover the moments of ˆ σinstead, given that we know\nthe moments of ˆh. To calculate the latter, we need to\nperform a measurement ˆS0with the signal mode ˆ σin\nvacuum, such that all the moments of ˆ σare 0, except for\nthe 0th order. This gives:\n⟨ˆhnˆh†m⟩=⟨ˆS†n\n0ˆSm\n0⟩ (E4)\nwhich can then be used to calculate the required moments\nof ˆσ.\nOur detection protocol involves driving the two qubits\nfor a certain duration through the waveguide, turning off\nthe drive, and measuring the fields emitted by the two\nqubits. Since the two qubits are not very distant, the\nemitted fields are completely overlapped in the temporal\ndomain. Instead, we note that the qubits are spectrally\nseparated for our experimental settings and hence ˆ σ1(t)\nand ˆσ2(t) rotate at different frequencies in Eq. (E1). This\nallows us to use a frequency multiplexed version of the\nabove protocol.\nExperimentally, a heterodyne detection in the mi-\ncrowave domain involves using an IQ mixer to down-\nconvert from RF to a frequency range that can be digi-\ntized by an ADC. This sampled version of the field can16\nthen be demodulated (along with weights for the tem-\nporal mode matching) at the appropriate frequency to\nobtain an I, Q pair. This I, Q pair represents the two\northogonal quadratures of the detected field S, such that\nS=I+iQ, at the demodulated frequency. But once\nthe signal has been downconverted and digitized, we can\ndemodulate the signal at two different frequencies simul-\ntaneously to obtain two I, Q pairs representing ˆS1and\nˆS2. These are then related to the two qubits ˆ σ1and ˆσ2\nand two noise modes ˆh1andˆh2in the same way that S\nis related to ˆ σandˆh, i.e.\nˆS1= ˆσ1+ˆh†\n1,\nˆS2= ˆσ2+ˆh†\n2.(E5)\nThe moments of the individual modes ˆ σ1and ˆσ2can then\nbe found in the same way as for ˆ σ. The measurements\nforˆh1andˆh2are done by measuring ˆS01andˆS02, where\nthere is no signal mode. The cross moments such as ˆ σ†\n1ˆσ2\ncan also be found in a by finding the cross-correlations\nbetween ˆS1andˆS2. For example:\n⟨ˆS†\n1ˆS2⟩=⟨(ˆσ†\n1+ˆh1)(ˆσ2+ˆh†\n2)⟩\n=⟨ˆσ†\n1ˆσ2⟩+⟨ˆh1ˆh†\n2⟩+⟨ˆσ†\n1ˆh†\n2⟩+⟨ˆh1ˆσ2⟩\n=⇒ ⟨ˆσ†\n1ˆσ2⟩=⟨ˆS†\n1ˆS2⟩ − ⟨ˆh1ˆh†\n2⟩ − ⟨ˆσ†\n1⟩⟨ˆh†\n2⟩ − ⟨ˆh1⟩⟨ˆσ2⟩\nagain under the assumption that the noise mode ˆh1,2are\nuncorrelated with the signal modes ˆ σ1,2.\n2. Mode-matched filtering\nOur field tomography protocol recovers the qubit state\nby heterodyne detection of the field emitted into the\nwaveguide. A finite time window (80 ns) is sufficient to\ncapture the qubit emission ( T1≈16 ns), and the emitted\nsignal is subsequently down-converted with an IQ mixer\nto an intermediate frequency (IF) and digitized. The dig-\nitized signal ( ˜I+i˜Q) is then demodulated to obtain the\nfield; the digital demodulation is detailed in the expres-\nsion below.\nS=I+iQ=X\nnf[n](˜I[n] +i˜Q[n])e−iωIFn(E6)\nHere, Scorresponds to the measured field, nis the index,\nandωIFis the IF. f[n] is the mode-matched filter func-\ntion (discussed previously as f(t) in Appendix E 1 and\nEq. (E2)) and is the focus of the following discussion.\nIn our experiments, qubits are driven via the waveg-\nuide; drives are turned off just prior to the start of the\nmeasurement window. We note that in this discussion,\nthe qubit and drive tone are detuned at the same settings\nas the dark state experiment (see Fig. 3a and Table I for\ndetails). However, we find that artifacts of the drive are\npresent in the measurement window - mainly due to drive\na\nc0\nMeasured\nBackground\nCrosstalk\n00.5\nRe(1)\nIm(1)\n|1|\n|1| /f_it\n0 20 40 60 80 100\nTime (ns)00.51\n|11|\n|11| /f_it0.20.40.6\n0 20 40 60 80\nTime (ns)00.2 Weight\nbFIG. 6. State tomography of an individual qubit a)\nMeasured field quadrature of Qubit 1 at the same detuned\nsetting as Fig. 2a, with Qubit 2 flux-tuned away (purple, S1).\nMeasured background field with both qubits flux-tuned away\n(green, background, S0). Measured field at the same detuned\nsetting as Fig. 2a with Qubit 1 flux-tuned away (orange,\ncrosstalk, S2). The background and crosstalk are minimal\ncompared to the signal. Inset: The mode-matched envelope\nof Qubit 1 (blue) and ideal exponential decay envelope (red).\nThe envelope maximizes the signal from the qubit while re-\njecting noise and crosstalk sources. Background subtracted\n(b,c) field (photon number) measurement of Qubit 1 using\nmode-matched filtering, under a detuned drive (same setting\nas Fig. 2a. Fits to master equation simulations yield Gain\n= 6.3±0.1×105.\nroll-off and drive reflections. First, drive roll-off leaks\ninto the measurement window because of dispersion in\nthe measurement lines. Second, repeated reflections of\nthe drive at microwave connections in the measurement\nchain create delayed copies of drive pulses. In our ex-\nperiment, we measure roll-off and reflections in the qubit\nemission window that are ≈20 dB lower than applied\nwaveguide drives. Parasitic artifacts from the drive are\nwithin ≈5 dB of the power emitted from qubits and over-\nlap temporally with the emission of interest. This effect\ntherefore cannot be ignored; it is further exacerbated by\nthe large qubit dissipation rate into the waveguide and17\nconcomitant short emission duration (Γ 1D/2π≈10 MHz,\nT1≈16 ns). The majority of qubit emission is contained\nin the first 16 ns of the measurement window, which is\nmost susceptible to the discussed parasitic effects. Cap-\nturing this drive signal in addition to qubit emission can\ncause un-wanted correlations between the measured and\nbackground signals, SandS0, compromising the mea-\nsurement fidelity (see Eq. (E3), Eq. (E5)).\nTo overcome this problem, we optimize the mode-\nmatching function f[n] to maximize the measured qubit\nemission while rejecting the parasitic drive signal in the\nmeasurement window. In other words, f[n] is optimized\nto be orthogonal to parasitic artifacts of the drive. For\nthis purpose, we measure 10 averaged samples of single\nqubit emission following a 100 ns drive pulse (denoted\nasS1). Each sample contains 105shots. Multiple aver-\naged samples are used to mitigate effects of fluctuations\nin gain or qubit emission over time. We repeat this mea-\nsurement with the qubit detuned away, such that the\nmeasurement window only captures parasitic drive (de-\nnoted as S0) and no qubit signal. We then use non-linear\nleast squares optimization to minimize the following loss\nfunction over the 10 samples and remove parasitic drive\nfrom the qubit signal.\nLbkg(f[n]) =|S0|\n|S1−S0|(E7)\nHere, minimizing |S0|in the numerator orthogonalizes\nthe filter function f[n] with respect to the parasitic drive.\nHowever, because of the substantial temporal overlap be-\ntween the parasitic drive and qubit signal, simply mini-\nmizing |S0|significantly reduces the qubit signal. There-\nfore,|S1−S0|is included in the denominator to maximize\nthe output qubit signal. Fig. 6a shows the result of this\noptimization for qubit 1. The purple plot denotes S1and\nthe green plot denotes S0. When mode-matching is ap-\nplied, we see that S0is nearly zero over a range of drive\npulse durations. The optimized mode-matched tempo-\nral envelope for qubit 1 is shown in the Fig. 6a inset.\nWe note that the filter function resembles an exponential\ndecay with reduced weights in the first 8 ns of the mea-\nsurement window. This occurs because parasitic drive\nartifacts are localized in this short time window.\nThe steps detailed are sufficient to remove parasitic\ndrive artifacts from the measurement window. Another\nsource of noise in the qubit measurement arises when\nboth qubits are measured simultaneously, as in our two-\nqubit dark state tomography. Fig. 3a shows the detuning\nprofile of qubit 1 and qubit 2 in the stabilization experi-\nment. While the two qubits are separated by ≈3Γ1Din\nfrequency ( |ωIF,1−ωIF,2|>3Γ1D), there is still the pos-\nsibility for qubit demodulation to capture a “crosstalk”\nsignal from the adjacent qubit. To remove this crosstalk,\nwe simultaneously optimize for a second loss function\n(Ltotal=Lbkg+Lcross).Lcross(f[n]) =|S2−S0|\n|S1−S0|(E8)\nHere, the S2signal denotes the adjacent qubit. To\nobtain S2, 10 qubit emission samples are taken with the\ndesired qubit detuned away and the adjacent qubit at the\nexperimental setting. S1andS0are un-changed. Simi-\nlar to the previous case, minimizing |S2−S0|(the adja-\ncent qubit) rejects the parasitic crosstalk from the adja-\ncent qubit. Maximizing |S1−S0|prevents reduction of\nthe desired qubit’s signal. Fig. 6a shows the minimized\ncrosstalk signal from qubit 2 (in orange), where qubit 1\nis the qubit of interest.\nIn the case of perfect mode matching, the mode-\nmatching efficiency is ηF= 1. For imperfect mode match-\ning,ηF<1; this has the effect of reducing the total de-\ntection efficiency and can be interpreted in a similar way\nto attenuation between the device and the first amplifier\n[73]. Ideally, qubit emission follows an exponentially de-\ncaying envelope, and a corresponding exponential filter\nfunction provides perfect mode matching. In our experi-\nment, the optimized filter functions are non-exponential\nin order to reject the parasitic noise sources discussed\nabove. Our experimental mode-matching efficiency to\nqubit emission is therefore imperfect; we find ηF= 0.59\n(0.50) for qubit 1 (2). The Fig. 6a inset shows both the\noptimized (qubit 1) filter function (blue) and the corre-\nsponding ideal filter function (red). Mode-matching effi-\nciency is calculated by taking the squared inner product\nbetween normalized ideal and optimized filter functions.\nWe note that using an ideal exponential filter function\n(where ηF= 1), we obtain ratios between qubit and\nnoise signal |S1−S0|/|S0| ≈1. This indicates the non-\ntrivial effect of parasitic drive features in the absence of\nmode-matching optimization. On the other hand, the\noptimized filter function yields |S1−S0|/|S0| ≈20 (see\nFig. 6a).\nLastly, we verify that our mode-matching scheme does\nnot affect the statistics of measured moments by calibrat-\ning against master equation simulations of single qubits,\nwhich is discussed in the following section.\n3. Gain calibration and noise temperature\nQubit emission from the device passes through an am-\nplifier chain and is then downconverted to an intermedi-\nate frequency, digitized, and digitally demodulated. The\nmeasured demodulated signal must be scaled properly to\naccurately recover the qubit state. We calibrate the gain\nof the measurement chain Gby fitting master equation\nsimulations of a single qubit to demodulated emission,\nshown in Fig. 6b,c for qubit 1. For this measurement,\nqubit 1 is flux-tuned to the dark-state experiment set-\nting (6.409 GHz) and a detuned drive (6.392 GHz) is\napplied. Digital demodulation is applied with ωIFcorre-\nsponding to the qubit frequency. The obtained |⟨ˆσ1⟩|meas18\nand|⟨ˆσ†\n1ˆσ1⟩|measare then fit simultaneously to a master\nequation simulation to obtain |⟨ˆσ1⟩|=√\nG|⟨ˆσ1⟩|measand\n|⟨ˆσ†\n1ˆσ1⟩|=G|⟨ˆσ†\n1ˆσ1⟩|meas. Gain calibration for qubit 1\nand 2 yield G= 6.3±0.1×105andG= 4.2±0.1×105,\nrespectively. Note that gain values include the contri-\nbution from mode-matching. Asymmetry between cali-\nbrated gain values is attributed to asymmetric crosstalk\ncaused by the Fano lineshape of each qubit. A qubit\nexperiencing crosstalk signal from the neighboring qubit\nwill have a reduced gain Gbecause mode-matched filter-\ning will less efficiently capture the desired qubit’s signal.\nWe also independently extract the noise photon num-\nber and temperature of the HEMT amplifier located at\nthe 4-K stage (see Appendix A) by referencing to a res-\nonance fluorescence measurement of a single qubit. This\nmeasurement yields a noise photon number of nHEMT =\n13 photons and temperature of THEMT = 4.1 K. Based on\ncalculated mode-matching efficiencies, the effective noise\nphoton numbers of our measurement are 22 (26) photons\nfor qubit 1 (2).\n4. Maximum Likelihood Estimation\nOnce we have all the moments of the photonic modes\nemitted by the qubit (see Appendix E 1), we use the stan-\ndard method of maximum likelihood estimation (MLE)\nto get the density matrices for the 2-qubit state [73, 74].\nSince we only have emission from the first excited state of\nthe transmon (second excited level is ∼300 MHz away),\nwe can work in the single excitation sector for each mode.\nWe can thus restrict our attention to the 16 moments of\nthe form: Aj= (ˆσ†\n1)n1ˆσm1\n1(ˆσ†\n2)n2ˆσm2\n2∀n1, n2, m1, m2∈\n{0,1}.\nFollowing [74], given a state ρ, the probability of mea-\nsuring ⟨¯Aj⟩as the mean of N measurements of a partic-\nular moment Ajis given by:\np(⟨¯Aj⟩|ρ)∝e−|⟨¯Aj⟩−Tr(Ajρ)|2/(vj/N)(E9)\nwhere vjis the variance of the moment. For large enough\nN, the law of large numbers tells us that we can approx-\nimate vjby the measured variance. One can then define\nthe log of a likelihood function:\n−logL(D|ρ) =16X\nj=1|⟨¯Aj⟩ −Tr(Ajρ)|2/(vj/N) (E10)\nwhere jenumerates all the different moments. This,\nalong with the constraints that Tr( ρ) = 1 and ρ > 0,\ndefines an objective function to be used for the minimiza-\ntion problem that finds the most likely density matrix for\nthe given set of measured moments.\nTo find the fidelity and concurrence bounds, we per-\nform MLE on resampled copies of the moments. We\nrecord the average of all the moments over 1 million shots,\nand perform this entire measurement 3000 times (a total\nof 3 billion averages). This gives us an approximatelyGaussian histogram of 3000 values, from which we ex-\ntract the variance and mean. We then resample each\nmoment from a Gaussian with the corresponding param-\neters to reconstruct 1000 resampled copies. For each re-\nsampled copy, we perform MLE and calculate the fidelity\nand concurrence. The standard deviation can be calcu-\nlated from this list of fidelities and concurrences, and the\n95% confidence interval is then bounded by the 25th and\n975th element of the sorted lists.\nAppendix F: Analysis of the decoherence sources\nWe observe in Fig. 4b that concurrences of the sta-\nbilized entangled state exceed the values predicted by\nqubit Purcell factors ( ∼10, extracted from fits to trans-\nmission traces). To investigate potential sources of this\ndiscrepancy, we perform measurements of qubit decoher-\nence caused by interaction with the waveguide bath. In\nparticular, we study effects of the drive power.\nOur dark state stabilization protocol involves applying\nstrong drives (Ω /Γ1D>1) to the qubits via the waveg-\nuide. Previous studies have found that decoherence dur-\ning a qubit’s driven evolution depends on the noise power\nspectral density (PSD) at the Rabi frequency, which has\nbeen shown to exhibit a 1 /fαdependence [51, 52, 75–77].\nWe investigate the noise PSD by measuring the driven\nevolution of single qubits using Rabi noise spectroscopy\n[75] and a spin-locking (SL) protocol [51].\n1. Driven evolution of a qubit\nThe decoherence of a qubit under driven evolution has\nnoise contributions from the qubit frequency ωq, the Rabi\ndrive frequency Ω R, and quasi-static noise [75]. We con-\nsider the evolution of a single qubit under a resonant\ndrive, which is described by the following Hamiltonian in\nthe laser frame, with δ=ωq−ωd= 0.\nˆH=δ\n2ˆσz+ΩR\n2ˆσy (F1)\nTo understand our noise spectroscopy measurement pro-\ntocols, We invoke an analogy between the free and driven\nevolution of a qubit [51]. A freely evolving qubit re-\nvolves around the z-axis of the Bloch sphere at a fre-\nquency δ(in the laser frame). The qubit will experi-\nence a longitudinal (Γ 1= 1/T1) and transverse decay\n(Γ2= Γ 1/2 + Γ ϕ= 1/T2, where Γ ϕis pure dephasing)\nwhen subjected to the environment. When the qubit is\ndriven on resonance in the y-direction, the qubit state\nrevolves about the y-axis with the Rabi frequency Ω R.\nIn this case, the driven dynamics can be interpreted as a\nfreely evolving spin quantized in the y-axis direction. In\nanalogy with the case of the non-driven qubit, longitudi-\nnal (˜Γ1) and transverse ( ˜Γ2) decay rates may be defined\n(given below for a resonant drive) [52].19\nWG\nWG\n101102\nDrive (MHz)102103Dephasing (kHz)Rabi\nSL3\n00.51x\n0 20 40 60 80 100 120\n (ns)00.51x\nba c\nFIG. 7. Individual qubit decoherence when driven via the waveguide a) Driven Rabi oscillations of qubit 2 with drive\nof Ω R/2π= 30 MHz. Extracted Γ ϕ= 133 kHz. Inset: Rabi spectroscopy pulse sequence. b) Spin-locking protocol decay signal\nfor qubit 2, with drive of Ω R/2π= 30 MHz. Extracted Γ ϕ= 93 kHz. Inset: Spin-locking protocol pulse sequence c) Extracted\ndephasing Γ νover a range of drive powers for the Rabi (blue) and SL (red) protocols, with 95% confidence intervals indicated\nby error bars. Dotted line indicates fit of spin-locking measurement dephasing v. drive power to 1 /fα, yielding α= 1.85. Data\ncorresponding to a,b) are shaded in gray.\n˜Γ1=1\n2Γ1+ Γν (F2)\n˜Γ2=3\n4Γ1+1\n2Γν (F3)\nHere, Γ ν=πSz(ΩR), is the pure dephasing under\ndriven evolution, where Sz(ΩR) is the noise PSD at the\nRabi frequency Ω R. (Γ ϕ=πSz(0) is the pure dephas-\ning under free evolution). By varying the qubit drive\nstrength and measuring decoherence rates, the noise PSD\nSz(ΩR) may be extracted. We measure ˜Γ2and˜Γ1in two\nindependent experiments: driven Rabi spectroscopy and\na modified spin-locking (SL) protocol (respectively). We\ndescribe these experiments in the following sections.\n2. Driven Rabi spectroscopy\nTo measure Rabi oscillations, we directly drive the\nqubit via the waveguide with variable pulse durations\n(τ) and measure the emitted field to obtain the qubit\nstate. The measurement pulse protocol is shown in the\ninset of Fig. 7a. For a qubit initially in the ground state\n⟨σz⟩(t= 0) = −1 and given Ω R≥ |Γ1−Γ2|/2, the driven\nevolution may be solved analytically, yielding\n⟨σx⟩(τ) =x∞−\u0012˜Γ2x∞−ΩR\nνRsin (νRτ)\n+x∞cos (νRτ)\u0013\nexp(−˜Γ2τ)(F4)with⟨σy⟩(τ) = 0. Here, νR=p\nΩ2\nR−(Γ1−Γ2)2/4 is\nthe effective Rabi oscillation frequency. The steady state\nvalue of ⟨σx⟩isx∞= Γ1Ω/(Γ1Γ2+ Ω2), and the envelope\ndecays at ˜Γ2. An example of measured Rabi oscillations\nof qubit 2 is shown in Fig. 7a, with a fit to Eq. (F4). In\nthese fits, Γ 1is set constant at 10.5 MHz (estimated from\ntransmission traces, see Table I), while all other param-\neters are allowed to vary. We note here that we exclude\nquasi-static noise in our analysis [75], which manifests\nin non-exponential decay (a feature not observed in our\nexperiment). Sweeping the drive power Ω Rfrom 10-37\nMHz gives an approximate 1 /fdependence for Γ ϕ(and\ntherefore Sz(ΩR)), shown in Fig. 7c. Extracted Γ νvalues\nrange from 0.1-1.1 MHz. We note that at drive powers\nabove 37 MHz, fitted Γ νvalues approach zero. This indi-\ncates that the driven Rabi measurement is not sensitive\nenough to detect Γ ϕ<0.1 MHz because large Gamma 1\ndominates the decay. This motivates the use of the spin-\nlocking protocol discussed next.\n3. Modified spin-locking protocol\nWe independently extract Γ νusing a modified spin-\nlocking (SL) protocol discussed in [51], which measures\nlongitudinal decay in the driven qubit frame ( ˜Γ1). Previ-\nous studies have demonstrated more sensitive noise spec-\ntroscopy with the SL protocol, citing robustness against\nlow frequency fluctuations in Rabi frequency (Ω R) due to\ninstrumentation and low-frequency qubit dephasing [51].\nThe protocol involves first using a resonant π/2 pulse\nin the x-direction (on the Bloch sphere) to initialize a20\nqubit state collinear with the y-axis. This is immedi-\nately followed by a continuous π/2 phase-shifted drive\nin the y-direction to begin the driven evolution. Ideally,\nthe drive and qubit state are parallel, so that the qubit\nstate relaxes to the center of the Bloch sphere at rate\n˜Γ1. This pulse sequence is shown in the Fig. 7b inset.\nThe qubit state is measured via waveguide emission as\nit decays. We note here that our measurement protocol\ndoes not include a second π/2 pulse used in [51] because\nwe do not use a readout resonator. An example of the\nexponential decay profile measured for qubit 2 is shown\nin Fig. 7b. Fits to the relaxation include fixed param-\neter Γ 1= 10.5 MHz and Γ νfree. Extracted Γ νranges\nfrom 60-800 kHz over a range of Rabi frequencies 15-48\nMHz. A fit to extracted Γ νreveals a 1 /fαdependence\nwith α= 1.85.\n4. Purcell factor under driven evolution\nThe observed reduction in Γ νat higher drive powers in-\ndicates that the true Purcell factor of our qubits is larger\nthan∼10 (extracted from VNA traces), which qualita-\ntively explains the increase in concurrence observed in\nFig. 4b. However, extracted concurrences do not indi-\ncate monotonically increasing Purcell factors with drive\npower (Fig. 4b), which would be expected for a purely\n1/fαtrend in dephasing. Instead, at low powers, concur-\nrences indicate Purcell factors exceeding 30, while at high\npowers, Purcell factors range from 10-30. Previous stud-\nies [51] have reported Lorentzian “bump-like” features in\nnoise PSD attributed to coherent two-level system (TLS)\nfluctuators. While this could explain the above discrep-\nancies, the large confidence bounds of extracted Γ νvalues\n(Fig. 7c) preclude a quantitative conclusion.\nAppendix G: Numerical modeling\nWe use a master equation solver in QuTip [78, 79] to\nsimulate our system. We use the same Hamiltonian as in\nAppendix B, but also include the third level (f) of trans-\nmons (see Appendix G 1). The simulation also includes\ndissipators for dephasing and intrinsic loss of the individ-\nual qubits (for the second level), as well as for correlated\ndecay into the waveguide between the two transmons.\nNote that we do not include any waveguide mediated in-\nteraction in the Hamiltonian as discussed in Appendix B.\nThe shaded regions in Fig. 3 and Fig. 4 were found via\nthese simulations, using different Purcell factors. The\nPurcell factor is varied by changing the intrinsic loss and\ndephasing rates of the individual transmons. Parameters\nsuch as the individual coupling rates of the qubits to the\nwaveguide are set to match the fits of experimental data\nat the operation frequencies, i.e. coupling rates of 8.7\nMHz for Qubit 1 and 10.5 MHz for Qubit 2.\na\nb\n2 4 6 8 101020304050\n0.00.10.20.30.40.50.60.70.8\nFidelity2 4 6 8 101020304050\n0.00.10.20.30.40.50.60.70.8\nFidelity\nδ (MHz)\nδ (MHz)\n/uni03A9 (MHz)/uni03A9 (MHz)FIG. 8. Achievable fidelities for different parameter regimes\nof drive strength (Ω and detunings ( δ), without the e-f transi-\ntion (a) and with the e-f transition (b) included in the master\nequation. Clearly, the inclusion of the third level makes a sig-\nnificant difference in the operational region, while the maxi-\nmum fidelity is roughly the same. See Appendix G 1 for more\ndetails of simulation.\n1. Effects of the e-f transition\nAn important aspect of the simulations was the inclu-\nsion of the third level of the transmons. We observed\nthat while the third level did not significantly impact\nthe maximum achievable fidelities, it did affect the op-\neration ranges for a given fidelity. As can be seen from\nAppendix B, for no dephasing and no intrinsic loss, the\nsinglet fraction, which is directly related to the fidelity for\na pure state, is only a function of the ratio of the detuning\nand the drive power. This would mean that increasing\nthe drive strength for any detuning would help increase\nthe fidelity of the steady state. Practically, this would21\nalso increase the settling time which competes with any\nintrinsic loss or dephasing of the qubits. Hence, there are\noptimal drive-to-detuning ratios for a true two-level sys-\ntem model. This can be seen in panel (a) of Fig. 8 which\nshows results from a simulation with a Purcell factor of\n200 for the qubits and equal couplings of 10 MHz to the\nwaveguide. We perform a sweep for various drives and\ndetunings, but do not include the e-f transition for panel\n(a). Here, we can clearly see regions of high fidelity even\nfor high drive powers. In contrast, once we include the\ne-f transition (panel (b)), the parameter space with high\nfidelities is restricted. Increasing the drive power for a\ngiven drive-to-detuning ratio does not necessarily help in\nthis case. We expect that this has to do with the e-f tran-\nsition allowing the system to escape out of the relevant\nHilbert space. Since the third level is populated more for\nhigher drive powers, we see a clear deviation from the\ntwo-level system model at these drives.\nAppendix H: Achievable fidelities in future\nexperiments\nIn this section we discuss the challenges that we faced\nin our experiment that limited our fidelities. We will\nthen discuss avenues to overcome these bottlenecks and\nuse simulations to show that it is possible to even reach\nfidelities of over 90% with this driven-dissipative stabi-\nlization protocol and experimentally viable parameters\nfor the transmons.\nThe first limitation of our experiment was in fact\nthe effect of the e-f transition. As mentioned in Ap-\npendix G 1, the highest achievable fidelity is not very\ndifferent between a true two-level system model and that\nof a transom, but the operation point does change and\ntends to move towards lower detunings. While we could\nin principle use such an operation point, the effects of\ncross-talk and drive roll-off (described in Appendix E 2)\nwere more pronounced at lower detuning and would have\naffected our measurements.\nThis effect could be avoided by using fast flux lines to\ntune the qubits far enough after state preparation to con-\nduct the measurement. Alternatively, dedicated readout\nresonators with other architectural changes to the de-\nsign could be used as well. In particular, the coupling of\nthe qubits to the waveguide would have to be tunable.\nWithout a tunable coupling, the qubits would emit into\nthe waveguide during readout which would reduce the\nfidelities of measurement. The fact that the qubits are\nstrongly coupled to the waveguide (order of 10s of MHz\nto increase the Purcell factors) would reduce the fideli-\nties further since they impose stringent limitations on the\nduration of the dispersive readout pulse.\nAnother limitation to our experiment was the imbal-\nance in Γ 1Dof the two qubits at their operation points\ndue to standing modes in the background causing Fano\nin the waveguide response. While this in itself is not very\ndetrimental to the fidelities, having a common drive and\n40 50 60 70 80\nTemperature (mK)02004006008001000 Purcell Factora\n0 200 400 600 800 1000\nPurcell factor0.50.60.70.80.9Maximum /f_idelity\nbFIG. 9. Maximum achievable fidelities a) Achievable fidelities\nfor different Purcell factors of the individual qubits. Dashed\nline marks the 50% threshold for entanglement. b) Estimates\nof maximum achievable Purcell factors under strong driving.\nUpper limit on Purcell factor set by waveguide temperature\nwhen dephasing is ignored, assuming internal losses of Γ′= 6\nkHz. At 39 mK, Γ 1D/Γ′= 730.\ndifferent Γ 1Dcauses a difference in the Rabi drives that\neach qubit sees. This imbalance in drive strength can be\nshown to prevent the singlet state from being truly dark\n[43], thus reducing the fidelities. Our experiment had an\nimbalance of around 4%, which along with the limitation\non detunings mentioned earlier, reduced our maximum\nachievable fidelities by around 6% (see blue shaded re-\ngion of Fig. 4c). The previously mentioned design of\nhaving tunable couplers can help mitigate this effect as\nthe coupling rates would then be in-situ tunable.\nWhile these two challenges did affect our fidelities, the\nprimary bottleneck in our experiment was in fact the Pur-\ncell factor of our qubits ( P1D= Γ1D/Γ′,Γ′= 2Γ 2−Γ1D=\nΓint+2Γ ϕ). This sets an upper bound on the fidelities in\nour stabilization protocol. Fig. 9a shows the maximum\nachievable fidelities as the Purcell factor is increased.\nFrom the measured transmission of our qubits, we es-\ntimate that our (undriven) Purcell factors are 10, which\nclearly limits our fidelities. Of course, in our protocol,\nthe continuous driving of the qubits reduces the pure de-\nphasing of the qubits (Appendix F 4) and hence we see\nhigher concurrences and fidelities than that estimated by\na Purcell factor of only 10.\nWe estimate that this scheme of generating entangle-\nment can reach fidelities exceeding 90% (concurrence ex-22\nceeding 0.86) for driven Purcell factors of 600 and above.\nTo our knowledge, the highest reported Purcell factor\nfor a qubit directly coupled to a waveguide is approxi-\nmately 200 [27]. Meanwhile, state-of-the-art aluminum\ntransmons exhibit lifetimes of several hundred microsec-\nonds [80], which correspond to Purcell factors exceeding\n1000. One of the causes of this apparent gap in perfor-\nmance is the thermal occupation of the waveguide in the\nformer case. Qubits directly coupled to waveguides ex-\nperience additional decoherence due to interaction with\nthermal photons residing in the photonic bath. This non-\nzero waveguide temperature degrades the Purcell factor.\nAgain, for our stabilization protocol, strong drives mit-\nigate this effect slightly due to a reduction in pure de-\nphasing (discussed in Appendix F 4). Nevertheless, the fi-\nnite waveguide temperature ultimately imposes an upper\nbound on achievable Purcell factors. Following the mas-\nter equation treatment of [27], Purcell factor in the pres-ence of a finite waveguide temperature is P1D= Γ1D/Γ′\nth,\nwhere Γ′\nthis the intrinsic qubit decoherence and is de-\nfined as Γ′\nth= 2Γ 2,th−Γ1D(Γ2,th= Γ 1,th/2 + Γ ϕ,\nΓ1,th= (2¯nth+ 1)Γ 1, ¯nthdenotes bath occupancy).\nUsing a reference measurement of Γ 1for a separate\nqubit chip (without any coupling to a waveguide), we\nmay estimate Γ′for the main device. This separate chip\ncontains a qubit coupled to a readout resonator and un-\ndergoes an identical fabrication procedure as the main\ndevice. We measure T1= 25 .5±1µs for the separate\nqubit device, corresponding to Γ′/2πof 6 kHz. Previ-\nously reported waveguide temperatures have been as low\nas 39 mK [72]. At 39 mK, the Purcell factor is limited to\n730 by the bath (for an estimate of Γ′/2π= 6 kHz), as\nshown in Fig. 9b. This would enable us to reach a fidelity\nof 91% and a concurrence of 0.88 with a stabilization rate\nof 250 kHz."    },    {        "title": "2402.15775v1.Optimal_frequency_resolution_for_spectral_proper_orthogonal_decomposition.pdf",        "content": "Optimal frequency resolution for spectral proper orthogonal decomposition\nLiam Heidt1a, Tim Coloniusb\naGraduate Aerospace Laboratories of the California Institute of Technology, California Institute of Technology, California, USA\nbDepartment of Mechanical and Civil Engineering, California Institute of Technology, California, USA\nAbstract\nWe demonstrate that accurate computation of the spectral proper orthogonal decomposition (SPOD) critically depends\non the choice of frequency resolution. Using both artificially generated data and large-eddy simulation data of a\nturbulent subsonic jet, we show that the optimal choice depends on how rapidly the SPOD modes change in space at\nadjacent frequencies. Previously employed values are found to be too high, resulting in unnecessarily biased results\nat physically important frequencies. A physics-informed adaptive frequency-resolution SPOD algorithm is developed\nthat provides substantially less biased SPOD modes than the standard constant resolution method.\nKeywords: Spectral proper orthogonal decomposition, SPOD, Spectral analysis\n1. Introduction\nSpectral proper orthogonal decomposition (SPOD) [1, 2] is a data-driven method used to extract coherent struc-\ntures from statistically stationary data and has been extensively used to study flow physics [3–5] and to develop\nreduced order models [6, 7]. SPOD is typically estimated using the Welch periodogram method, which requires the\nchoice of frequency bin width consistent with the amount of available data. Modes are subsequently computed using\nfrequency-domain data averaged over this bin width. This frequency resolution is crucial due to the well-known bias-\nvariance tradeo ff[8], and this resolution is typically chosen such that spectral peaks, if any, are su fficiently resolved.\nThis work demonstrates that the choice of frequency resolution also depends on how rapidly the SPOD modes\nchange in space from one frequency to the next. We first illustrate this using artificially generated data and then on a\npreviously analyzed turbulent subsonic jet. We show previously employed values of bin width are too high, resulting\nin unnecessarily biased results at physically important frequencies. We then develop a physics-informed adaptive\nfrequency-resolution algorithm that provides substantially less biased SPOD modes than the standard constant res-\nolution method. Lastly, we demonstrate that the alignment metric widely used to quantify the similarity between\ndifferent SPOD modes (or between SPOD and resolvent analysis [9]) is sensitive to bias and the degree of statistical\nconvergence, so care must be taken when inferring conclusions.\n2. Computation of SPOD and the influence of ∆f\nAn overview of SPOD is provided here; the reader is referred to [1, 8, 10] for details. Let qkbe a possibly\ncomplex vector-valued snapshot of a statistically stationary flow at time tkon the spatial domain Ω. The snap-\nshot length Nequals the number of variables multiplied by the number of spatial locations. Assume we have Nt\nequally spaced snapshots available tk+1=tk+ ∆t. To employ Welch’s method, we obtain the overlapping data matrix\nQ(n)=[q(n)\n1,q(n)\n2,···,q(n)\nNs], where q(n)\nk=qk+(n−1)(Ns−N0),Nsis the number of snapshots per block, N0is the number\nof snapshots that overlap, and Nbis the number of blocks. Assuming ergodicity, each block is an independent flow\nrealization. Next, the Fourier transform of each windowed block ˜Q(n)=Q(n)·w=[w1q(n)\n1,w2q(n)\n2,···,wNsq(n)\nNs] is\ncomputed ˆQ(n)=FFT (˜Q(n),Nf), where wis a window that reduces spectral leakage and FFT (Q(n),Nf) is the Nf\nlength Fourier transform of ˜Q(n). IfNs<Nf, the data is zero-padded. This computes SPOD at Nffrequencies with a\n1Corresponding author: lheidt@caltech.edu\nPreprint submitted to Elsevier February 27, 2024arXiv:2402.15775v1  [physics.flu-dyn]  24 Feb 2024spectral resolution ∆f=1/(Ns∆t), where the SPOD estimate at fkaverages the data in f=[fk−∆f/2,fk+∆f/2]. The\nfrequency vector is given by fk=k−1\nNf∆t,k=[1,Nf]. Alternatively, a non-FFT method can be used, like the Goertzel\n[11] method, where specific discrete frequencies are chosen. The cross-spectral density (CSD) at fkis estimated by\nSfk=ˆQfkˆQ∗\nfk, where ˆQfk=√κ˜Q(n),κ= ∆t/(||w||2Nb), and∗is the Hermitian transpose. SPOD is computed by\nSfkWΨfk=ΨfkΛfk, (1)\nwhere Wis a weighting matrix. The estimated SPOD modes are given by the columns of Ψfkand are ranked by their\nenergyλj. The CSD matrix, Sfk, is an N×Nmatrix, far too large for typical problems. Thus, to practically compute\nSPOD, we employ the exact method-of-snapshot approach [12], giving\nˆQ∗\nfkWˆQfkΘfk=ΨfkΘfk, (2a)\nΨfk=ˆQfkΘfkΛ−1/2\nfk. (2b)\nThis estimate converges as the number of blocks Nband the number of snapshots in each block Nsare increased\ntogether [13, 14]. For a fixed amount of data Nt, asNsincreases, Nbdecreases. This leads to an estimate with a\ngreater variance but reduced bias. The vice-versa is also true. Practitioners attempt to determine a Ns(and thus Nb)\nthat results in the most accurate SPOD modes. It is well known that for flows exhibiting a spectral peak, such as\nan open cavity, a greater Nsis required to ensure minimal bias. This is because when computing SPOD at a discrete\nfrequency fk, the energy /data from [ fk−∆f/2,fk+∆f/2] is averaged. Thus, if the energy /flow in [ fk−∆f/2,fk+∆f/2]\nvaries substantially, the discrete estimate of the SPOD modes (and energy) at fkwill not accurately reflect the true\nmodes.\n3. Artificial problem\nIn addition to the bias being high if ∆fis large compared to the bandwidth of a spectral peak, we hypothesize\nthat a small ∆fis required when the SPOD modes change rapidly as a function of f. By that, we mean if the “true”\nSPOD modes change substantially between fk±∆f/2, then the estimated SPOD at fkwill contain a high amount of\nbias (i.e. error). To verify this, we construct artificial data with a known SPOD spectrum /modes and then analyze the\nimpact of ∆fas a function of how rapidly the known SPOD modes vary.\nBy definition, a random statistically-stationary process q(x,t) with a SPOD spectrum λj(f) and modes ψj(x,f)\nhas a CSD, S(x,x′,f)=P∞\nj=1λj(f)ψj(x,f)ψj(x,f)∗. Data with these properties can be generated by\nq(x,t)=Z∞\n−∞ˆq(x,f)e−i2πf td f (3a)\n=Z∞\n−∞˜Fn\nG\u0010\n0,S(x,x′,f)\u0011\n,fo\ne−i2πf td f, (3b)\nwhere g(x,t)=G(0,S(x,x′,f)) is zero-mean Gaussian white noise with a covariance kernel of S(x,x′,f) (i.e. a\ncovariance kernel equal to the CSD of q(x,t) at frequency f) and ˜F\bg(x,t),f′\tis a Kronecker delta filter such that\nˆg(x,f)=0∀f,f′. The dominant SPOD modes are defined as\nψ1(x,f)=e−x2/(2c2\n1)e1ix f c 2/(π1/4c1/2\n1), (4)\nwhereπ1/4c1/2\n1is a normalization constant ensuring ⟨ψ1(x,f),ψ1(x,f)⟩=1,c1is a constant that defines the spatial\ndistribution, and c2defines how rapidly the modes change from one frequency to the next. Increasing c2results in\nmore rapidly changing SPOD modes. Additionally, λ1(f)=1,λ2(f)=0.5, andψ2(x,f)=r(x,f), where r(x,f) is a\nrandomly spatially distributed mode orthogonal to ψ1(x,f).\nNext, we generate discrete data using our analytical SPOD modes, numerically compute SPOD using varying ∆f,\nand then compute the alignment between the numerical (n) and analytical (a) SPOD modes α=⟨ψ1,a(x,f),ψ1,n(x,f)⟩.\nSince this process is stochastic, we repeat this process to obtain a converged averaged alignment. In this artificial\ncase, each frequency is independent, allowing us to also average over di fferent frequencies. For all results, we confirm\n2sufficient convergence (not shown). We employ an equally spaced grid x=[−20,20] with Nx=801,∆t=1,c1=5,\nc2=10,20,40,80,160, Nt=5000,10000,20000,40000,80000,160000, and an SPOD block overlap of 67%.\nIn figure 1 (a), we display an example mode R{ψ1(x,f)}atf=0,0.01 and the alignment α=|⟨ψ1(x,0),ψ1(x,0.01)⟩|.\nWe see that as c2increases, the modes vary more rapidly. In figure 1 (b), we display the average alignment between\nthe analytical and numerical SPOD modes as a function of ∆ffor the di fferent values of c2andNt=5000. For a given\nc2, we find that as ∆fincreases, the alignment first increases and then decreases. This is the standard bias-variance\ntradeo ffwith high variance at low ∆fand high bias at high ∆f. We clearly note that as c2increases, the optimal ∆f\ndecreases, confirming our hypothesis that rapidly changing SPOD modes require a smaller ∆f. In figure 1 (c), we\nshow the average alignment for a constant c2=160 as a function of Nt. As expected, the error decreases for larger Nt,\nand the optimal ∆falso decreases for increasing Nt. Here, we find that if a too-high ∆fis chosen, the error does not\ndecrease with increasing Nt(since the error is dominated by bias and not variance). Thus, selecting an appropriate ∆f\nis crucial, and an appropriate value varies greatly depending on how rapidly the SPOD modes vary.\na)\n b)\n c)\nFigure 1: Real component of the analytical artificial problem SPOD modes (a) and average alignment between numerical and analytical SPOD\nmodes as a function of ∆ffor varying c2(b) and Nt(c).\n4. Turbulent jet\nWe now perform a similar investigation on an LES of an isothermal Mach 0.4 round jet with a Reynolds number\nRej=ρjUjD/µj=4.5×105, whereρj,Uj,D,µjare the jet density, velocity, diameter, and dynamic viscosity. For\ndetails, we refer the reader to [3, 15]. We simulate tsimc∞/D=16000 flow through times saving Ntot=80000\nsnapshots (4−8×previous studies of similar turbulent jets [3, 16, 17]), with ∆t=0.2. We look at the axisymmetric\nfluctuating components only.\nWe now compute SPOD using a variety of NsandNtfor a range of f(using either method described in §2). Since\nthe true SPOD modes are not known, we compare them to resolvent analysis modes generated using an identical\nmethod as in Heidt et al. [16]. While resolvent analysis modes are not exactly SPOD modes for turbulent jets, the\nalignment has been shown to be high over the range of important frequencies. In addition, any error due to bias or\nvariance is not likely to align in the direction of the resolvent mode. Thus, improvement in the alignment can be\nconsidered improvements to the accuracy of the corresponding SPOD mode.\nIn figure 2 (a, b), we display the alignment between the Nt=10000 SPOD case for several ∆S t. For Nt<Ntot,\nmultiple sets of SPOD are computed with 50% overlap between datasets, which are used to compute the average\nalignment. We see that for higher S t(>0.5), increasing ∆S tresults in better alignment (i.e. decreases variance).\nFor low S t(≈0), increasing ∆S tdecreases alignment due to increasing bias. Using a constant ∆S t=0.05 (a\ntypical value when studying subsonic jets) results in severely biased results when investigating S t→0, which is an\nimportant regime that contains more energy than any other frequencies. In the intermediate region ( S t∈[0.1,0.5]), the\nalignment initially increases with increasing ∆S tdue to decreasing variance and then decreases due to increasing bias.\nThis occurs because the wavelength of the dominant mode scales approximately inversely with S tdue to the constant\nphase speed of the wavepackets. Thus, modes at S t=0.05,0.075 (resolvent mode alignments |⟨u1(S t=0.05),u1(S t=\n0.075)⟩|=0.4) vary more rapidly than modes at S t=1.5,1.525 (|⟨u1(S t=1.5),u1(S t=1.525)⟩|=0.85). Thus, based\non §3, we expect a lower /higher ∆S tto be required at low /high S t, respectively. In figure 3, we display the dominant\n3resolvent and SPOD modes at S t=0.05 for ∆S t=0.0125,0.025,0.05. Here, we find that as ∆S tincreases, the mode\ninitially becomes less noisy, indicating less variance. In contrast, the mode shape changes substantially for ∆S t=0.1,\nwith the wavelength greatly reducing due to increasing bias.\na)\n b)\nFigure 2: Alignment between resolvent and SPOD for Nt=10000 for various ∆S t(a) (zoomed in (b)).\nFigure 3: Dominant resolvent and SPOD modes at S t=0.05 for several ∆S tatS t=0.05.\nWe now create an adaptive resolution SPOD algorithm to vary ∆ffor improved alignment. We employ a similar\ncost function as Yeung and Schmidt [18] and begin with a small ∆fjand increase ∆fj+1until\nJ=1−|⟨ψ1,∆fj(fk),ψ1,∆fj+1(fk)⟩|<e, (5)\nwhereψ1,∆f(fk) is the dominant SPOD mode at frequency fkcomputed using a frequency bin width of ∆fandeis\na convergence tolerance. We also find that at frequencies where the SPOD modes vary slowly, a better convergence\ncan be reached for a fixed amount of data (as seen in §3). Thus, we vary e=e(f) using known physics. For jets,\ndue to the constant phase speed, increasing frequency results in more slowly varying SPOD modes. Thus, we assume\ne(f)=e2/f, such that\nJ=1−|⟨ψ1,∆fj(fk),ψ1,∆fj+1(fk)⟩|<e2/fk, (6)\nIn figure 4 (a, b), we show the alignment between the dominant SPOD and resolvent modes for several ∆S tand the\ntwo adaptive procedures described, using a convergence constant of e=0.5,e2=0.1, for Nt=5000,20000. Here,\nboth adaptive methods switch from a lower ∆S tat low S tto a higher ∆S tat higher S t. Using a fixed convergence\ntolerance results in the optimal ∆S tdecreasing with increasing data lengths (consistent with results found in §3\nsince the variance decreases and ∆S tbecomes smaller to reduce the bias). Both adaptive methods result in greatly\n4improved SPOD modes compared to employing a fixed ∆S t. In particular, the modified adaptive method provides\nexcellent alignment across all frequencies and data lengths.\nIn general, we note that no value of e,e2can be considered universal, and for each case, practitioners must use\nphysical intuition to determine the best values of ∆f. In general, we recommend increasing ∆fwhile the mode shape\nstays similar but becomes less noisy (i.e. decreasing variance). If the mode shape changes, ∆fis likely too high, and\nlarge bias errors will occur.\nLastly, in figure 4 (c), we show the alignment as a function of the data length and ∆S t=0.025. We also display\nthe standard deviation of the alignment for Nt=10000, where we find that the alignment is sensitive and varies\ngreatly from one set of snapshots to another. We also find that the alignment increases substantially with increasing\nNt(particularly at lower frequencies). This shows that the alignment metric requires a significant amount of data\nto converge, even once the modes look visually similar, which we show for S t=0.2 in figure 5. Thus, due to\nthe high variance and slow convergence of the alignment metric, caution is advised when inferring results from (or,\nin particular, comparing between) alignments. With respect to the modeling of turbulent flows, and in particular\nturbulent jets, the substantial increase in alignment with increasing Nt(with the alignment increasing by up to ≈0.2)\ndemonstrates that resolvent analysis can model coherent structures even better than previously determined [19].\na)\n b)\n0 0 . 5 1\nS tN t = 5 0 0 0\nN t = 1 0 0 0 0\nN t = 2 0 0 0 0\nN t = 4 0 0 0 0\nN t = 8 0 0 0 0 c)\nFigure 4: Alignment between dominant SPOD and resolvent mode for Nt=5000 (a), Nt=20000 (b) jet for varying ∆S tand the two adaptive\nmethods. (c) Alignment between dominant SPOD and resolvent mode for ∆S t=0.025 for varying Ntalong with standard deviation of alignment\nforNt=10000 (shaded region).\n5. Conclusion\nThis work demonstrates that the choice of frequency resolution ∆fis vital to obtaining accurate SPOD modes,\nand the optimal choice critically depends on how rapidly the SPOD modes change in space at adjacent frequencies.\nWe show that previously employed frequency resolution values are too high, which results in unnecessarily biased\nresults at physically important frequencies. We demonstrate this using artificial data and data from a turbulent jet. We\nalso develop a physics-informed adaptive frequency-resolution algorithm that provides substantially superior SPOD\nmodes than the standard constant resolution method. Lastly, we demonstrate that the alignment metric widely em-\nployed to quantify the similarity between di fferent SPOD /resolvent modes is highly sensitive to bias and the level\nof statistical convergence, and caution should be taken when inferring conclusions. Our results generalize to other\nspectral decomposition techniques such as Cyclostationary SPOD (CS-SPOD) [20], an extension of SPOD to flows\nwith harmonic forcing such as externally forced flows or naturally harmonic phenomena such as vortex-shedding,\nBispectral Mode Decomposition [21] which investigates triadic interactions, and Phase-Conditioned Localized SPOD\n[22] that investigates the evolution of high-frequency coherent structures on a low-frequency mean field.\nAcknowledgments: The authors gratefully acknowledge support from the United States O ffice of Naval Research under con-\ntract N00014-20-1-2311 with Dr. S. Martens as program manager and the Federal Aviation Administration under grant 13-C-AJFE-\nUI. This work was supported in part by high-performance computer time and resources from the DoD High Performance Com-\n5Figure 5: Dominant resolvent and SPOD modes at S t=0.1 for varying Ntwith∆S t=0.025.\nputing Modernization Program. This work used Stampede2 at Texas Advanced Computing Center through allocation CTS120005\nfrom the Advanced Cyberinfrastructure Coordination Ecosystem: Services & Support (ACCESS) program, which is supported by\nNational Science Foundation grants #2138259, #2138286, #2138307, #2137603, and #2138296.\nReferences\n[1] J. L. Lumley, The structure of inhomogeneous turbulent flows, Atmospheric turbulence and radio propagation (1967) 166–178.\n[2] J. L. Lumley, Stochastic tools in turbulence, Courier Corporation, 2007.\n[3] O. T. Schmidt, A. Towne, G. Rigas, T. Colonius, G. A. Br `es, Spectral analysis of jet turbulence, J. Fluid Mech. 855 (2018) 953–982.\n[4] D. B. Araya, T. Colonius, J. O. Dabiri, Transition to blu ff-body dynamics in the wake of vertical-axis wind turbines, J. Fluid Mech. 813\n(2017) 346–381.\n[5] L. I. Abreu, A. V . Cavalieri, P. Schlatter, R. Vinuesa, D. S. Henningson, Spectral proper orthogonal decomposition and resolvent analysis of\nnear-wall coherent structures in turbulent pipe flows, J. Fluid Mech. 900 (2020).\n[6] T. Chu, O. T. Schmidt, A stochastic spod-galerkin model for broadband turbulent flows, Theor Comput Fluid Dyn 35 (2021) 759–782.\n[7] A. Cammilleri, F. Gu ´eniat, J. Carlier, L. Pastur, E. M ´emin, F. Lusseyran, G. Artana, Pod-spectral decomposition for fluid flow analysis and\nmodel reduction, Theor Comput Fluid Dyn 27 (2013) 787–815.\n[8] O. T. Schmidt, T. Colonius, Guide to spectral proper orthogonal decomposition, AIAA journal 58 (2020) 1023–1033.\n[9] B. McKeon, A. Sharma, A critical-layer framework for turbulent pipe flow, J. FluidMech 658 (2010) 336382.\n[10] A. Towne, O. T. Schmidt, T. Colonius, Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and\nresolvent analysis, J. Fluid Mech. 847 (2018) 821–867.\n[11] G. Goertzel, An algorithm for the evaluation of finite trigonometric series, American Math. Monthly 65 (1958) 34–35.\n[12] L. Sirovich, Turbulence and the dynamics of coherent structures. i. coherent structures, Quarterly of applied mathematics 45 (1987) 561–571.\n[13] P. Welch, The use of fast fourier transform for the estimation of power spectra: a method based on time averaging over short, modified\nperiodograms, IEEE Transactions on audio and electroacoustics 15 (1967) 70–73.\n[14] J. S. Bendat, A. G. Piersol, Random data: analysis and measurement procedures, John Wiley & Sons, 2011.\n[15] G. A. Br `es, P. Jordan, V . Jaunet, M. Le Rallic, A. V . Cavalieri, A. Towne, S. K. Lele, T. Colonius, O. T. Schmidt, Importance of the nozzle-exit\nboundary-layer state in subsonic turbulent jets, J. Fluid Mech. 851 (2018) 83–124.\n[16] L. Heidt, T. Colonius, A. Nekkanti, O. Schmdit, I. Maia, P. Jordan, Analysis of forced subsonic jets using spectral proper orthogonal\ndecomposition and resolvent analysis, in: AIAA Aviation 2021 Forum, 2021, p. 2108.\n[17] A. Nekkanti, I. Maia, P. Jordan, L. Heidt, T. Colonius, , O. T. Schmidt, Triadic nonlinear interactions and acoustics of forced versus unforced\nturbulent jets, in: Twelveth International Symposium on Turbulence and Shear Flow Phenomena (TSFP12), Osaka, Japan (Online), July\n19-22, 2022.\n[18] B. C. Yeung, O. T. Schmidt, Adaptive spectral proper orthogonal decomposition of tonal flows, arXiv preprint arXiv:2312.02385 (2023).\n[19] E. Pickering, G. Rigas, O. T. Schmidt, D. Sipp, T. Colonius, Optimal eddy viscosity for resolvent-based models of coherent structures in\nturbulent jets, J. Fluid Mech. 917 (2021).\n[20] L. Heidt, T. Colonius, Spectral proper orthogonal decomposition of harmonically forced turbulent flows, arXiv preprint (2023).\n[21] O. T. Schmidt, Bispectral mode decomposition of nonlinear flows, Nonlinear Dynamics 102 (2020) 2479–2501.\n[22] L. Franceschini, D. Sipp, O. Marquet, J. Moulin, J. Dandois, Identification and reconstruction of high-frequency fluctuations evolving on a\nlow-frequency periodic limit cycle: application to turbulent cylinder flow, J. Fluid Mech. 942 (2022).\n6"    },    {        "title": "2402.15787v1.Grid_Peeling_of_Parabolas.pdf",        "content": "Grid Peeling of Parabolas\nG¨ unter Rote, Moritz R¨ uber, Morteza Saghafian\nFebruary 27, 2024\nAbstract\nGrid peeling is the process of repeatedly removing the convex hull vertices of the grid-\npoints that lie inside a given convex curve. It has been conjectured that, for a more and\nmore refined grid, grid peeling converges to a continuous process, the affine curve-shortening\nflow, which deforms the curve based on the curvature.\nWe prove this conjecture for one class of curves, parabolas with a vertical axis, and we\ndetermine the value of the constant factor in the formula that relates the two processes.\nContents\n1 Introduction 2\n1.1 History and background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3\n1.1.1 Peeling with random sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . 3\n1.1.2 Homotopic curve shortening. . . . . . . . . . . . . . . . . . . . . . . . . . 4\n1.1.3 Equivariance under affine transformations. . . . . . . . . . . . . . . . . . . 4\n1.2 Conics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4\n1.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5\n2 The grid parabola 5\n2.1 The horizontal period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\n2.1.1 Periodic continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7\n3 Grid peeling for parabolas 7\n3.1 Evaluating the horizontal period Ht. . . . . . . . . . . . . . . . . . . . . . . . . 8\n3.2 Distance between the grid parabola and the reference parabola . . . . . . . . . . 9\n3.3 Comparison to true parabolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10\n3.4 Refined grid peeling for parabolas, proof of Theorem 1 . . . . . . . . . . . . . . . 11\n4 Proof of Theorem 2 about the period of the grid parabola 12\n5 Future research 18\nA Alternative expressions for the horizontal period Ht 19\nB Proof of Lemma 4 about the points on the grid parabola 20\nC Minimum-area convex lattice polygons and grid parabolas 23\nD Experiments with grid peeling for parabolas 23\nD.1 Results of the experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26\nD.2 The average speed at the critical values of a. . . . . . . . . . . . . . . . . . . . . 27\nD.3 The time period at the critical values of a. . . . . . . . . . . . . . . . . . . . . . 29\nD.4 Deviation from the parabolic shape . . . . . . . . . . . . . . . . . . . . . . . . . . 29\n1arXiv:2402.15787v1  [cs.CG]  24 Feb 2024E Vertical difference between the grid parabola and the reference parabola 30\n1 Introduction\nIn 2017, Eppstein, Har-Peled, and Nivasch [9] observed a remarkable connection between a\ncontinuous deformation of a curve in the plane, the affine curve-shortening flow (ACSF), and a\ndiscrete process, grid peeling .\nIn the affine curve-shortening flow, a smooth curve is deformed by moving every point toward\nthe direction in which the curve bends, at a speed of κ1/3in the normal direction, where κis the\ncurvature at that point at the current point of time, see Figure 1. The left part of Figure 2 shows\na few snapshots of this inward-growing process, starting from a semicircle. (The semicircle is\nnot a smooth curve, but the definition of the flow can be extended to cover piecewise smooth\ncurves.)\nBy contrast, grid peeling is a process that is discrete both in space and in time. Given a\nconvex curve, we start by finding the convex hull of all points of a uniform square grid inside\nthe curve. Then we iteratively remove the vertices of the convex hull, and take the convex hull\nof the remaining grid points.\nFigure 1: Left: The Affine Curve-Shortening Flow (ACSF). The velocity is indicated by arrows,\nwhose length is proportional to κ1/3. Right: A convex curve and the first three steps of grid\npeeling.\nEppstein, Har-Peled, and Nivasch observed that, as the underlying grid is refined, grid peeling\napproximates the ACSF process, as can be seen in Figure 2. More specifically, they conjectured\nthat, for a more and more refined grid of spacing 1 /n, the m-th convex layer of a convex curve\nconverges to the ACSF after time T, ifmis chosen as\nm=⌊cgTn4/3⌋ (1)\nFigure 2: ACSF (left) and grid peeling (right) of a semicircle of diameter 1. The left figure\nshows 10 snapshots of ACSF with regular time increments; the right figure shows every 2714th\nconvex layer for a grid of spacing 1 /5000. The increment 2714 corresponds to the conjectured\nformula (1) with a value cg≈1.587. (From [9, Figure 3], by permission from the authors.)\n2for an appropriate constant cg, which was experimentally determined to be approximately 1 .6\n[9, Conjecture 1].\nWe prove this conjecture for the special case when the curve is a parabola with vertical axis,\nand we find the precise value of the constant cg:\ncg=3s\nπ2\n2ζ(3)≈1.60120980542577 , (2)\nwhere\nζ(s) = 1 +1\n2s+1\n3s+1\n4s+···\nis the Riemann zeta function, with values ζ(2) = π2/6≈1.644934 and ζ(3)≈1.2020569.\nTheorem 1. For the parabola Π:y=ax2/2 +bx+c, the ACSF is a vertical translation with\nvelocity a1/3. Thus, at time T >0, it becomes the parabola ΠT:y=ax2/2 +bx+c+Ta1/3.\nIf we apply grid peeling to Πwith a grid of spacing 1/nform=⌊cgTn4/3⌋steps, then, as\nn→ ∞ , the vertical distance between the resulting grid polygon and ΠTis bounded by\nO\u0012(Ta2/3+a−2/3) logn\na\nn1/3\u0013\n.\nFor fixed Tanda, this error bound goes to 0 as n→ ∞ .\nWe can extend this theorem to any parabola whose axis has a rational slope a/b: A uni-\nmodular transformation with a suitable matrix (a−bu v) of determinant 1 will leave the grid\nunchanged and make the axis vertical, and then Theorem 1 can be applied.\n1.1 History and background\nThe ACSF process was first studied in the 1990s in the area of computer vision and image\nprocessing, by Alvarez, Guichard, Lions, and Morel [1] and by Sapiro and Tannenbaum [16].\nOne way to understand the ACSF is to regard it as a limit of affine erosions , as shown by\nF. Cao [7, Theorem 6.22]. An affine erosion with parameter εremoves the union of all pieces\nof area εthat can be cut off by a straight line. (In convex geometry, this is also called the wet\npart; it plays a role in estimating the area and the number of vertices of the convex hull of a\nrandom sample of points [4, 3].) Repeating this process makes the shape rounder and rounder,\nlike a pebble rolling in water. Letting εgo to zero leads to the ACSF as the continuous limit.\nThe other process that we study is formed by the convex layers oronion layers of a point\nset. They have their origin in computational geometry and statistics: The innermost convex\nlayer provides a robust estimate of the “center” of a distribution. The special case where the\npoint set is a grid was first investigated by Har-Peled and Lidick´ y [10], who showed that the\nn×nsquare grid has Θ( n4/3) convex layers. For a box in three and higher dimensions, the\nasymptotic number of layers is not known, see for example [8] and the references given there.\nSee [2, 6, 7] for more background and references to the literature, both on the ACSF and on\ngrid peeling.\n1.1.1 Peeling with random sets.\nMore recently, Calder and Smart [6] investigated the related process where the grid is replaced\nby a random point set. More precisely, the refined grid of spacing 1 /nis replaced by a Poisson\npoint set of density 1 /n2. In this setting, they could prove an analogous statement:\nThere exists a constant cr≈1.3 such that the m-th convex layer, for m=⌊crTn4/3⌋,\napproximates the ACSF at time T. Since the underlying process is random, this statement\nrequires some probabilistic qualification; see [6, Theorem 1.2] for the precise statement, which\n3is quite strong and general: It is valid in arbitrary dimension, and convergence holds (with\nhigh probability) uniformly for all T. The density can be nonuniform, which corresponds to an\nACSF with a location-sensitive speed. There is no precise formula for the value of the random-\nset peeling constant cr, not even a conjectured one. Since cr< cg, random-set peeling proceeds\nfaster than grid peeling at the same density.\n1.1.2 Homotopic curve shortening.\nAvvakumov and Nivasch [2] extended peeling to nonconvex and even self-crossing curves, in-\ntroducing the concept of homotopic curve shortening . Both for grid peeling and for random-set\npeeling, the observed relation with the ACSF process persists also in this setting.\n1.1.3 Equivariance under affine transformations.\nIt is easy to check that the ACSF is equivariant under area-preserving affine transformations, a\nproperty that gave rise to the term “ affine curve-shortening flow.” (Arbitrary affine transfor-\nmations, which are not necessarily area-preserving, can be accommodated by scaling the time\nparameter.) The relation between ACSF and grid peeling is the more surprising as grid peeling\ndoes not have this property. Grid peeling is equivariant only under a special class of affine trans-\nformations, namely those that also preserve the grid (unimodular transformations), a property\nthat we will often use. (Peeling with random sets, on the other hand, is clearly equivariant\nunder area-preserving affine transformations.)\n1.2 Conics\nAs stated in Theorem 1, the ACSF for a parabola is just a translation at constant speed. This\nspecial behavior is shared, to a certain extent, by the other types of conics: They are scaled\nunder ACSF but otherwise maintain their shape [17, Lemma 8]. More specifically,\n•an ellipse (or a circle) shrinks toward the center, and eventually collapses to a point;\n•a parabola is translated parallel to the axis;\n•a hyperbola expands from its center.\nAmong the conics, parabolas appear most attractive for investigation, because they don’t even\nneed to be scaled. Also for the case of random-set peeling, the peeling of a parabola lies at\nthe core of the proof of Calder and Smart [6], forming what they call the cell problem . As\nregards experiments, the downside of parabolas, as opposed to ellipses, is that a parabola is an\nunbounded curve, and even the first step of grid peeling is not obvious to compute. However,\nas we shall see, for parabolas with rational coefficients, we can make use of a certain periodicity\nalong the curve, which reduces grid peeling to a finite computation. Once the sequence of\npeelings goes into a loop, one has a complete overview of the whole infinite grid peeling process.\nGrid peeling has been investigated also for hyperbolas , in a sense. If one starts with the\nupper-right quadrant R+×R+, the ACSF develops into positive branches of hyperbolas xy=c.\nEppstein, Har-Peled, and Nivasch [9, Theorem 5] investigated the convex layers of N×Nand\nproved that the m-th convex layer is sandwiched between two hyperbolas:\nc1m3/2≤xy≤c2m3/2, (3)\nexcept that the lower bound does not hold within a strip of width O(√mlog2m) around the\naxes.\nThe constants c1andc2are not computed explicitly, but some values can be worked out from\nthe proof. We add two side remarks regarding the exception near the axes that the theorem\nmakes. Firstly, some exception of this sort has to be made, because the m-th layer goes through\n4the points ( m,0) and (0 , m), and no hyperbola xy= const can be squeezed below these points.\nTo give the hyperbola some chance in principle to squeeze under, we might peel the quadrant\nofpositive grid points, or equivalently, we allow the hyperbola to be centered at ( −1,−1) (or\nsome other fixed point), but this would still not suffice for (3). Secondly, the claim [9, Theorem\n5] is stated with an “exception strip” around the axes whose width is only O(√m); however,\nthere is a small gap in the proof [9, p. 315, right column]: In the proof of Lemma 18, when\napplying Lemma 7 for the rectangle spanned by the points q/2 and q, an error term of the form\n±O(NlogN) from Lemma 7 is ignored. When the error term is taken into account, the proof\ngoes through with the larger margin of exception claimed above.\n1.3 Overview\nIn Section 2, we define a family of specific curves, the so-called grid parabolas Pt. Grid peeling\nreproduces them with a vertical shift after tsteps (or t+1, depending on the parity). This is our\nmain technical result (Theorem 2), whose proof is postponed to Section 4. Based on Theorem 2,\nwe prove Theorem 1, our main theorem about grid peeling of parabolas, in Section 3. Sections A\nand B analyze the quantities that arise in the construction of the grid parabola, using arguments\nfrom elementary number theory. The final Section D reports computer experiments with grid\npeeling for parabolas. These experiments were the source the discoveries expressed in Theorem 2\nbelow, and its consequence, our main Theorem 1. We also describe some interesting phenomena\nbeyond those that are discussed and proved in the first part of the paper.\n2 The grid parabola\nOur object of investigation is a special infinite polygonal chain Pt, which depends on a positive\ninteger parameter t. It is defined as follows:\n1. Let Stbe the set of all rational numbers s=a/bwith 0 < b≤t. We call these elements\ntheslopes . We will always assume that the fractions a/brepresenting slopes are reduced.\n2. For each slope s=a/b∈St, take the longest integer vector of the form\n\u0000x\ny\u0001\n=q\u0000b\na\u0001\n(q∈Z)\nwith 0 < x≤t. Let Vtdenote the set of these vectors. Figure 3 shows Vtfort= 11.\n3. Form the chain P=Ptby concatenating these vectors in order of increasing slope.\nFigure 4 shows a section of the grid parabola P5.\nWe can make a few simple observations: It is clear that for every vector ( x, y)∈Vtwith a\npositive slope, there is a corresponding vector ( x,−y)∈Vtwith negative slope. Thus, the curve\nPis symmetric with respect to a vertical axis. The lowest points on Pform a horizontal edge\nof length t. We place the origin Oof our coordinate system at the center of this edge, so that\nthe symmetry axis becomes the y-axis. When tis odd, this implies that the vertices of Phave\nhalf-integral x-coordinates. Nevertheless, we will refer to the points of the square unit grid on\nwhich the vertices of Plie (as shown in Figure 4) as the grid.\nFigure 5 applies a few grid peeling steps to the grid parabola P5. We can see that P5\nreproduces itself after 5 iterations, translated up by 1 unit. From then on, the process repeats\nad infinitum. Our central technical result is that this is always the case.\nTheorem 2. For odd t, the chain Ptrepeats after tpeeling steps, one unit higher.\nFor even t, the chain Ptrepeats after t+ 1peeling steps, one unit higher.\nTheorem 2 and the special construction of the grid parabola were suggested by experiments,\nwhich are reported in Appendix D. In Appendix C we mention that our grid parabolas play a\nrole in the convex lattice n-gons of minimum area.\n5t= 11Figure 3: The set V11of vectors ( x, y) from which P11is formed, shown as green dots. The\nvector with slope s= 2/5 is highlighted. The points of V11extend indefinitely to the top and to\nthe bottom. The picture shows the range −1≤y≤9.\n2.1 The horizontal period\nWhile Pis an infinite object, we will argue that it is sufficient to look at a finite section, because\nthis section “repeats periodically” in a certain sense.\nWe partition the vectors Vt=··· ∪ V(−2)\nt∪V(−1)\nt∪V(0)\nt∪V(1)\nt∪V(2)\nt∪ ··· according to the\nintegral part of their slope into the sets\nV(i)\nt:={(x, y)∈St|i <y\nx≤i+ 1}\nfori∈Z. The vectors in V(0)\ntleadPfrom the origin to an edge with vector ( t, t) of slope 1.\nMore precisely, in the way we have defined V(0)\nt, these vectors lead from the right endpoint of\nthe horizontal edge to the upper-right endpoint of the edge ( t, t). However, we prefer to select\nthe midpoint of the edge ( t, t), and we place a reference point Qat this point. We define the\nhorizontal period Htas the horizontal distance between the origin OandQ.\nHtis the sum of the x-coordinates of the vectors in V(0)\nt. (Only half of the vector ( t, t)∈V(0)\nt\ncontributes to Ht, but this is compensated by including half of the vector ( t,0)/∈V(0)\nt.) The first\nvalues of HtareH1, H2, . . .= 1,4,11,22,43,64,107,150,etc. In Section 3.1, we will evaluate\nthis quantity and see that Ht≈0.24t3(Lemma 4).\nWe claim that the segment OQhas slope1\n2, and therefore the y-coordinate of QisHt/2.\nProposition 1. 1. The segment OQhas slope1\n2.\n2.Htis even if and only if tis even.\nProof. The set Vtis symmetric with respect to the shearing operation\u0000x\ny\u0001\n↔\u0000x\nx−y\u0001\n, which keeps\nthe “mirror line” y=x/2 fixed and inverts the orientation of every vertical line, see the inset\nof Figure 4. Thus, every vector in Vtwith slope >1\n2can be matched with a vector with slope\n<1\n2, so that the sum of these two vectors has slope1\n2. The set V(0)\ntis slightly asymmetric with\nrespect to this mirror operation because it contains the edge of slope 1 but not the corresponding\nedge of slope 0. This asymmetry is taken care of by including half of both edges in the vector\nfrom OtoQ. Thus, the slope between OandQaverages out to1\n2.\nTo see statement 2, note that the vectors that are matched in a pair sum to a vector with\nan even x-coordinate. The unmatched vectors in V(0)\ntare the vector of slope s=1\n2, which has\nan even x-coordinate, and the vector\u0000t\nt\u0001\n, whose parity therefore decides the parity of Ht.\n6V(0)\n5\nV(1)\n5\nV(\u00001)\n5\nt= 5\nQ= (Ht;Ht\n2)\nHt= 43\n\u0005t:y=x2\n2Ht\u0000\r\nO\ny=x\n2\nx\ny\nxFigure 4: The grid parabola Ptfort= 5. The inset in the upper left corner shows some vectors\nof the set V5, at a slightly enlarged scale. Π tis the reference parabola defined by y=x2\n2Ht. Here\nit is shifted down by some offset γ.\n2.1.1 Periodic continuation\nThe mapping ( x, y)7→(x, y+ix) maps V(0)\nttoV(i)\nt; hence it is sufficient to know V(0)\nt; all other\nsetsV(i)\ntare copies of V(0)\ntwhere the slope of each vector is modified by an integer constant,\nleaving the x-coordinate fixed. This means that the continuation of Pbeyond the arc from Oto\nQis in some sense periodic: the same sequence of edges will appear again and again, only with\nmodified slopes. The mapping ( x, y)7→(x+Ht, y+Ht/2 +x) maps OtoQ, and it maps the\ncurve Pto itself. The midpoints of the edges with integer slopes appear regularly at intervals\nof length Ht. The midpoint of the edge with slope iis (iHt, i2Ht/2). These points lie on a\nparabola, which we call the reference parabola\nΠt:y=x2/(2Ht),\nand the polygonal chain Ptfollows Π twith bounded local deviations. We summarize these\nconsiderations in the following lemma, whose proof is straightforward.\nLemma 1. The affine transformation\n\u0012x\ny\u0013\n7→\u0012x+Ht\ny+x+Ht/2\u0013\nmaps the grid to itself, and in addition, it maps both Ptand the reference parabola Πt, or any\nvertical translate of it, to itself.\n3 Grid peeling for parabolas\nWe start with the simple observation that grid peeling preserves inclusion:\nObservation 1. LetU⊆U′⊂Z2be two sets of grid points that are upward closed :(x, y)∈\nU=⇒(x, y+ 1)∈U. Let fdenote one peeling step. Then f(U)⊆f(U′).\n7Figure 5: Consecutive peelings of P5. Since consecutive peelings share many vertices, it is not\neasy to distinguish the curves. In the lower part, we have therefore vertically separated the\nconsecutive peelings. This has the effect that some grid points appear in several copies with\nsmall vertical offsets, and horizontal grid lines get a curved appearance.\nObservation 1 implies that if CandDare two convex x-monotone curves that extend from\nx=−∞tox= +∞(e.g. grid curve or an arbitrary smooth or piecewise smooth curve), and C\nlies everywhere (weakly) below D, then this relation is maintained by grid peeling.\nLemma 2. PandΠadvance at the same limiting speed.\nProof. As already argued, Papproximates Π in a global sense, while locally, there might be\ndeviations. This implies that we can shift Π vertically down by some integer amount γand\nensure that the shifted parabola, denoted by Π −γ, lies completely below P. Figure 4 shows\nthe parabola for γ= 8, but actually, γ= 1 should already be sufficient to push Π below P.\nNow imagine that we start grid peeling simultaneously with Pand with Π −γ, or more\nprecisely, with the convex hull of the grid points on or above Π −γ. Applying the monotonicity\nproperty of Observation 1, we conclude that the evolution of Π −γalways remains below the\nevolution of P. It can never overtake P, and in particular, the limiting speed of Π −γ, which is\nthe same as the limiting speed of Π, is at most the limiting speed of P.\nWe can push Π upward and start with a parabola Π + γ′that lies above Peverywhere, and\nargue in the same way that the evolution of Pcan never overtake the evolution of Π + γ′, and\nthus, the limiting speed of Π is at least the limiting speed of P.\n3.1 Evaluating the horizontal period Ht\nWe have seen that Htis the sum of the x-coordinates of the vectors in V(0)\nt. It is thus given by\nthe following expression:\nHt=X\n(x,y)∈V(0)\ntx=X\n0<y≤x≤t\ngcd(x,y)=1\u0016t\nx\u0017\nx\n8\u0000x0\ny0\u0001\n\u0000x1\ny1\u0001\n\u0000x0\n^y\u0001\n\u000b\n\u000b\n\u0005t\nPt\n\u0000\u0001x\n\u0001y\u0001Figure 6: The vertical distance between Ptand Π t\nThis sequence appears in the Online Encyclopedia of Integer Sequences [12, A174405]. It starts\nwith the values\nH1, H2, . . .= 1,4,11,22,43,64,107,150,211,274,385,462,619,748,895,1066,1339, . . .\nThe sequence has been investigated by S´ andor and Kramer [18], who showed the following\nasymptotic estimate:\nLemma 3.\nHt∼2ζ(3)\nπ2t3≈0.243587656 t3\nThis can also be derived as a consequence of the more general Lemma 4, which is stated\nbelow. Section A gives several alternative expressions for Ht.\n3.2 Distance between the grid parabola and the reference parabola\nWe want to analyze the deviation between the grid parabola and the reference parabola. While\nthe reference parabola has an explicit expression, the grid parabola is given by a multistep\nprocess, as described in Section 2. For getting from the origin to an arbitrary vertex of Pt, we\nneed to sum the vectors whose slope is at most some threshold α:\nUα\nt:=X\n0≤x≤t\n0<y≤αx\ngcd(x,y)=1\u0016t\nx\u0017\u0012x\ny\u0013\nMore precisely, since the sum includes only vectors with y >0, it measures the distance from\nthe right endpoint of the horizontal segment of Ptand not from the origin. Lemma 4, whose\nproof will be given in Appendix B, gives an asymptotic expression for this sum:\nLemma 4. Let0≤α≤1. Then\nUα\nt=2ζ(3)\nπ2\u0012t3α+O(t2logt)\nt3α2/2 +O(t2logt)\u0013\n,and Ht=2ζ(3)\nπ2t3+O(t2logt).\nThe second expression is obtained from the first one by setting α= 1 and looking only at\nthex-coordinate.\nProposition 2. The vertical distance between the grid parabola Ptand the reference parabola\nΠtis bounded by O(t2logt).\n9Proof. By the periodic behavior of Ptand Π t(Lemma 1), it suffices to look at the interval\n0≤x≤Ht. Pick a point x0in this interval, see Figure 6. The corresponding point\u0000x0\ny0\u0001\non\nΠthasy0=x2\n0/(2Ht), and the slope at this point is α:=x0/Ht≤1. As a first step, we find\nthe point\u0000x1\ny1\u0001\nonPtwith the same slope α. We will show that it deviates from\u0000x0\ny0\u0001\nby at most\nO(t2logt) in each coordinate: By construction the grid parabola contains the vertex\n\u0012x1\ny1\u0013\n=\u0012t/2\n0\u0013\n+Uα\nt.\nThe correction term t/2 accounts for the fact that Uα\ntdoes not include the vector\u0000t\n0\u0001\nand thus\nmeasures the distance from the right endpoint of the horizontal segment of Ptand not from the\norigin. Applying both parts of Lemma 4, we get\n\u0012x1\ny1\u0013\n=\u0012t/2\n0\u0013\n+Uα\nt=2ζ(3)t3\nπ2\u0012α\nα2/2\u0013\n+\u0012O(t2logt)\nO(t2logt)\u0013\n+\u0012t/2\n0\u0013\n=\u0000\nHt+O(t2logt)\u0001\u0012α\nα2/2\u0013\n+\u0012O(t2logt)\nO(t2logt)\u0013\n=Ht\u0012α\nα2/2\u0013\n+\u0012O(t2logt)\nO(t2logt)\u0013\n=\u0012x0\ny0\u0013\n+\u0012∆x\n∆y\u0013\nwith ∆ x,∆y=O(t2logt). In the range 0 ≤x≤Ht, the slope of Ptis bounded by 1. So when\nwe move from x1tox0=x1+ ∆xonPt, we arrive at a point ( x0,ˆy) with |ˆy−y1| ≤ |∆x|, and\nthus, the vertical distance |ˆy−y0|between Ptand Π tis at most |∆y|+|∆x|=O(t2logt).\nFigure 20 in Appendix E shows the actual difference Pt−Πtfor a few selected values of t.\n3.3 Comparison to true parabolas\nLety=ax2/2 +bx+c. We are interested in the average (vertical) speed in which the curve\nmoves upwards. If we start grid peeling with a parabola y=ax2+bx+cfor rational coefficients\naandb, we can show that, after some irregular “preperiod”, it will enter a periodic behavior:\nAfter a certain number ∆ mof steps, the same curve reappears, translated upward by ∆ y. We\ncall ∆ mthetime period and ∆ ythevertical period (to be distinguished from yet another period,\nthe horizontal period H, which was introduced at the beginning). The average (vertical) speed\nis then ∆ y/∆m. However, if aorbis irrational, we no longer have a periodic behavior. For a\nmore general curve, like y=ex, different parts of the curve will move at different speeds, and\na common average speed will not exist. For this reason, we define the lower andupper average\nspeed.\nDefinition 1 (average speed) .LetCbe the graph of a convex function on R. We denote by\nC+γthe copy of Cvertically translated by γ(upwards for γ >0, downwards for γ <0).\nFor another such curve D, we write C≤Dif no point of Clies above D.\nLetf(m)denote msteps of grid peeling.\nThelower average speed v−=v−(C) is defined as follows:\nv−= lim inf\nm→∞sup{γ|C+γ≤f(m)(C)}\nm\nTheupper average speed v+=v+(C) is defined similarly:\nv+= lim sup\nm→∞inf{γ|f(m)(C)≤C+γ}\nm\nIfv−andv+coincide, we call it simply the average speed v=v(C).\n10\u0005:y=ax2+bx+cFigure 7: Upper and lower approximation of the parabola Π by “integer” parabolas\nWe have the obvious inequalities 0 ≤v−≤v+≤ ∞ . By approximating the parabola\ny=ax2/2 +bx+cfrom above and below by appropriate grid parabolas (see Figure 7), to which\nwe apply Theorem 2, we arrive at the following result:\nTheorem 3. 1. If1\nHt< a <1\nHt−1andtis odd (ora >1\nH1= 1), then v= 1/t.\n2. If1\nHt< a <1\nHt−1andtis even, then1\nt+1≤v−≤v+≤1\nt−1.\n3. If a=1\nHtandtis odd, then1\nt+2≤v−≤v+≤1\nt.\n4. If a=1\nHtandtis even, then1\nt+1≤v−≤v+≤1\nt−1.\nThe experiments in Section D suggest that the first statement also holds for even t, and for\na=1\n2Ht,vexists always and lies in the range1\nt+1≤v≤1\nt.\nProposition 3. LetCbe the graph of a convex function on R. The upper and lower average\nspeed is not changed by\n•horizontal translation by an integer distance,\n•or arbitrary vertical translation.\nProof. It is clear that a translation by an integer vector does not change anything.\nConsider an arbitrary vertical translation C+γ. Then (1) The integer translates C0=C+⌊γ⌋\nandC1=C+⌊γ⌋+ 1 have the same upper and lower average speeds as C, (2) they maintain\na constant vertical distance of 1 during grid peeling, and (3) C+γis sandwiched between C0\nandC1, and it maintains this relation during grid peeling. It follows that the upper and lower\naverage speeds of Cmust agree with those of C1andC2.\nOne might be tempted to believe that a small horizontal translation should also not change\nthe vertical speed. However, Section D.2 reports examples where translations cause the vertical\nspeed to change, see Figure 15.\n3.4 Refined grid peeling for parabolas, proof of Theorem 1\nWe prove our main theorem, Theorem 1 about the relation between grid peeling and ACSF for\nparabolas y=ax2/2 +bx+c. We use ( x, y) for the original coordinates, with a grid of spacing\n1/n, and (ˆ x,ˆy) = ( nx, xy ) for the scaled coordinates, with a unit grid. The curvature at the\nvertex of the parabola is a; thus the vertical speed of ACSF at this point (and thus everywhere,\nby affine invariance) is a1/3.\n11At time Twe have y=ax2/2+bx+c+Ta1/3, and ˆ y=a\nnˆx2/2+bˆx+cn+Tna1/3. Determine\ntsuch that1\nHt≤a\nn≤1\nHt−1. Soa\nn≈1\nHtand Lemma 4 implies\nn\na≈Ht=2ζ(3)t3\nπ2·(1 +O(logt/t)) =\u0010t\ncg\u00113\n·(1 +O(logt/t)),\nwhich yields\nt=cg3rn\na\u0000\n1 +O(logt/t)\u0001+O(1) = cg3q\nn\na·\u0000\n1 +O(logt/t)\u0001\n= Θ\u0010\n3q\nn\na\u0011\n.\nHere, the O(1) term accounts for rounding tto an integer, and it also covers the uncertainty of\nTheorem 3, where tis sometimes replaced by t−1,t+ 1, or t+ 2. This additive error term is\nabsorbed in the multiplicative error term 1 + O(logt/t). By Theorem 3, the lower and upper\naverage speed is\nv≈1\nt=1\ncg3q\na\nn·\u0000\n1 +O(logt/t)\u0001\nAfter m=⌊cgTn4/3⌋steps, the vertical distance that the curve has moved up is therefore\nmv+O(1) = Ta1/3n\u0000\n1 +O(logt/t)\u0001\n+O(1) = Ta1/3n\u0000\n1 +O(3q\na\nnlogn\na)\u0001\nThe difference to the movement of Π, which is Tna1/3, is\nO\u0000\nTa1/3n3q\na\nnlogn\na\u0001\n=O(Ta2/3n2/3logn\na).\nTo this, we must add the distance between Ptand the reference parabola Π tfrom Proposition 2,\nthat is, O(t2logt) =O((n\na)2/3logn\na). Dividing by n, we conclude that the error term in terms\nof the original y-coordinates is\n\u0000\nO(Ta2/3n2/3logn\na) +O((n\na)2/3logn\na)\u0001\n/n=O\u0000\n(Ta2/3+a−2/3)/n1/3logn\na\u0001\n.\n4 Proof of Theorem 2 about the period of the grid parabola\nWhen we speak of the curve , we mean the grid parabola Ptafter some iterations of peeling.\nLets=a/b∈Stbe a fixed slope. We consider the supporting line gwith slope s, and we\nstudy how it evolves during the peeling process, see Figure 8 for an illustration.\nDefinition 2. The strip of slope sis the vertical strip that bounds the segment of slope sin\nPt. It goes from x=Lstox=Rs. We denote by ℓ=Rs−Lsthewidth of the strip.\nTheextended strip of slope sincludes an additional margin of ⌊t/2⌋on both sides. It goes\nfrom ¯Ls=Ls− ⌊t/2⌋to¯Rs=Rs+⌊t/2⌋.\nWe state some obvious properties of the peeling process:\nObservation 2. Throughout the whole peeling process:\n(i) The supporting line gintersects the curve in a line segment, which might degenerate to a\npoint.\n(ii) If the segment contains k≥1grid points, its horizontal length is (k−1)b.\n(iii) At every peeling step, the two endpoints of this line segment or the single point is peeled.\n(iv) As long as the segment contains at least 3 grid points, the supporting line does not change,\nand the number kof grid points on the segment decreases by 2. In this case, we say that\nthe segment shrinks .\n12bt=2c= 8\n`= 15\nbt=2c= 8\nLs\n\u0016Ls\nRs\n\u0016RsFigure 8: 17 consecutive iterations of the development of slope s= 2/5 for t= 16, starting\nwith the curve Pt. The uppermost part shows the true situation. In the middle part, the\nsuccessive curves are separated for better visibility, as in Figure 5. In the lowest part, an affine\ntransformation has been applied to make the segment of interest horizontal; this allows the\ndifferent slopes to be distinguished more easily. The segment of slope sis highlighted in red on\neach curve. The initial segment on P16has horizontal length ℓ=⌊16\n5⌋ ×5 = 15. The region\nbetween the dashed lines is the extended strip.\n13(v) If the segment contains only 1 or 2 grid points, the supporting line changes. We say that\nthere is a jump for slope s.\nWe use the following terminology: The left endpoint of slope sis the left endpoint of the\nsegment of slope s; in case the segment degenerates to a single point, it is that point. In other\nwords, it is the leftmost point where the supporting line of slope stouches the curve. The right\nendpoint is defined analogously. We will only deal with horizontal offsets, lengths, positions,\nand distances, and thus we will often omit the word horizontal .\nIn the following crucial lemma, Properties 1 and 2 predict what happens when a jump\noccurs. In particular, Property 2 characterizes the possible locations of the vertices after a\njump. Property 3 describes the final position of the segment before the jump. This statement\nallows us to predict when a jump occurs. Property 3 can be easily worked out, assuming that\nthe initial position after the previous jump satisfies Property 2. Properties 4 and 5 describe the\nsituation when two consecutive slopes are involved.\nLemma 5. The following properties hold throughout the peeling process:\n1. (No grid line is skipped.) Whenever there is a jump, the supporting line gof slope s\nadvances to the next grid line with slope s.\n2. (The filling property.) After the supporting line ghas advanced, the curve will contain\nprecisely those grid points on gthat lie within the extended strip of s. In other words, the\nsegment fills the extended strip as much as possible. (See Figure 8 and 9.)\n3. (Jump position) A jump of slope soccurs if and only if the left endpoint of slope slies in\nthe range\nLs+⌊t/2⌋ −(b−1), . . . , L s+⌊t/2⌋. (4)\nA symmetric property holds for the right endpoint.\n4. (There are no gaps.) For any two consecutive slopes s=a/bands′=a′/b′from St,\nwith s < s′, the right endpoint of slope scoincides with the left endpoint of slope s′. In\nparticular, no edge has an intermediate slope between sands′. This implies that only\nslopes from the set Stappear in the curves.\n5. (Breakpoint position) The breakpoint between two consecutive slopes s, s′is in the inter-\nval between X−t/2andX+t/2, where X=Rs=Ls′is the boundary between the\ncorresponding strips. (See Figure 10.)\nProperties 1–3, which were discovered experimentally, are strong enough to predict the be-\nhavior of the supporting segment of slope sduring the peeling process in a purely local manner,\nwithout looking at the whole curve. The proof that this is the evolution that actually takes place\namounts to checking whether these local characterizations fit together when considering differ-\nent slopes. In particular, we will look at two consecutive slopes (Properties 4 and 5). This will\ninvolve checking some cases, but with the rigid structure provided by the strong properties 1–3,\none cannot really avoid to come up with the proof.\nProof. We will show inductively that the claimed properties are maintained as invariants for all\nslopes throughout the peeling process.\nWe rely on the following properties of two consecutive slopes s=a\nbands′=a′\nb′, which follow\nfrom the definition of the slope set St(Section 2):\n(I) The denominators bandb′are bounded by b, b′≤t, and their sum is b+b′> t.\n(Otherwise, the vector\u0000b\na\u0001\n+\u0000b′\na′\u0001\nwould give rise to a slope in Stbetween sands′.)\n14`+ 2bt=2c= 31\n\u0001\nr= 1\nrange for a jump\nrange for the left endpoint of slope s\nb\u00001\n2bt=2c= 16\nextended strip: range for the segments of slope s\nb= 5\n\u0016Ls\n\u0016Rs\nb\u00001\n(range for a jump for the right endpoint of slope s)Figure 9: The b= 5 grid lines from Figure 8. t= 16 = 3 b+ 1 = qb+r. The initial segment on\neach line, immediately after the jump, is shown in red. The lines are considered according to the\noffset ∆ of the leftmost grid point from the left edge of the extended strip. We have highlighted\nthe last remaining single point or pair of points before the jump occurs.\n(II) The two vectors\u0000b\na\u0001\nand\u0000b′\na′\u0001\nform a lattice basis of the unit grid.\n(Otherwise, they would span a parallelogram that contains interior points, and some of\nthese points would lead to vectors with an intermediate slope between sands′inSt.)\nAs the basis for the induction, it can be seen without computation that the “original” segment\nof slope sofPtfalls in the pattern of analysis leading to Property 3: Indeed, it lies centrally in\nthe extended strip. Thus, when extending it as much as possible within the extended strip and\nstarting the peeling process, the starting segment will appear during this process, by symmetry.\nAlso, the endpoints of consecutive segments match on Ptby construction, establishing Property 4\nand Property 5 at the beginning.\nFor the induction step we consider two consecutive slopes s, s′and make sure that no matter\nif they make a jump or not, the properties of Lemma 5 hold. Thus we have four cases. In each\ncase we prove properties 1,2, 4, 5, and then assuming properties 1,2, 4, 5, we prove Property 3\nall at once.\nCase 1: sand s′both shrink: We show that this is not possible. Assume that s, s′\nboth shrink. Since no jump occurs for s′, by Property 3 the rightmost possible position of the\nleft endpoint of s′isLs′+⌊t/2⌋ −b′. Symmetrically, the leftmost possible position of the right\nendpoint of sisRs− ⌊t/2⌋+b, and therefore Rs− ⌊t/2⌋+b≤Ls′+⌊t/2⌋ −b′. Since Ls′=Rs,\nit follows that −⌊t/2⌋+b≤ ⌊t/2⌋ −b′, orb+b′≤2⌊t/2⌋ ≤t, which contradicts (I).\nCase 2: sjumps and s′shrinks: We claim that the endpoint of the shrunken segment\nfors′arrives at the next grid line with slope s, and its position on this line matches the position\nfor the right endpoint of spredicted by Property 2. The left endpoint of s′moves by the vector\u0000b′\na′\u0001\n, and since\u0000b′\na′\u0001\nand\u0000b\na\u0001\nform a lattice basis (II), the supporting line of slope swill indeed\njump to the next grid line of slope s, establishing Property 1 in this case.\nWe claim that this new left endpoint of s′is indeed the rightmost grid point on this line in\nthe extended strip for s(establishing Property 2). We show that it lies in the extended strip of\ns, but the point after this point on the grid line of slope sis already outside the extended strip.\nTo show the former, consider the rightmost possible position for the left endpoint of s′before\nshrinking. It is Ls′+⌊t/2⌋−b′by Property 3. After shrinking, it is Ls′+⌊t/2⌋=Rs+⌊t/2⌋=¯Rs,\n15Rs=X=Ls0\ns0=a0\nb0=5\n13\ns=a\nb=3\n8\nsmakes a jump\nbt=2c= 8\n\u0016Rs\n\u0016Ls0\ns0makes a jump\nbt=2c= 8\nb0\u00001 = 12\nb\u00001 = 7\nslope2\n5\nslope4\n11\nA\nA+\nU\nU0Figure 10: The transition between slope s= 3/8, with vector\u000016\n6\u0001\n∈Vt, and s′= 5/13, with\nvector\u000013\n5\u0001\n∈Vt, for t= 16. The figure is drawn after an affine transformation, following the\nconventions of the lower part of Figure 8. The right endpoint of slope salways coincides with\nthe left endpoint of slope s′, and it varies in the interval between X− ⌊t/2⌋andX+⌊t/2⌋.\nand thus it lies in the extended strip of s. To show the latter, consider the leftmost possible\nposition for the left endpoint of s′before shrinking. It is ¯Ls′=X− ⌊t/2⌋by Property 3. After\nshrinking, it is X− ⌊t/2⌋+b′. The point after this point is at offset b, and X− ⌊t/2⌋+b′+b >\nX− ⌊t/2⌋+t≥X+⌊t/2⌋=¯Rs, by (I), and thus this point is already outside the extended\nstrip.\nTherefore, the new left endpoint of s′coincides with the new right endpoint of s, establishing\nProperty 4 and Property 5 in this case.\nCase 3: sshrinks and s′jumps: The situation is symmetric to Case 2.\nCase 4: sand s′both jump: By Property 3 the position xof the breakpoint A=\u0000x\ny\u0001\nbefore the jump is in the interval\nX+⌊t/2⌋ −b′< x < X − ⌊t/2⌋+b (5)\nLetUbe the previous point on the grid line of slope sthrough A, and let U′be the next point on\nthe grid line of slope s′through A. Construct the parallelogram UAU′A+. Such a parallelogram\nis shaded in Figure 10. We know that both supporting lines of slope sands′must advance at\nleast to the next grid line. Those two grid lines intersect at the fourth point A+=\u0000x+\ny+\u0001\nof this\nparallelogram with x+=x−b+b′. We will show two things:\n(a) The point A+is not peeled until after the jump. Thus it will be the common right endpoint\n16of slope sand left endpoint of slope s′.\n(b) It is indeed the rightmost grid point on the grid line of slope sin the extended strip of s.\n(Symmetrically, it is the leftmost point on the grid line in the extended strip of s′.)\nTo prove (a), assume w.l.o.g. that b≤b′, so that the segment AU′lies below A+. We are\ndone if we show that this segment is part of the boundary when Ais peeled. Assume otherwise.\nThen Awould be not only the left endpoint of slope s′, but also the right endpoint of slope s′\nwhen it is peeled. According to Property 3, this means that x≥Rs′−⌊t/2⌋. Let ℓs′≥b′denote\nthe length of the strip of slope s′. Then, with b≤b′≤ℓs′we get a contradiction to (5):\nx≥Rs′− ⌊t/2⌋ ≥Rs′−ℓs′− ⌊t/2⌋+b=X− ⌊t/2⌋+b > x\nTo prove (b), note that A+lies in the extended strip for sbecause x+=x−b+b′<\nX−⌊t/2⌋+b′≤X−⌊t/2⌋+t, by the right inequality of (5), and hence x+≤X+⌊t/2⌋. On the\nother hand, the next grid point on the line of slope shasx-coordinate x++b=x+b′> X+⌊t/2⌋,\nby the left inequality of (5), and this is outside the extended strip for s.\nNow that we have established the first four invariants in all cases, we prove Property 3,\nassuming Property 2 has been true so far.\nLett=qb+r, with 0 ≤r < b . Then qb=ℓis the (horizontal) length of the vector in Vt\nthat forms the segment of slope sonPt. We have defined it as the width of the strip.\nTo illustrate Property 2, Figure 9 shows the possible cases how the segment of slope scan\nlie on the grid line, immediately after a jump occurs, according to this property. On every grid\nline, the grid points form an arithmetic progression with (horizontal) increment b. The different\ngrid lines are distinguished by the offset ∆ of the leftmost grid point from the left edge of the\nextended strip. There are bpossibilities, 0 ≤∆< b.\nFor the sake of the following analysis, we have sorted the lines by ∆ in Figure 9. (This\nis not the order in which they occur from bottom to top. The true order in this example is\n∆ = 0 ,2,4,1,3,0, . . ., see Figure 8.)\nFor simplicity, we focus on the case when tis even [and put the odd case into brackets].\nLet us start with the case ∆ = 0 (the topmost line in Figure 9). In a strip of width t, we\ncan fit qsegments of length b, with q+ 1 points, leaving a remainder of length r. The extended\nstrip has width t+ℓ[t−1 +ℓ]. Since the extra length ℓis filled precisely by qsegments, we can\nfitqadditional segments of length b, for a total of 2 q+ 1 points. [For odd t, the last claim holds\nonly when r >0.]\nSince the number of points is odd, the last peeled segment on this line before the jump is a\nsingleton, after qsteps and at distance qb=ℓfrom the left boundary ¯Lsof the extended strip,\nor distance ℓ− ⌊t/2⌋from Ls.\nWe can increase ∆ up to r[r−1] without changing the situation:\n•For ∆ = 0 ,1, . . . , r [∆ = 0 ,1, . . . , r −1], the number kof points is odd, and for the last\npoint that is peeled, the distance from Lsis in the range\nℓ−t/2, . . . , ℓ −t/2 +r[ℓ− ⌊t/2⌋, . . . , ℓ − ⌊t/2⌋+r−1].\nSince ℓ+r=t, this range simplifies to ℓ− ⌊t/2⌋, . . . ,⌊t/2⌋.\nStarting from ∆ = r+ 1 [∆ = r], the situation changes. We have now an even number 2 qof\npoints, and the last peeled segment is a proper segment with a pair of points. The left peeled\npoint is at distance ∆ + ( q−1)b= ∆ + ℓ−bfrom ¯Ls, or at an offset ∆ + ℓ−b− ⌊t/2⌋from Ls\n(This offset may be negative, in which case it denotes an offset to the left.)\n17•For ∆ = r+ 1, . . . , b −1 [∆ = r, . . . , b −1], the number kof points is even, and for left\npoint of the last peeled pair, the distance from Lsis in the range\nr+ 1 + ℓ−b−t/2, . . . , b −1 +ℓ−b−t/2 [r+ℓ−b− ⌊t/2⌋, . . . , b −1 +ℓ−b− ⌊t/2⌋].\nSince ℓ+r=t, this range simplifies to ⌊t/2⌋ −b−1, . . . , ℓ − ⌊t/2⌋ −1.\nCombining the ranges for two cases establishes Property 3, and this concludes the proof of\nLemma 5.\nProposition 4. The left endpoint of slope sis at distance at most ⌊t/2⌋from Ls(on the left\nor on the right), see Figure 9. Every position in this range occurs.\nProof. Property 3 describes bpossibilities before a jump, one value for each of the bresidue\nclasses modulo b, and Property 2 suggests bpossibilities after a jump, namely ∆ = 0 ,1, . . . , b −1.\nSince for every ∆, the jump must occur at some point, the range (4) uniquely characterizes this\npoint.\nBy Property 3, the left endpoint of slope scan never deviate more than ⌊t/2⌋from Lsto\nthe right, and by Property 2, it cannot deviate more than ⌊t/2⌋from Lsto the left. Thus, the\nleft endpoint of slope sis at distance at most ⌊t/2⌋from Ls. In fact, since there is a grid line of\nslope spassing through every such point, and no grid line is skipped (Property 1), every point\nin this range will be peeled once as the left endpoint of slope s.\nThere are 2 ⌊t/2⌋+ 1 different offsets at distance at most ⌊t/2⌋from Ls, and exactly one of\nthem is always peeled. Therefore, after 2 ⌊t/2⌋+ 1 steps the same segment of slope srepeats,\none unit higher. This is true for any slope, so after 2 ⌊t/2⌋+ 1 steps the same chain repeats one\nunit higher. It means the peeling process is periodic with period 2 ⌊t/2⌋+ 1, and this concludes\nthe proof of Theorem 2.\n5 Future research\nThe obvious open problem is to prove the relation between grid peeling and the ACSF for\narbitrary convex curves. As a first challenge, one might try the case of a circle. The natural\napproach is to leverage Theorem 1 by locally approximating the curve by parabolas.\nSome of the phenomena that were revealed by the experiments described in Section D are\nstill awaiting an explanation.\nReferences\n[1] Luis Alvarez, Fr´ ed´ eric Guichard, Pierre-Luis Lions, and Jean-Michel Morel. Axioms and\nfundamental equations of image processing. Arch. Rational Mech. Anal. , 123:199–257, 1993.\ndoi:10.1007/BF00375127 .\n[2] Sergey Avvakumov and Gabriel Nivasch. Homotopic curve shortening and the affine curve-\nshortening flow. Journal of Computational Geometry , 12:145–177, 2021. doi:10.20382/\njocg.v12i1a7 .\n[3] Imre B´ ar´ any, Matthieu Fradelizi, Xavier Goaoc, Alfredo Hubard, and G¨ unter Rote. Random\npolytopes and the wet part for arbitrary probability distributions. Annales Henri Lebesgue ,\n3:701–715, 2020. arXiv:1902.06519 ,doi:10.5802/ahl.44 .\n[4] Imre B´ ar´ any and David G. Larman. Convex bodies, economic cap coverings, random\npolytopes. Mathematika , 35:274–291, 1988. doi:10.1112/S0025579300015266 .\n18[5] Imre B´ ar´ any and Norihide Tokushige. The minimum area of convex lattice n-gons. Com-\nbinatorica , 24(11):171–185, 2004. doi:10.1007/s00493-004-0012-0 .\n[6] Jeff Calder and Charles K. Smart. The limit shape of convex hull peeling. Duke\nMathematical Journal , 169(11):2079–2124, 2020. arXiv:1805.08278 ,doi:10.1215/\n00127094-2020-0013 .\n[7] Fr´ ed´ eric Cao. Geometric Curve Evolution and Image Processing , volume 1805 of Lecture\nNotes in Mathematics . Springer, Berlin, Heidelberg, 2003. doi:10.1007/b10404 .\n[8] Travis Dillon and Narmada Varadarajan. Explicit bounds for the layer number of the grid,\n2023. arXiv:2302.04244 .\n[9] David Eppstein, Sariel Har-Peled, and Gabriel Nivasch. Grid peeling and the affine curve-\nshortening flow. Experimental Mathematics , 29(3):306–316, 2020. arXiv:1710.03960 ,doi:\n10.1080/10586458.2018.1466379 .\n[10] Sariel Har-Peled and Bernard Lidick´ y. Peeling the grid. SIAM Journal on Discrete Math-\nematics , 27(2):650–655, 2013. arXiv:1302.3200 ,doi:10.1137/120892660 .\n[11] G. H. Hardy and E. M. Wright. An Introduction to the Theory of Numbers . Oxford Science\nPublications, 1979.\n[12] The On-Line Encyclopedia of Integer Sequences. URL: http://oeis.org/ .\n[13] Ch. Radoux. Note sur le comportement asymptotique de l’indicateur d’Euler. Ann. Sci.\nBruxelles, Ser. I , 91:13–18, 1977.\n[14] Marko Riedel. Answer to Euler phi function, number theory . Mathematics Stack Exchange,\n2014. version: 2014-05-08. URL: https://math.stackexchange.com/q/787082 .\n[15] Moritz R¨ uber. Die Gittersch¨ alung und der affine Kurvenfluss auf Parabeln. M. Ed. thesis,\nFreie Universit¨ at Berlin, Department of Computer Science, November 2021.\n[16] Guillermo Sapiro and Allen Tannenbaum. Affine invariant scale-space. Int. J. Comput.\nVision , 11:25–44, 1993. doi:10.1007/bf01420591 .\n[17] Guillermo Sapiro and Allen Tannenbaum. On affine plane curve evolution. Journal of\nFunctional Analysis , 119(1):79–120, 1994. doi:10.1006/jfan.1994.1004 .\n[18] J. S´ andor and A. V. Kramer. ¨Uber eine zahlentheoretische Funktion. Mathe-\nmatica Moravica , 3:53–62, 1999. URL: http://www.moravica.ftn.kg.ac.rs/Vol_3/\n10-Sandor-Kramer.pdf .\nA Alternative expressions for the horizontal period Ht\nThe following long chain of equations and estimates, which we will discuss step by step, includes\nseveral different expressions for the quantity Ht. We denote by P={(i, j)∈Z×Z|gcd(i, j) =\n191}the set of primitive vectors .\nHt:=X\n0<y≤x≤t\n(x,y)∈P\u0016t\nx\u0017\nx (6)\n=X\n1≤x≤tX\n0<y≤x\n(x,y)∈P\u0016t\nx\u0017\nx=X\n1≤x≤tϕ(x)\u0016t\nx\u0017\nx (7)\n=X\n1≤j≤i≤ti\ngcd(i, j)(8)\n=X\n1≤i≤tX\n1≤j≤ii\ngcd(i, j)(9)\n=X\n1≤i≤tX\nd|idϕ(d) (10)\n=2ζ(3)\nπ2t3+O(t2logt) (11)\n∼2ζ(3)\nπ2t3(12)\nAfter partitioning the double sum over xandyinto two nested sums, the last expression in (7)\nexpresses the inner sum, in which the summation variable ydoes not appear in the summand, in\nterms Euler’s totient function ϕ(x), the number of residue classes ymodulo xthat are relatively\nprime to x.\nThe asymptotic expression (12) is due to S´ andor and Kramer [18]. They define the function\nthat they investigate by the expression (9), using the notation ψ1(i) for the inner sum in (9),\nand they establish equality between (9) and the last expression in (7) [18, §6, formula (13) with\nα= 1. Formula (13′′′), which should apply here, has an obvious typo]. The expressions (8) and\n(9) are obviously equivalent. We indicate why (8) is equal to (6), thus establishing equality of\nall expressions (6–9):\nIn the sum (6), only primitive vectors appear, but every primitive vector ( x, y)∈Pis taken\nwith multiplicity ⌊t/x⌋. If we look at the multiples ( i, j) = (kx, ky ) of each primitive vector ( x, y)\nfork= 1,2, . . . ,⌊t/x⌋, we get allinteger vectors in the triangle 0 < j≤i≤t. In the sum (8),\neach vector ( i, j) contributes the x-coordinate of the primitive vector ( x, y) = ( i/k, j/k ), since\nk= gcd( x, y). Thus, in total, the vector ( x, y) is taken ⌊t/x⌋times in (8), and we get the same\nsum.\nFormula (10) follows from (9) by splitting the integers j= 1, . . . i according to the value of\ni/gcd(i, j) =d, cf. [18, §2, formula (1)].\nThe asymptotic expression (12) is given in [18, equation (23), p. 61]. A different proof,\nusing the Wiener-Ikehara Theorem, was sketched by Marko Riedel on Mathematics Stack Ex-\nchange [14]. The slightly stronger statement (11) with the explicit error term follows from\nLemma 4.\nB Proof of Lemma 4 about the points on the grid parabola\nThe proof follows standard ideas; in particular, it uses an easy adaptation of the arguments that\nwere used for the special case α= 1 [18], as stated in Lemma 4. Following [18, p. 61], we reduce\nthe computation of Uα\ntto the estimation of the vector Rα(u), which is defined as follows for any\n20real parameter u:\nRα(u) :=X\n1≤x≤u\n0<y≤αx\n(x,y)∈P\u0012x\ny\u0013\n(13)\nWe start from the definition of Uα\ntand regroup and reformulate sums:\nUα\nt=X\n1≤x≤t\n0<y≤αx\n(x,y)∈P\u0016t\nx\u0017\u0012x\ny\u0013\n=X\n1≤x≤t\n0<y≤αx\n(x,y)∈PX\nk≥1\nxk≤t\u0012x\ny\u0013\n=X\nk≥1X\n1≤x≤t\nxk≤t\n0<y≤αx\n(x,y)∈P\u0012x\ny\u0013\n=X\nk≥1X\n1≤x≤t/k\n0<y≤αx\n(x,y)∈Px\n=X\nk≥1Rα(t\nk) (14)\nForRα(u), we will use the following asymptotic estimate, whose proof is given below:\nLemma 6. For0≤α≤1,\nRα(u) =2\nπ2\u0012u3α+O(u2logu)\nu3α2/2 +O(u2logu)\u0013\n(15)\nasu→ ∞ .\nSubstitution of (15) in (14) gives our claimed asymptotic formula for Uα\nt:\nUα\nt=X\nk≥1Rα\u0000t\nk\u0001\n=X\nk≥12\nπ2k3\u0012t3α\nt3α2/2\u0013\n+X\nk≥1\u0012O\u0000t2\nk2logt\nk\u0001\nO\u0000t2\nk2logt\nk\u0001\u0013\n=2ζ(3)\nπ2\u0012t3α\nt3α2/2\u0013\n+\u0012O(t2logt)\nO(t2logt)\u0013\nProof of Lemma 6. We mention that the special case α= 1 and the x-coordinate of R1(u) is\ngiven (without the error bound) in S´ andor and Kramer [18, p. 61, (20)]:\nX\n1≤y≤x≤u\n(x,y)∈Px=X\n1≤x≤uX\n1≤y≤x\n(x,y)∈Px=X\n1≤x≤uxϕ(x)∼2u3\nπ2\nThere, this asymptotic expression is derived as an easy consequence of a general statement of\nRadoux [13] about the sumPu\nx=1f(x\nu)ϕ(x) being asymptotically equal to 6 u2/π2R1\nz=0zf(z)dz\nfor an arbitrary function f, provided that zf(z) is continuous. (The deeper reason behind this\nstatement is that the fraction of primitive vectors among the integer vectors in anysufficiently\nlarge “well-behaved” region is 1 /ζ(2). Examples of well-behaved regions for which this statement\nholds are convex regions with a limit on the ratio between incircle and circumcircle, or dilates\nof a fixed convex region.)\nThe following proof of Lemma 6 adapts the textbook derivation of the similar formula\nX\n1≤y≤x≤u\n(x,y)∈P1 =⌊u⌋X\nx=1ϕ(x) =u2\n2ζ(2)+O(ulogu)∼3u2\nπ2\n21in [11, Theorem 330], a result that goes back to Mertens in 1874.\nWe use the notation Tα(u) for the sum (13) without the condition that ( x, y) should be\nprimitive:\nTα(u) :=X\n1≤x≤u\n1≤y≤αx\u0012x\ny\u0013\n(16)\n=⌊u⌋X\nx=1\u0012x⌊αx⌋\n⌊αx⌋(⌊αx⌋+ 1)/2\u0013\n(17)\n=⌊u⌋X\nx=1\u0012αx2+O(x)\nα2x2/2 +O(x)\u0013\n(18)\n=\u0012α\nα2/2\u0013\u0012⌊u⌋3\n3+⌊u⌋2\n2+⌊u⌋\n6\u0013\n+\u0012O(u2)\nO(u2)\u0013\n(19)\n=\u0012α\nα2/2\u0013u3\n3+\u0012O(u2)\nO(u2)\u0013\n(20)\nWe claim that the O(u2) error terms in (20) can be explicitly bounded by\n\r\r\r\rTα(u)−\u0012α\nα2/2\u0013u3\n3\r\r\r\r\n∞≤E(u) :=(\nu3\n3for 0≤u <1\n5u2\n3foru≥1(21)\nThe bound for the first case, u <1, is trivial because Tα(u) =\u00000\n0\u0001\nin this case. For the second\ncase, u≥1, we accumulate the error bounds for the successive steps of the above derivation.\nTheO(x) error term in the transition from (17) to (18) is bounded (in absolute value) by xin\nthe first coordinate and by αx/2≤xin the second coordinate. Thus, we can bound each of the\nO(u2) terms in (19) by u(u+ 1)/2≤u(2u)/2 =u2.\nThe change in the factor between the term A(u) :=⌊u⌋3\n3+⌊u⌋2\n2+⌊u⌋\n6in (19) and the factor\nu3/3 in (20) is bounded by 2 u2/3, as can be checked by an easy computation:\nA(u) =⌊u⌋3\n3+⌊u⌋2\n2+⌊u⌋\n6≤u3\n3+u2\n2+u\n6≤u3\n3+u2\n2+u2\n6\nA(u) =⌊u⌋3\n3+⌊u⌋2\n2+⌊u⌋\n6≥(u−1)3\n3+(u−1)2\n2+u−1\n6=u3\n3−u2\n2+u\n6≥u3\n3−u2\n2−u2\n6.\nand therefore, |A(u)−u3\n3| ≤2u2\n3. Adding the two contributions together gives the claimed bound\nofu2+ 2u2/3 = 5 u2/3.\nComparing (13) with the sum (16), for which we have an explicit formula, we want to exclude\nthe vectors ( x, y) that are not primitive, i.e., where both xandyare multiples of one of the\nprimes 2 ,3,5,7,11, . . .. By the inclusion-exclusion formula, we have to subtract the contribution\nof those vectors that are divisible by 2, by 3, by 5, etc., add the vectors that are divisible by\ntwoprimes, subtract the vectors that are divisible by three primes, etc. The contribution of\nthe vectors that are divisible by nisnTα(u\nn), as these vectors are all integer vectors from the\nsmaller region 0 < x≤u/n, 0≤y≤αx, scaled by n. We obtain:\nRα(u) =Tα(u)−\u0000\n2Tα(u\n2) + 3Tα(u\n3) + 5Tα(u\n5) +···\u0001\n+\u0000\n2·3·Tα(u\n2·3) + 2·5·Tα(u\n2·5) +···\u0001\n−\u0000\n2·3·5·Tα(u\n2·3·5) +···\u0001\n+···\n=∞X\nn=1µ(n)nTα(u\nn),\n22In the last line, we have expressed the alternating sum in terms of the M¨ obius function\nµ(n) =\n\n+1,ifnis the product of an even number of distinct primes,\n−1,ifnis the product of an odd number of distinct primes,\n0,otherwise, i.e., if nis not square-free.\nWe now use the approximation (20) for Tα(u\nn)\nRα(u) =∞X\nn=1µ(n)nTα(u\nn) =∞X\nn=1µ(n)u3\n3n2\u0012α\nα2/2\u0013\n+E0 (22)\nand bound the error E0by (21):\n∥E0∥∞≤∞X\nn=1\f\fµ(n)nE(u\nn)\f\f≤∞X\nn=1nE(u\nn)\n≤⌊u⌋X\nn=15u2\n3n+∞X\nn=⌊u⌋+1u3\n3n2=5u2\n3·O(logu) +u3\n3·O(1\nu) =O(u2logu)\nThe first term in (22) can be expressed in terms of the zeta-function using the well-known\nrelation [11, Theorem 287] (which also derives from the inclusion-exclusion formula)\n∞X\nn=1µ(n)\nn2=\u0012\n1−1\n22\u0013\u0012\n1−1\n32\u0013\u0012\n1−1\n52\u0013\u0012\n1−1\n72\u0013\n···\n=1\n1+1\n22+1\n42+1\n82+····1\n1+1\n32+1\n92+1\n272+····1\n1+1\n52+1\n252+1\n1252+··��···\n=1P∞\nn=11\nn2=1\nζ(2)=6\nπ2.\nWe obtain\nRα(u) =∞X\nn=1µ(n)u3\n3n2\u0012α\nα2/2\u0013\n+E0=6u3\n3π2\u0012α\nα2/2\u0013\n+E0=2\nπ2\u0012u3α+O(u2logu)\nu3α2/2 +O(u2logu)\u0013\n.\nC Minimum-area convex lattice polygons and grid parabolas\nOur grid parabolas Ptmake their appearance in a different context: the convex lattice n-gons\nof minimum area for a given number nof vertices, see Figure 11 for an example. B´ ar´ any and\nTokushige [5] showed that, as nincreases, the smallest n-gon resembles a more and more oblong\nellipse-like shape, whose “axes” grow like n2andn, respectively, see the schematic drawing in\nFigure 12. It follows from their analysis (although this is not explicitly stated) that, after a\nunimodular transformation, the optimal n-gons are composed of pieces of the grid parabolas\nP1, P2, . . . , P mwith horizontal axes. There is a global finite bound mon the number pieces.\nThere is strong numerical evidence that m= 15, like shown in the picture, and thus, every such\npolygon consist of (at most) 58 pieces, taken from the grid parabolas P1, P2, . . . , P 15.\nD Experiments with grid peeling for parabolas\nOur work was initiated by experimentally exploring the grid peeling process for parabolas\nΠ:y=ax2+bx, i.e., parabolas with a vertical axis. These experiments were carried out\nin the M. Ed. thesis of Moritz R¨ uber [15], and we report some of the findings from this thesis.\n23area = 7307 :5\n107\n90Figure 11: The lattice 75-gon with smallest area\nP1\nP2\nP15\nP15\nP3\n\u0002(n)\n\u0002(n2)\nFigure 12: As nincreases, the minimum-area lattice n-gon becomes more and more oblong.\n24y=1\n20x2\nO\nQFigure 13: An affine grid-preserving transformations maps the origin Oto the point Qand it\nmaps the parabola to itself. The horizontal period His 10 in this example.\nWe tried different rational values of a, sometimes in combination with various values of b, and\nstarted the grid peeling process. We let the process run until it reached a curve that was a\ntranslate of a previous curve, and the process became periodic. Then we could estimate the\naverage vertical speed as well as various other quantities.\nIn order to carry out these experiments, we needed to restrict the computations to a fi-\nnite range, as mentioned in the introduction, and therefore we had to compute the horizontal\nperiod H: The smallest horizontal translation that, in combination with a shearing operation,\nmaps both the parabola Π and the integer grid to itself. The following lemma, which is analogous\nto Lemma 1 for the grid parabola, gives a formula for H:\nProposition 5. Consider the parabola\ny=aN\naDx2+bN\nbDx+c (23)\nwith reduced fractionsaN\naDandbN\nbD, and let ¯H:= lcm( aD, bD).\nThe horizontal period His either equal to ¯Hor to ¯H/2. In particular\n(i) If aD≡0 (mod 4) , then H= lcm( aD/2, bD).\n(ii) If aD≡2 (mod 4) andbD≡2 (mod 4) , then H= lcm( aD/2, bD/2).\n(iii) In all other cases, H= lcm( aD, bD).\nProof. See Figure 13 for an example. Since vertical lines should remain vertical and vertical\ndistances should remain unchanged, the affine transformation that we are looking for has the\nform \u0012x\ny\u0013\n7→\u0012x +H\nzx+y+w\u0013\n(24)\nfor suitable parameters H, z, w . Substituting the right-hand sides of (24) for xandyshould\nleave the parabola equation (23) unchanged. This leads to the following equations for zandw:\nz= 2aN\naDH (25)\nw=aN\naDH2+bN\nbDH (26)\n25The additional requirement is that (24) should map integer grid points to integer grid points.\nThis boils down to requiring that H, z, and ware integral.\nThus, the horizontal period that we are looking for is the smallest positive integer Hfor\nwhich the two quantities zandwdefined above are integers.\nObviously, the choice H=¯H:= lcm( aD, bD) makes both (25) and (26) integral. The\nremainder of the proof, which is elementary but goes into case distinctions, only amounts to\nchecking whether some smaller value of Halso does the job.\nWe start with an easy observation:\nObservation A: If Handzare integers and zis even, then wis an integer iff His a multiple\nofbD.\nThis can be seen after rewriting the first term of (26), using (25) to express one factor Hin\nterms of z,\nw=zH\n2+bN\nbDH (27)\nIfzHis even, then the first termzH\n2in (27) is an integer. Therefore, wis an integer iff the\nsecond term is also an integer, which is the case iff His a multiple of bD.\nWe make a case distinction based on the parity of aD:\nCase 1: aDis even: Then, from (25), z=aN\naD/2His an integer iff His a multiple of aD/2.\nCase 1.1: aDis a multiple of 4: Then His even, and Observation A implies that wis an\ninteger iff His a multiple of bD. Part (i) follows.\nCase 1.2: aD≡2 (mod 4). Then aNmust be odd, becauseaN\naDis reduced.\nWe distinguish two subcases, depending on the (unknown) value H.\nCase 1.2.1: His odd. In this case, z=aNH\naD/2is also odd, and the first termzH\n2in (27) is\na half-integer (a non-integer multiple of 1 /2). For wto be an integer, the second termbNH\nbDmust also be a half-integer. This is only possible if the factor 2 is contained in the denominator\nbD. Then bNmust be odd, becausebN\nbDis reduced. If bDwere divisible by 4, then, since the\nnumerator bNHis odd,bNH\nbDcould not be a half-integer. Hence the only remaining possibility is\nbD≡2 (mod 4). ThenbNH\nbDis a half-integer iffbNH\nbD/2is an integer, which holds iff His a multiple\nofbD/2. The value lcm( aD/2, bD/2) = ¯H/2 is odd, and therefore it is indeed the smallest viable\nvalue of Hin this case.\nCase 1.2.2: His even. Then Hmust be a multiple of lcm(2 , aD/2) = aD. Observation A\nimplies that wis an integer iff His also a multiple of bD. Thus Hmust be a multiple of\n¯H= lcm( aD, bD).\nSumming up the two subcases, we see that the value ¯H/2 for Case 1.2.1 is indeed the smallest\nvalue of Hwhen this case is possible, i.e., when bD≡2 (mod 4). This establishes Part (ii) of the\nProposition. When Case 1.2.1 does not apply, the value H=¯His in accordance with Part (iii).\nCase 2: aDis odd: Then zis an integer iff His a multiple of aD, and zwill be even. As\nabove, Observation A implies that wis an integer iff His also a multiple of bD. The smallest\npossible value His¯H= lcm( aD, bD), and this is in accordance with Part (iii).\nD.1 Results of the experiments\nParabola y=ax2+bxfor various rational values aandb. We computed the convex hull of the\ngrid points above the parabola and started to peel. After some preperiod, the peeling process\nwill settle into a cyclic behavior: After a certain number ∆ mof steps, the same curve reappears,\ntranslated upward by ∆ y. We call ∆ mthetime period and ∆ ythevertical period . The average\n(vertical) speed vis then ∆ y/∆m. Figure 14 shows the average vertical speed depending on a\n(for various values of b).\nThese experiments show a very clear picture. For almost all a, the average speed takes one\nof the values 1 ,1\n2,1\n3, . . ., and it does not depend on b. The sharp transitions between the values\noccur at critical values ofa, which we can recognize, with hindsight, as 1 /(2Ht). In fact, this is\n261\n22\n1\n2\n1\n8\n1\n44\n1\n86\naverage vertical speed v\na\n0:00\n0:05\n0:10\n0:15\n0:20\n0:25\n0:30\n0:35\n0:40\n0:45\n0:50\n0:55\n0:60\n0:65\n0:2\n0:3\n0:4\n0:5\n0:6\n0:7\n0:8\n0:9\n1:0Figure 14: The average vertical speed vof the parabola y=ax2+bxversus the coefficient a\n(for various values of b).\nhow the phenomenon described in Section 2 was discovered: The sequence of denominators of\nthe critical values1\n2,1\n8,1\n22,etc., after clearing the common factor 2, appears in the O.E.I.S. [12,\nA174405], where various formulas are given, including (9) and (10) in Section A. By trying\nto find an interpretation of these formulas that would make sense in the context of a convex\ncurve, we were led to the discovery of the grid parabolas Pt. Experiments soon revealed their\nremarkable behavior under grid peeling, as described in Theorem 2.\nD.2 The average speed at the critical values of a\nAt the critical values a=1\n2Ht, the average vertical speed varies between the consecutive fractions\n1\nt+1and1\nt. Figure 15 shows the average vertical speed for the first four critical values a,\ndepending on b. We see a very regular, piecewise linear dependence on b, filling the range\nbetween1\nt+1and1\nt.\nThe horizontal periodicity of these graphs can be explained easily:\nProposition 6. The parabolas y=ax2+bxandy=ax2+b′xwithb′=b+ 2ahave the same\naverage speed.\nProof. See Figure 16. The “shape” of the parabola depends only on a. Variation of bcan be\ninterpreted as a translation, placing the vertex of the parabola to a different location. More\nprecisely, the vertex of y=ax2+bxlies at the minimum of the function, x=−b\n2a. Thus,\nincreasing bby 2aincurs an amount of horizontal translation that is integral. By Proposition 3,\nthe resulting parabola has then the same average speed.\nWe see that the graph consists of Htrepetitions of a sawtooth. Proposition 6 explains the\nperiodicity with period 1 /Ht. It is also clear exchanging bwith−bperforms only a reflection at\nthey-axis and thus has no effect on the average vertical speed. This mirror symmetry, together\nwith the periodicity with period 1 /Htimplies that the whole graph is determined by the part\nin the interval 0 ≤b≤1\n2Ht.\nWhat remains is the remarkable fact that the average speed grows linearly in this interval.\nWe have no explanation for this phenomenon.\nThe acute observer may notice irregular gaps in the dotted lines of Figure 15. These gaps\nare, however, only an artifact of the way how the values bwere chosen: We took all reduced\nfractions b=bN\nbDwith 0 ≤bN≤bD≤50. Such a set leaves gaps around values with small\ndenominator like 1 ,1\n2,1\n3,2\n3, etc., while it fills other intervals more densely. (In Figure 18 below,\n27a=1\n2H1=1\n2\na=1\n2H2=1\n8\na=1\n2H3=1\n22\na=1\n2H4=1\n44\n0:00\n0:25\n0:50\n0:75\n1:00\nb\n0:00\n0:25\n0:50\n0:75\n1:00\nb\n0:00\n0:25\n0:50\n0:75\n1:00\n0:00\n0:25\n0:50\n0:75\n1:00\naverage vertical speed v\naverage vertical speed v\n0:20\n0:22\n0:24\n0:35\n0:40\n0:45\n0:50\n0:5\n0:6\n0:7\n0:8\n0:9\n1:0\n0:26\n0:28\n0:30\n0:32\n0:34Figure 15: Average vertical speed for critical values a=1\n2Htfort= 1,2,3,4, depending on b\n\u00000\n0\u0001\ny=ax2\ny=ax2+bx\nb\n2a\n\u0000\u0000b=2a\n\u0000b2=4a\u0001\nslope b\nx\ny\nFigure 16: Changing the parameter bof the parabola y=ax2+bxamounts to a translation.\n28this choice of parameters is also the reason why the lower parts of the figure do not exhibit the\nperiodicity of Proposition 3, but rather show the data points arranged along various curves.)\nD.3 The time period at the critical values of a\nNote that an average speed like 0 .4 = 2 /5 (as for the parabola y=1\n8x2+1\n5x, for example) means\nthat we can no longer have a vertical period of 1: To get the fraction 2 /5, the vertical period\n∆ymust be a multiple of 2, and the time period ∆ mmust be a multiple of 5. In this example,\nthe true periods are ∆ y= 6 and ∆ m= 15, see Figure 17. We can see that the curve reappears\nalready after 3 iterations, combined with a horizontal shift by 4 units. Some characteristic\npieces of the curves are marked in red in Figure 17 to highlight this repetition. Note that the\nappearance as a translation is due to the special drawing style of Figure 17, which makes also\nthe horizontal period Happear as a translational symmetry. What appears as a translation is\nactually an affine transformation.\nThe horizontal period is H= 20, and thus, after 20 /4 = 5 shifts (and 5 ×3 = 15 iterations),\nthe shifted curve coincides with the original curve, completing a cycle.\nThis pattern seems to hold for most examples: The curve returns after a “subperiod” of\u0000t+1\n2\u0001\nsteps, shifted by a multiple of Ht, and all multiples of Htmodulo Happear as shifts.\nThus, the time period ∆ m, under these assumptions, is equal to\n∆m=\u0012t+ 1\n2\u0013H\nHt=t(t+ 1)H\n2Ht, (28)\nwhere the horizontal period His given by Proposition 5. Figure 18 shows the time period ∆ m\nfor the same set of parabolas as Figure 15, on a logarithmic scale.\nFor the cases when bis a multiple of a, where the vertical speed takes the “extreme” values\n1\nt+1or1\nt, the time period is t+ 1 or trespectively, with ∆ y= 1. For all remaining cases\nwith one exception, the estimate (28) predicts the correct value. The exception is the parabola\ny=1\n44x2+1\n5xand its symmetric twin y=1\n44x2+4\n5xwith ∆ m= 25, whereas (28) would give\n∆m= 50. The reason behind this exceptional behavior is that the curve reappears already after\n5 iterations (with a horizontal shift of 44) instead of\u0000t+1\n2\u0001\n=\u00005\n2\u0001\n= 10 iterations. We suspect\nthat such exceptions will turn up more frequently when more values of band higher values of t\nare tested.\nIn any case, the experiments revealed interesting patterns in the grid peeling process for\nparabolas (extending the results stated for grid parabolas in Theorem 2 and Lemma 5), whose\nprecise structure remains to be discovered and analyzed.\nWe tried to see whether the grid parabola Ptappears when peeling is started with an ap-\npropriate parabola. To obtain the time period predicted by Theorem 2, as suggested by the\ndata of Figure 18, we tried the parabolas y=1\n2Htx2+1\n2Htx+γfor odd tandy=1\n2Htx2+γ\nfor even t, introducing some vertical shift γ. These parabolas have also the correct horizontal\nperiod H=Htby Proposition 5, using the fact that Htis even iff tis even (Proposition 1.2).\nWe found that, although the average speed does not change with γ, the periodic cycle which\nthe process enters can change. For t≤5, we always found Ptamong one of the periodic cycles,\nwith an appropriate choice of γ.\nIf the piecewise linear dependence of the average speed that is shown by the experiments\nholds for all values of b, this means that there must be parabolas with an irrational average\nspeed, namely when bis irrational. This would imply that the peeling process is necessarily\naperiodic (a fact that can of course not be confirmed experimentally).\nD.4 Deviation from the parabolic shape\nFor a curve Cthat arises in the peeling process of Π we can compute the deviation from the\nshape of the starting parabola Π as follows: We find the highest translate Π+ γthat lies below C\n29H= 20\n\u0001m= 15\n\u0001y= 6Figure 17: Grid peeling of the parabola Π: y=1\n8x2+1\n5x=1\n2Htx2+1\n5xfort= 2, after the\nperiodic behavior has started. Each curve Cis regarded as a function C(x) ofx, and we draw\nthe function C(x)−(1\n8x2+1\n5x) instead of C(x). Straight segments have turned into downward-\ncurving parabolic arcs. The original parabola Π would appear as a horizontal line. (This is the\nsame drawing convention that was used for the grid parabolas in Figure 20.)\nand the lowest translate Π + γ′that lies above C, such that Cis contained in a parabolic tube\nof (vertical) thickness γ′−γ.\nWe looked at all parabolas y=ax2+bxwith rational coefficients aandbin the interval\n0≤a, b < 1, where the denominator of aranges between 1 and 100, and the denominator of b\nbetween 1 and 10. For each parabola, we observed the maximum tube thickness of all curves\narising during the periodic part of the peeling process. Figure 19 plots the maximum tube\nthickness on a logarithmic scale and the parameter a. The dependence on bis not shown.\nMost of the time, the tube thickness is near 1. Interestingly, it rises to high values when a\nis close to one of the critical values1\n2Ht, seemingly approaching a vertical asymptote. At the\ncritical values themselves, the tube thickness is, however, lower than usual.\nE Vertical difference between the grid parabola and the refer-\nence parabola\nFor some selected values of t, we have computed the vertical difference Pt−Πtas a function\nofx, and we show the result in Figure 20.\nWe observe that Ptis always above Π t, with the exception of the horizontal segment of length\ntaround the origin. We can clearly discern local minima near Ht/2,Ht/3 and Ht/4, and less\nmarked minima near other rational multiples of Ht.\n30a=1\n2\na=1\n8\na=1\n22\na=1\n44\n0:0\n0:2\n0:4\n0:6\n0:8\n1:0\n0:0\n0:2\n0:4\n0:6\n0:8\n1:0\n0:0\n0:2\n0:4\n0:6\n0:8\n1:0\n1\n10\n100\n3\n10\n100\n1\n10\n100\n100\n10\n0:0\n0:2\n0:4\n0:6\n0:8\n1:0\nb\nb\ntime period \u0001 m\ntime period \u0001 m\n3\noutliersFigure 18: Time period at the critical values a, as a function of b. The orange point in the graph\nfora=1\n44shows where the value should be if formula (28) were true.\n0:0\n0:2\n0:4\n0:6\n0:8\n1:0\n0:1\n1\n10\na\nmaximum tube thickness\nFigure 19: The tube thickness versus the coefficient a, for various values of b.\n31Ht=2\n1:25\n6\n9\n21:25\n40:5\n25\nt= 10\nt= 20\nt= 40\nt= 60\nt= 80\nt= 100Figure 20: Vertical difference Pt−Πtfort= 10,20,40,60,80,100. The horizontal axis ranges\nfrom the origin to the point x=Ht/2, where both Ptand Π thave slope1\n2. The boundaries of\nthis interval, marked by dashed lines, act as vertical mirrors for the function Pt−Πt. The curves\nare vertically shifted so that they can be nested under each other, and the vertical scale is chosen\nindependently for each curve: The vertical double-headed arrows show the range between 0 and\nthe difference at x=Ht/2. (The outlier value 40 .5 for t= 80 is no mistake. The curve for\nt= 80 rises indeed to a higher maximum than for t= 100.)\n32"    },    {        "title": "2402.15794v1.Hagedorn_transitions_in_exact_U_q_su_2___S_matrix_theories_with_arbitrary_spins.pdf",        "content": "arXiv:2402.15794v1  [hep-th]  24 Feb 2024Hagedorn transitions in exact Uq(su2)S-matrix theories\nwith arbitrary spins\nChangrim Ahn1, Tommaso Franzini2, and Francesco Ravanini3,4\n1Department of Physics\nEwha Womans University\n52, Ewhayeodae-gil, Seoul 03760, S. Korea\n2Department of Physics, Astronomy and Mathematics\nUniversity of Hertfordshire\nHatfield, AL10 9AB, United Kingdom\n3Department of Physics and Astronomy\nUniversity of Bologna\nVia Irnerio 46, 40126 Bologna, Italy\n4Istituto Nazionale di Fisica Nucleare, Sezione di Bologna\nVia Irnerio 46, 40126 Bologna, Italy\nAbstract\nGeneralizing the quantum sine-Gordon and sausage models, we cons truct exact S-matrices\nfor higher spin representations with quantum Uq(su2) symmetry, which satisfy unitarity,\ncrossing-symmetry and the Yang-Baxter equations with minimality a ssumption, i.e. without\nany unnecessary CDD factor. The deformationparameter qis related toa coupling constant.\nBased onthese S-matrices, we derive the thermodynamic Bethe ansatz equations f orqa root\nofunityintermsofauniversal kernel wherethenodesareconnec tedbygraphsofnon-Dynkin\ntype. We solve these equations numerically to find out Hagedorn-like singularities in the free\nenergies at some critical scales and find a universality in the critical e xponents, all near 0.5\nfor different values of the spin and the coupling constant.1 Introduction\nIntegrable quantum field theories (QFTs) in two dimensions are valua ble models of under-\nstanding various non-perturbative properties. Thanks to an infin ite number of conserved\ncharges, multi-particle scattering processes are purely elastic so that the number and mo-\nmenta of the asymptotic particles are preserved and their amplitud es are factorized into\na product of two-particle S-matrices. These S-matrices are often determined completely\nby Yang-Baxter equations (YBEs) along with unitarity and crossing symmetry, and other\nanalyticbootstrap constraints such as bound state poles in the physical strip if they e xist.\nConsider thesine-GordonQFTasanexample. Althoughit isdefined by a Lagrangianofa\nself-interacting scalar field, there is a regime of its coupling constan t (the so-called repulsive\nregime)wheretheasymptoticparticlesarenotbosonic, butadoub letofasoliton-antisoliton\npair, whose S-matrices have been obtained in [1] by the bootstrap method and YB E. The\nS-matrix is proportional to the Rmatrix of the quantum group Uq(su2) [2] where the soliton\npair belongs to a fundamental spin 1 /2 representation and the deformation parameter qis\nrelated to the coupling constant. The S-matrix satisfies the YBE as the Rmatrix does\nby construction. An overall scalar function S0(θ), which cannot be fixed by the YBE, is\ndetermined by unitarity, crossing symmetry and additional analytic properties. Instead,\nwhen the coupling constant is in the attractive regime, the overall s calar function should\nhave additional factors with poles in the physical strip 0 <ℑm θ < π which correspond to\n“breathers”, the soliton-antisoliton bound states. Strictly spea king, the S-matrix bootstrap\nmethod does not completely fix the scalar function since one can alwa ys multiply the so-\ncalled “CDD factors” which satisfy all the S-matrix axioms but without new bound state\npoles in the physical strip. It is customary to adopt a “minimality” ass umption, which\namounts to include no such CDD factors.\nTheS-matrices constructed in this way can provide computational tools for integrable\nQFTs even when conventional Lagrangians are either not given or g iven in terms of scale\ndependent effective parameters. Althoughthe S-matricesareon-shell quantities, they canbe\nused to compute finite-size effects using the Thermodynamic Bethe ansatz (TBA) [3] method\nand even off-shell quantities like correlation functions in terms of th e so-called form factor\nexpansions. The “inverse scattering program” is a philosophical fr amework in which the\nexactS-matrices play a fundamental role. This point of view has been succe ssfully applied\nto many theories in 2D relativistic integrable QFTs such as the sausag e model, a deformed\nnon-linear sigma model on an effective background metric [4], as well a s higher dimensional\n1gauge theories related to AdS/CFT duality [5].\nNotwithstanding, we want to raise a question on the inverse scatte ring program: can\nthere be a scale where the S-matrices are not valid anymore? In QFTs, it is not uncommon\nthat the fundamental degrees of freedom change depending on a scale. For example, quarks\nreplace colour singlet hadrons as fundamental degrees of freedo m when the temperature\nscale goes beyond Λ QCDin the strong interactions, which is the so-called Hagedorn phase\ntransition [6]. A similar transition is also predicted in string theories [7]. Even for integrable\nQFTs, it is not excluded in principle that the particles may either chang e into others or\neven lose their validity as asymptotic states at a certain energy sca le. This possibility may\nconflict apparently with the inverse scattering program and its too ls such as TBA which\nassume the S-matrices are valid at all scales. If this happens, such fundamenta l building\nblocks of integrability as symmetry, asymptotic particle spectrum, and their S-matrices will\nall be in trouble.\nRecent developments on TTdeformations have addressed this question. If an integrable\nQFT is deformed by irrelevant operators built from the energy-mom entum tensor compo-\nnentsTand¯Tand their descendants, Tsand¯Ts, theS-matrix gets extra CDD factors, which\nin turn lead to Hagedorn-like singularities in the free energies [8, 9, 10 , 11, 12]. In excep-\ntional cases where these TTdeformations are fine-tuned, the deformed theories and their\nparticle spectrum can avoid the singularities and reach UV complete t heories, which inter-\npolate between UV and IR conformal QFTs by exact RG flows [13]. But general irrelevant\ndeformations cannot avoid the singularities and meet the Hagedorn transitions at a certain\nscale. Even though the TTdeformations provide interesting examples of the transitions,\none cannot exclude that other irrelevant operators may play a cen tral role in the singular\nbehaviours of the Hagedorn transition. It fails to identify which inhe rent features of the orig-\ninal scattering theories trigger the phase transitions. To unders tand this, it will be useful to\nfind QFTs which show the singularities without any additional irrelevant deformations .\nIn this paper, we will address this question based on a new class of ex act integrable\nfactorised scattering theories. The integrable QFTs we are consid ering here are a generaliza-\ntion of the sine-Gordon and the sausage models to higher spin repre sentation of the quantum\nUq(su2) group. The deformation parameter qis related to a coupling constant. A rational\nversion of this scattering theory for q→1 (zero coupling limit) has been explored before in\n[14]. Similar R-matrix-related constructions for higher spin representations a ppear in the\ncontext of statistical lattice models like the integrable XXZ higher sp in chains introduced\nby a fusion method in [15].\n2In section 2, we derive exact S-matrices for spin smultiplets of quantum Uq(su2) group\nand solve the S-matrix bootstrap conditions exactly. We emphasize that our S-matrices are\n“minimal”, inthe sense thatthey do not have any additional CDDfact or. Asfar aswe know,\ntheseS-matrices are the first ones for higher spin particles with non-trivia l interactions. We\nwill derive the TBA equations rigorously in the form of a universal ker nel and use them to\nstudy the free energy using the S-matrices in section 3 and appendix A. The TBA systems\nare non-linear integral equations which normally cannot be solved an alytically except for the\nUV and IR limits. We therefore need, in section 4, to rely on numerical analysis of the TBA\nto find the critical scale where the Hagedorn singularities occur and numerically estimate\nthe critical exponents. We find that while the critical scales depend both on the values of\nspin and on the coupling constants, the critical exponents are ver y close to 1 /2 for all spins\nand couplings, which suggests a universal behaviour with square ro ot singularity. We will\nsummarize and discuss some open questions in the concluding section 5. An attempt to\nunderstand the analytic mechanisms underlying the phase transitio ns in the TBA systems\nis reported in appendix B, where we analyze a simpler TBA system by us ing a toy kernel to\nfind out exact transition temperature.\n2 Exact S-matrix for particles with an arbitrary spin\nWe first consider a completely factorized scattering theory betwe en particles that belong\nto spinsirreducible representation of the su2algebra generated by J±,J3obeying the\ncommutation relations\n[J±,J3] =±J±,[J+,J−] = 2J3 (2.1)\nand having Casimir operator Q=J2= (J+J−+J−J+)/2 +J2\n3. The spin sare non-\nnegative integers or half-integers for finite-dimensional irreducib le representations. The on-\nshell particles can be denoted by Am(θ) where θis the rapidity which parametrizes the\nenergy-momentum E=mcoshθ, p=msinhθandmis the magnetic quantum number of\nthesu2algebra with m=s,s−1,...,−s.1If the scattering theory is integrable, multi-\nparticle scattering amplitudes are decomposed into two-particle ela sticS-matrix element\nSm′\n1m′\n2m1m2(θ1−θ2) :Am1(θ1)Am2(θ2)→Am′\n2(θ2)Am′\n1(θ1). (2.2)\nThisS-matrix satisfies the Yang-Baxter equations along with unitarity an d crossing sym-\nmetry.\n1Please note the difference between the symbol m(mass) and m, the magnetic su2quantum number.\n3We can decompose this two-particle S-matrix into projectors\nS(θ) =P2s/summationdisplay\nJ=0f[J](θ)P[J], (2.3)\nwherePis the permutation matrix and P[J]the projector into the spin- Jrepresentation,\nP[J]=J/summationdisplay\nM=−J|J,M/an}bracketri}ht/an}bracketle{tJ,M|, (2.4)\nwhich satisfies\n2s/summationdisplay\nJ=0P[J]=I,and/parenleftbig\nP[J]/parenrightbig2=P[J]. (2.5)\nTheir matrix elements are written in terms of the Clebsch-Gordan co efficients\nP[J]m′\n1m′\n2\nm1m2=J/summationdisplay\nM=−J/an}bracketle{ts,m′\n1;s,m′\n2|J,M/an}bracketri}ht/an}bracketle{tJ,M|s,m1;s,m2/an}bracketri}ht. (2.6)\nThe Yang-Baxter equation determines the scalar functions\nf[J](θ) =J/productdisplay\nk=1iπk−θ\niπk+θ(2.7)\nup to an overall function which can be fixed by unitarity and crossing symmetry. This\nS-matrix has been studied in [14].\nWe extend this “rational” S-matrix to the “trigonometric” one Sby introducing certain\ninteractions in terms of a coupling constant which is related to a defo rmation parameter\nq∈Cof the quantum group symmetry algebra Uq(su2), generated by J±,q±J3such that\n[J±,J3] =±J±,[J+,J−] = [2J3] (2.8)\nand with Casimir operator\nQ=J+J−+/bracketleftbigg\nJ3−1\n2/bracketrightbigg\n=J−J++/bracketleftbigg\nJ3+1\n2/bracketrightbigg\n(2.9)\nwhere\n[λ]≡qλ/2−q−λ/2\nq1/2−q−1/2(2.10)\nThe asymptotic massive particles form a spin- srepresentation of Uq(su2).\n4ThisS-matrix can be expressed similarly as in (2.3),\nS(θ) =σ/parenleftigg\nP2s/summationdisplay\nJ=0f[J]\nq(θ)P[J]\nq/parenrightigg\nσ−1, (2.11)\nbut now with some trigonometric scalar functions f[J]\nq(θ),q-deformed projectors P[J]\nqof\nUq(su2), and some gauge transformation σ. All these ingredients of the S-matrices will be\ndetermined completely by imposing constraints such as the Yang-Ba xter equation, unitarity,\nand crossing symmetries.\nFor a generic q, not a root of unity, the Lusztig-Rosso theorem states that the repre-\nsentations of the Uq(su2) are in one to one correspondence to those of su2, and labelled by\ninteger or half-integer J[18]. The tensor products of two irreducible representations are d e-\ncomposed into a direct sum of other irreducible ones in the same way a s the usual addition\nof two angular momenta in su2. The coefficients of this decomposition are now the quan-\ntum Clebsch-Gordan coefficients (qCGs), from which it is possible to c onstruct the quantum\nprojectors:\nP[J]\nqm′\n1m′\n2\nm1m2=J/summationdisplay\nM=−J/an}bracketle{ts,m′\n1;s,m′\n2|J,M/an}bracketri}htq/an}bracketle{tJ,M|s,m1;s,m2/an}bracketri}htq. (2.12)\nHere,|J,M/an}bracketri}htis an eigenvector of the Uq(su2) Casimir operator Qand of J3\nQ|J,M/an}bracketri}ht=/bracketleftbigg\ns+1\n2/bracketrightbigg\n|J,M/an}bracketri}ht,J3|J,M/an}bracketri}ht=m|J,M/an}bracketri}ht. (2.13)\nWe need explicit expressions of the qCGs to write down concrete S-matrices. They are given\nby [17, 19, 20]\n/an}bracketle{ts,m1;s,m2|J,M/an}bracketri}htq=f(J)·q(2s−J)(2s+J+1)/4+s(m2−m1)/2(2.14)\n× {[s+m1]![s−m1]![s+m2]![s−m2]![J+M]![J−M]!}1/2/summationdisplay\nν≥0(−1)νq−ν(2s+J+1)/2\nDν,\nwhere\nDν= [ν]![2s−J−ν]![s−m1−ν]![s+m2−ν]![J−s+m1+ν]![J−s−m2+ν]!,\nf(J) =/braceleftbigg[2J+1]([J]!)2[2s−J]!\n[2s+J+1]!/bracerightbigg1/2\n. (2.15)\nHere we use a convention of the q-factorial for a positive integer n\n[n]! = [n][n−1]···[1],[0]! = 1,[−n]! =∞. (2.16)\n5The summation over νis bounded above since Dν=∞if any argument of q-factorials in\nDνis negative. From these expressions, one can compute the q-projectors straightforwardly.\nNotice that there may be problems in this qCG expression when qis an-th root of unity:\nq=q(r,n) =e2πir/nwithn∈Z>0andr= 1,...,n−1. Then for any integer kmultiple\nofnthe corresponding quantum number [ k] = 0. This fact would create diverging factors\nin the expressions (2.14) and (2.15). To avoid them, one has to reso rt to the more general\nformulae illustrated in [19], where suitable finite expressions for the q CG coefficients are\nrecovered by a careful choice of normalisation of the states and b y a limiting procedure\nwhere a generic (non-root of unity) value of qapproaches a root of unity value q→q(r,n).\nBy this procedure, the root of unity cases leads to an expression f or the quantum projectors\nrelated to that of generic qby continuity. So the formulae below can be taken valid for any\nvalue ofqon the unit circle, be it a root of unity or not.\nFrom the Yang-Baxter equation, the scalar functions f[J]\nq(θ) in (2.11) are obtained as\nf[J]\nq(θ) =S0(θ)/bracketleftiggJ/productdisplay\nk=1qk−qθ/2πi\nqkqθ/2πi−1/bracketrightigg\n, J= 0,1,···,2s, (2.17)\nwhere an overall function S0(θ) is still not fixed.\nWe define the charge conjugation Cby\nC(Am) = (−1)2s+mA−m, m=s,s−1,···,−s (2.18)\nwhich can be represented by the following matrix\nC= (−1)s\n0 0 ···0 1\n0 0 ··· −1 0\n...............\n0 (−1)2s−1···0 0\n(−1)2s0···0 0\n,C2=1. (2.19)\nWith this, the crossing symmetry is expressed as\nSt1(θ) =C1·S(iπ−θ)·C1,C1=C⊗1, (2.20)\nwhere t 1stands for the matrix transpose on the first vector space. To sa tisfy this relation,\nwe need the gauge transformation σas noticed for the sine-Gordon model in [2]. For our\ncase, it is given by\nσ=qJ3θ1/2πi⊗qJ3θ2/2πi, (2.21)\n6whereθ1andθ2are defined in (2.2). In addition to the crossing symmetry, one can s how\nthat this S-matrix is also invariant under the charge conjugation C, a parity P, and a time\nreversalT:\nScd\nab=S¯c¯d\n¯a¯b=Sdc\nba=Sab\ncd. (2.22)\nNow we introduce a coupling constant γby\nq=e2πiγ. (2.23)\nThe scalar functions now can be expressed as\nf[J]\nq(θ) =S0(θ)J/productdisplay\nk=1sinh[γ(ikπ−θ)]\nsinh[γ(ikπ+θ)], J= 0,1,···,2s. (2.24)\nNext, we fix the overall scalar function S0(θ) following a standard procedure. By requiring\nunitarity and crossing symmetry, this function should satisfy\nS0(θ)S0(−θ) = 1, S 0(iπ−θ) =2s/productdisplay\nk=1sinh[γ(i(k+1)π−θ)]\nsinh[γ(ikπ+θ)]S0(θ).(2.25)\nThe standard procedure for fixing S0is to express this as an infinite product of factors that\nsatisfy the crossing symmetry and unitarity alternatingly as follows :\nS0(θ) =2s/productdisplay\nk=1/bracketleftigg\nsinh[γ(iπk+θ)]\nsinh[γ(iπk−θ)]/parenleftigg∞/productdisplay\nℓ=1sinh[γ(iπ(k+ℓ)−θ)]sinh[γ(iπ(k−ℓ)−θ)]\nsinh[γ(iπ(k+ℓ)+θ)]sinh[γ(iπ(k−ℓ)+θ)]/parenrightigg/bracketrightigg\n.(2.26)\nWhensis an integer, i.e. even 2 s, this infinite product is very much simplified to\nS0(θ) =s/productdisplay\nm=1sinh[γ(θ+i2mπ)]\nsinh[γ(θ−i2mπ)]. (2.27)\nTheS-matrix element Sss\nssdescribing scattering between Asparticles with J3=scan be\nread off from (2.24) since only the projector P[2s]contributes:\nSss\nss(θ) =s/productdisplay\nm=1sinh[γ(θ−i(2m−1)π)]\nsinh[γ(θ+i(2m−1)π)]. (2.28)\nThis reproduces the S++\n++element of the sausage model for s= 1.\n7For a half-integer s,i.e.odd 2s, one can convert the infinite products of trigonometric\nfunctions into products of Γ-functions\nS0(θ) =2s/productdisplay\nm=1/braceleftigg\n1\niπsinh[γ(θ+imπ)]Γ/bracketleftbigg\n1−γ(m−1)+iγθ\nπ/bracketrightbigg\nΓ/bracketleftbigg\n1−γm−iγθ\nπ/bracketrightbigg\n×\n×∞/productdisplay\nn=1/bracketleftigg\nR[s,m]\nn(θ)R[s,m]\nn(iπ−θ)\nR[s,m]\nn(0)R[s,m]\nn(iπ)/bracketrightigg/bracerightigg\n, (2.29)\nR[s,m]\nn(θ) =Γ/bracketleftbig\nγ(4sn−4s+2m−1)−iγθ\nπ/bracketrightbig\nΓ/bracketleftbig\n1+γ(4sn−2m+1)−iγθ\nπ/bracketrightbig\nΓ/bracketleftbig\nγ(4sn−2s+2m−1)−iγθ\nπ/bracketrightbig\nΓ/bracketleftbig\n1+γ(4sn−2s−2m+1)−iγθ\nπ/bracketrightbig.(2.30)\nAgain, the S-matrix element between Asparticles becomes\nSss\nss(θ) =2s/productdisplay\nm=1/braceleftigg\n1\niπsinh[γ(θ−imπ)]Γ/bracketleftbigg\n1−γ(m−1)+iγθ\nπ/bracketrightbigg\nΓ/bracketleftbigg\n1−γm−iγθ\nπ/bracketrightbigg\n×\n×Γ[γm]\nΓ[1−γ(m−1)]∞/productdisplay\nn=1/bracketleftigg\nR[s,m]\nn(θ)R[s,m]\nn(iπ−θ)\nR[s,m]\nn(0)R[s,m]\nn(iπ)/bracketrightigg/bracerightigg\n. (2.31)\nAlthough (2.28) for an integer sand (2.31) for a half-integer slook very different, it turns\nout that both have exactly the same integral representation for alls\nSss\nss(θ) = exp/integraldisplay∞\n−∞dk\nksinh(πks)sinhπk(s−1\n2γ)\nsinhπk\n2γsinhπkeikθ. (2.32)\nFrom this representation, one can notice that\nSss\nss(θ) = 1,whenγ=1\n2s. (2.33)\nThis can be thought of as a kind of free point. For s= 1/2, this expression reduces to the\nprefactor of the sine-Gordon S-matrix in [1].\nIn terms of this scalar factor, the S-matrix can be written as\nS(θ) =Sss\nss(θ)·Smat(θ),Smat(θ)≡σ/parenleftigg\nP2s/summationdisplay\nJ=0/bracketleftigg2s/productdisplay\nk=J+1sinh[γ(ikπ+θ)]\nsinh[γ(ikπ−θ)]/bracketrightigg\nP[J]\nq/parenrightigg\nσ−1.(2.34)\nIn the interval\n0≤γ≤1\n2s(2.35)\n8theS-matrix does not present any pole in the physical strip 0 ≤Imθ≤πfor anys, i.e.\nthere are no bound states. We are in a repulsive regime. An analyisis o f theseS-matrices\nfor attractive regime ( γ >1/2s) should be very interesting, but it is out of the scope of the\npresent paper.\nWe present explicit expressions for the next simplest s= 3/2S-matrix which has 4\nparticles Am,m= 3/2,1/2,−1/2,−3/2 withC(Am) =Am=A−m. Denoting these particles\nwith index 1 ,2,3,4, hence ¯1 = 4,¯2 = 3, non-vanishing S-matrix elements are given by the\nprefactor in (2.32) multiplied by the following matrix elements:\nS11\n11= 1,S12\n12=(0)\n(3),S21\n12=s3\n(3),S13\n13=(0)(−1)\n(2)(3),S22\n13=s2/radicalbig\ns3/s1(0)\n(2)(3),\nS31\n13=(s1s4+2s2)(0)\n(2)(3),S22\n22=f1\n(2)(3),S14\n14=(0)(−1)(−2)\n(1)(2)(3),S23\n14=s3(0)(−1)\n(1)(2)(3),\nS32\n14=s2s3(0)\n(1)(2)(3),S41\n14=s1s2s3\n(1)(2)(3),S23\n23=(0)f1\n(1)(2)(3),S32\n23=s2f2\n(1)(2)(3),(2.36)\nandthose related by C,P,Ttransformationsgiven in (2.22). Wehave used theshort notation\n(n)≡2sinh[γ(θ−iπn)], sn≡2sinh(inπγ),\nf1= 2cosh[ γ(2θ−iπ)]+s10\ns5−2s2\ns1, f2= 2s2\ns1cosh[γ(2θ−iπ)]+s2\n2−2s2\n1−4.\n3 Thermodynamic Bethe ansatz\nAt a finite temperature, a large number of asymptotic particles can be created from the\nheat bath, carrying all possible momenta and J3quantum numbers. During elastic scat-\ntering processes, these particles will reach a thermal equilibrium wh ere the momenta are\ndistributed in such a way that the free energy of the system is minimiz ed. This condition for\nthe equilibrium is the thermodynamic Bethe ansatz (TBA) equations. The main technical\ndifficulty arises from the fact that the S-matrix is nondiagonal. Many different “magnons”\ncan appear in diagonalizing transfer matrices. To simplify the analysis , we consider, in this\npaper, 1/γto be integers only, i.e. we are at values of qcorresponding to primitive roots\nof unity. The more generic case should be approached by an adapta tion of the Takahashi\nSuzuki decomposition methods [21], on which we intend to return in fu ture.\n93.1 Bethe-Yang equation\nIfanumber Nofon-shellparticlesarecreatedatafinitetemperature T, eachofthemomenta\ncarried by these particles should satisfy a periodic boundary condit ion, sometimes called the\nBethe-Yang equation. When the S-matrix is nondiagonal, this equation is given by a large\ntensor product of NS-matrices, called in [22] “color” transfer matrix T, formally equivalent\nto the “inhomogeneous” transfer matrix of an XXZ integrable spin c hain with higher spins\n[15]\neiLmsinhθjT(θj|{θi}) = 1, (3.1)\nT(θj|{θi})m′\n1,···,m′\nNm1,···,mN=/summationdisplay\nn1,···,nNSn2m′\n1n1m1(θ1−θj)Sn3m′\n2n2m2(θ2−θj)···Sn1m′\nNnNmN(θN−θj).(3.2)\nHereLis the volume of (infinite) one-dimensional space and we will eventually take the\nL→ ∞limit.\nAsshownin(2.34), thecolortransfermatrixisfactorizedintoapro ductofscalarfunctions\nSss\nssand a matrix part Smat. The matrix part has been diagonalized by analytic Bethe ansatz\nin [15, 16] for the spin sXXZ chain and its generalization to the inhomogeneous case is\nstraightforward. The resulting Bethe-Yang equation for the eige nvalues of Tis given by\neiLmsinhθjN/productdisplay\nk=1,k/negationslash=jSss\nss(θj−θk)M/productdisplay\nℓ=1e2s(θj−λℓ) = 1, (3.3)\nwhere we define short notations\nen(θ)≡sinhγ(θ+iπn/2)\nsinhγ(θ−iπn/2),gn(θ)≡coshγ(θ+iπn/2)\ncoshγ(θ−iπn/2). (3.4)\nThe parameters (Bethe roots) λℓ, often called “magnonic rapidities” in this context, must\nsatisfy the Bethe ansatz equations (BAEs)\nN/productdisplay\nj=1e2s(λℓ−θj) =M/productdisplay\nk=1,k/negationslash=ℓe2(λℓ−λk). (3.5)\nIn the thermodynamic limit where we take L→ ∞andN,M → ∞ , the magnons\norganize into “strings” of length n, where the nrapidities have the same real part but\ndifferent imaginary values as\nλ(n)\nj,α=λ(n)\nj+iπ\n2(n+1−2α), α= 1,2,···,n, (3.6)\n10with the “center” of the string λ(n)\njto be real. If the deformation parameter γis irrational,\nthere is no limit on the length n, hence we need to consider infinitely many different lengths\nof strings. This makes the analysis of TBA equations very complicate d.\nFor simplicity, we will restrict our considerations in this paper to\nγ=1\nN, N∈Z, N≥2s+1, (3.7)\nwhich make the functions in (3.5) periodic in the imaginary direction with periodπN/2.\nFollowing [21], two types of strings are allowed, defined as follows:\n•Type I:λ(n)\nj,αas in (3.6) with n= 1,2,···,N−1\n•Type II: λ(N)\nj=λj+iπN/2.\nTheMmagnons can be reorganized into a Mnnumber of type I strings of length n=\n1,2,···,N−1 and aMNnumber of type II strings. The formation of strings rearranges\nthe Bethe-Yang equation (3.3) as\neiLmsinhθjN/productdisplay\nk=1,k/negationslash=jS00(θj−θk)N−1/productdisplay\nn=1/bracketleftiggMn/productdisplay\nℓ=1S0n(θj−λ(n)\nℓ)/bracketrightiggMN/productdisplay\nk=1S0N(θj−λ(N)\nk) = 1,(3.8)\nwhere\nS00(θ) =Sss\nss(θ), (3.9)\nS0n(θ) =Sn0(θ) =n/productdisplay\nα=1e2s(θ−iπ\n2(n+1−2α)) =min(n,2s)/productdisplay\nj=1e|n−2s|+2j−1(θ),(3.10)\nS0N(θ) =SN0(θ) =g2s(θ). (3.11)\nThe BAEs in (3.5) should be also rearranged in terms of the strings in a similar way\nN/productdisplay\nk=1Sn0(λ(n)\nj−θk)N−1/productdisplay\nm=1/bracketleftiggMm/productdisplay\ni=1,i/negationslash=jSnm(λ(n)\nj−λ(m)\ni)/bracketrightiggMN/productdisplay\nℓ=1SnN(λ(n)\nj−λ(N)\nℓ) = 1,(3.12)\nN/productdisplay\nk=1SN0(λ(N)\nj−θk)N−1/productdisplay\nn=1/bracketleftiggMm/productdisplay\ni=1SNn(λ(N)\nj−λ(n)\ni)/bracketrightiggMN/productdisplay\nℓ=1,ℓ/negationslash=jSNN(λ(N)\nj−λ(N)\nℓ) = 1,(3.13)\nwhere (n,m= 1,···,N−1)\nSnm(θ) =Smn(θ) =n/productdisplay\nα=1m/productdisplay\nβ=1e−2(θ−iπ\n2(n−m−2(α−β)))\n11=\ne|n−m|(θ)en+m(θ)min(n,m)−1/productdisplay\nj=1e|n−m|+2j−1(θ)2\n−1\n, (3.14)\nSnN(θ) =SNn(θ) = [gn−1(θ)gn+1(θ)]−1, (3.15)\nSNN(θ) =e−2(θ). (3.16)\nEqs.(3.8) and (3.12) can be interpreted as diagonal Bethe-Yang eq uations in terms of\nthe original asymptotic massive particles and Nspecies of magnons, that do not transport\nany energy or momentum, but only internal degrees of freedom (c olors) of the multiplets of\nasymptotic particles, whose effective “diagonalized” S-matrices are given by eqs. (3.9 -3.11)\nand (3.14- 3.16).\n3.2 Derivation of thermodynamic Bethe ansatz equations\nWe rewrite eq. (3.8) and eq. (3.12) into\neimLsinhθjN/productdisplay\nk=1,k/negationslash=jS00(θj−θk)N/productdisplay\nn=1/bracketleftiggMn/productdisplay\nℓ=1S0n(θj−λ(n)\nℓ)/bracketrightigg\n= 1, (3.17)\nN/productdisplay\nk=1Sn0(λ(n)\nj−θk)N/productdisplay\nm=1/bracketleftiggMm/productdisplay\ni=1,i/negationslash=jSnm(λ(n)\nj−λ(m)\ni)/bracketrightigg\n= 1, n= 1,···,N.(3.18)\nInL→ ∞limit, the densities of the particles and magnons are defined by\nσn(θ) =1\nLdnn\ndθ, n= 0,1,···,N, (3.19)\nwherednnis the number of the massive particles ( n= 0) or the magnons of type I and II\nwhich carry the rapidities between θandθ+dθ.\nIn terms of these densities, we can rewrite the Bethe-Yang equat ions by taking logarithms\non both sides,\nσn(θ)+ ˜σn(θ) =δn0mcoshθ−νnN/summationdisplay\nm=0Knm⋆σn(θ), n= 0,1,···,N, (3.20)\nwhere we used a short notation\nνn=/braceleftigg\n1, n= 0,N\n−1, n= 1,···,N−1(3.21)\n12and a standard convolution notation ( ⋆)\nf ⋆g(θ) =/integraldisplay∞\n−∞f(θ′)g(θ−θ′)dθ′, (3.22)\nalong with the kernels defined by\nKnm(θ) =1\n2πid\ndθlnSnm(θ). (3.23)\nThe densities of “holes” ˜ σnare defined similarly as (3.19) for “unoccupied” states.\nThe system of equations (3.20) can be simplyfied further by taking in to consideration\ncertain identites on the Fourier transforms of the kernels. The ex plicit expressions of these\nkernels and the technical details of the derivation of the relations a mong them can be found\nin appendix A. The result is a set of simplyfied Bethe-Yang equations, expressed by a single\n“universal” kernel with a simple structure of couplings among densit ies.\nThe TBA equations can be derived in a standard procedure by minimizin g the free energy\nfor a finite temerature T= 1/Rwith the universal Bethe-Yang equations (A.27)–(A.31) as\nconstraints. They are given by coupled nonlinear integral equation s for pseudo-energies ǫn\nǫn(θ) =δn,0mRcoshθ−N/summationdisplay\nm=0Inmp⋆log/parenleftbig\n1+e−ǫm/parenrightbig\n(θ), n= 0,1,···,N, (3.24)\nwhere we have introduced the pseudo-energies\nǫ0(θ) = log˜σ0\nσ0, ǫ n(θ) = logσn\n˜σn, n= 1,...,N−1, ǫ N(θ) = log˜σN\nσN,(3.25)\nand the universal kernel\np(θ) =1\n2πcoshθ. (3.26)\nHereInmare the matrix elements of the incidence matrix2of the graphs in figs. 1 and 2 when\n2s < N−1 and 2s=N−1, respectively.\nAt finite temperature T, the free energy per unit length is obtained by the pseudo-energy\nǫ0using\nf(T)\nT=−/integraldisplay∞\n−∞m\n2πcoshθln/parenleftbig\n1+e−ǫ0(θ)/parenrightbig\ndθ. (3.27)\n2It is the matrix whose element m,nis 1 when the nodes nandmare connected, 0 otherwise.\n13122s−1\n02s2s+1\nN−3N−2N−1\nN\nFigure 1: Dynkin-like structure of the TBA equations for 2 s < N−1. Note that the graph is a\nproperDN+1Dynkin diagram only for 2 s= 1 and an extended one ˆDN+1for 2s= 2.\n12\nN−30N��2N−1\nN\nFigure 2: Structure of the TBA equations of 2 s=N−1.\n4 Numerical Analysis\nThe TBA is based on the idea that there are two equivalent ways to qu antise the theory\nalong different channels. This allows to identify the free energy per u nit length of (3.27)\nwith the Casimir energy of the mirror theory,\nE0(T) =f(T)\nT. (4.1)\nIt is customary to parameterize the vacuum energy as E0(T) =−/tildewidec(r)Tπ/6, by introducing\nthe so-called scaling function /tildewidec(r), withr=mRbeing a dimensionless parameter. From\neq. (3.27) one finds\n/tildewidec(r) =3\nπ2/integraldisplay∞\n−∞rcosh(θ)L0(θ)dθ (4.2)\nwhereL0(θ) = log(1+ e−ǫ0(θ)). In the limit r→0, the ultraviolet (UV) limit, this function\nencodes all the relevant data of the underlying conformal field the ory, since\nlim\nr→0/tildewidec(r) =c−24∆min, (4.3)\nwherecis the central charge and ∆ minis the lowest eigenvalue of the zero-th Virasoro\ngenerator.\nThe TBA equations (3.24) are a system of non-linear integral equat ions for which is in\ngeneral very difficult to find an analytical closed solution. Sometimes , however, it is possible\n14-60 -40 -20 0 20 40 6000.511.522.533.54\n-60 -40 -20 0 20 40 60012345678910\nFigure 3: The functions L0(θ) for spin s= 1/2 (left) and s= 1 (right) with γ= 1/7, for different\nvalues of r. One can see that for smaller values of rthe plateau starts to form.\nto do so. For example, in the UV limit r→0, it is well known that for some theories\nit is possible to find an explicit expression for the central charge cin terms of sums of\nRogers dilogarithms. The fundamental property shared by these theories is the fact that\nasrapproaches 0, the functions log(1 + e−ǫ(θ)), develop a plateau of width ∼2log(2/r),\nas first noticed by Al. Zamolodchikov [3]. In the family of scattering t heories we have\nintroduced above, we have two well-known examples of this behaviou r: the sine-Gordon\nmodel, corresponding to spin s= 1/2, and the sausage model for s= 1. In these cases, the\nplateaus start to form for small values of r, see e.g. fig. 3, and therefore one can explicitly\ncompute the value of the central charge using dilogarithms obtainin gc= 1 and c= 2,\nrespectively, independently from the value of γ= 1/N.\n0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 500.10.20.30.40.50.60.70.80.91\n0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 500.20.40.60.811.21.41.61.82\nFigure 4: Scaling functions for spin s= 1/2 (left) and s= 1 (right).\n15As pointed out above, it is usually difficult to find a closed solution for ge nericr: for this\nreason, it becomes very useful to perform a numerical analysis to study the behaviour of\nthese theories. The method that has proven to be more effective is by solving the system\nof equations via successive iterations. The idea is to start from the initial guess ǫ(0)\nn=\n(rcoshθ,0,...,0) forn= 0,...,Nand then define the k-th iterative solution, with k≥0,\nas\nǫ(k+1)\nn(θ) =δn,0rcosh(θ)−N/summationdisplay\nm=0Inm(p∗L(k)\nm)(θ), n= 0,1,...,N, (4.4)\nwhereL(k)\nn(θ) = log(1+ e−ǫ(k)\nm(θ)). In general, this process is not guaranteed to converge, but\nif it does, one is then able to find with arbitrarily high accuracy the valu es of the pseudo-\nenergies and the corresponding Ln(θ) (an extensive study of this convergence problem has\nbeen done in [23]).\nThis allowsustocomputenumerically theintegral(4.2)atdifferent va luesofr, findingthe\nvalue of the scaling function and, possibly, of the central charge o f the underlying conformal\ntheory. The cases of s= 1/2 ands= 1 are shown in fig. 4.\n4.1 Higher spins and Hagedorn transition\nHaving a natural generalization of the S-matrix for higher values of the spin, s≥3\n2, and of\nthe corresponding TBA equations, it is natural to ask what kind of t heories they describe.\nPerforming the same iterative procedure as above, we observe an unexpected behaviour as\nthe ground state energy E0(r) diverges at a positive finite value r∗and, correspondingly,\nthat the functions Ln(θ) do not develop a plateau, but rather become more peaked around\nθ= 0 as they approach the singular value, as shown in fig. 5.\nExtending the numerical analysis to different values of the spin and o f the coupling con-\nstant, we see that the critical value r∗is a function of both sandN= 1/γ. Some values of\nr∗are listed in table 1.\nMoreover, as can be seen from fig. 6, the critical values r∗are finite even for vanishing\ncoupling constant γ= 1/N→0 fors≥3/2. This means that the phase transitions do occur\nalso at the su(2) symmetric points and the UV limit does not exist even in those case s. It is\ninteresting to notice that a Hagedorn transition of N= 4 SYM at finite temperature exists\nin the limit where the coupling constant vanishes, as shown by [24].\n160 0.5 1 1.5 2 2.5 3-25-20-15-10-50\n-10 -5 0 5 1001234567\nFigure 5: Left: the vacuum energy E0(r) as it approaches the singular point r∗= 0.21628(2);\nright: the kernel L0(θ) at different values of r. Both were obtained for s= 5/2 andN= 12.\ns= 3/2s= 2 s= 5/2s= 3\nN= 40.06024(4) - - -\nN= 50.01683(2) 0.22505(9) - -\nN= 60.00722(5) 0.09996(5) 0.40380(3) -\nN= 70.00392(8) 0.05976(6) 0.21628(2) 0.57301(7)\nN= 80.00248(7) 0.04195(5) 0.14665(8) 0.34110(6)\nN= 90.00174(9) 0.03255(2) 0.11269(7) 0.24773(3)\nN= 100.00132(7) 0.02699(9) 0.09349(6) 0.19958(2)\nN= 110.00106(6) 0.0234(5) 0.08157(4) 0.17123(0)\nN= 120.00089(4) 0.02106(7) 0.07367(8) 0.15307(2)\nTable 1: Some values of the critical scale r∗for different values of sandγ= 1/N.\nA similar behaviour has been recently studied in TT-deformed theories with detailed\nnumerical analysis [25]. In this case, the vacuum energy develops a s quare root singularity\nE0(r)∼r→r∗c0+c1/2√\nr−r∗. (4.5)\nIn this setting, the singularity ultimately appears as a consequence of the presence of a\nCDD factor and it has been regarded as the appearance of a Haged orn-type phase transition.\nRemarkably, it has been shown that by finely tuning the parameters of the deformation, one\ncan ultimately remove the singularity, as described in [13].\nThe theories we have introduced in this work present some similar asp ects, but they are\ncrucially different. Indeed, the S-matrices we consider are not obtained as a deformation\nof some known theory but are genuinely obtained by imposing the defi ning properties of a\n170 5 10 15 20 25 3010-410-310-210-1100\nFigure 6: Valueofthesingularpoint r∗, fordifferentvaluesofspinandcouplingconstant γ= 1/N.\nThe values are computed with precision to the 6th decimal dig it. Ther∗-axis is log-scaled.\nscattering theory in two dimensions, as explained in section 2. As a re sult, the singularities\nare in a sense more “fundamental”, as they cannot be removed by a fine-tuning of the\nparameters. We analysed the behaviour of these models close to th e singularity, for different\nvalues of the spin, at different values of the coupling constant. Mor e explicitly, we have\ngenerated a number of points in a close neighbourhood of width ∼1% of the singular points\nof table 1. We then used these data and fitted the curves, as show n in fig. 7 with a fitting\nfunction given by\nEfit\n0(r) =b(r−r∗)a+c0. (4.6)\nIn table 2 we present the values of the critical exponent aobtained from the numerical\nanalysis, and in fig. 8 we show a fit of these points.\nN\n4 5 6 7 8 9 10 11 12 13 14 15\ns= 3/2 0.486 0.487 0.487 0.482 0.484 0.482 0.482 0.484 0.480 0.479 0.482 0.477\ns= 2 — 0.485 0.490 0.486 0.485 0.480 0.483 0.487 0.487 0.491 0.487 0.485\ns= 5/2 — — 0.506 0.502 0.503 0.509 0.503 0.505 0.498 0.505 0.503 0.509\ns= 3 — — — 0.499 0.492 0.496 0.498 0.495 0.499 0.499 0.496 0.498\nTable 2: Values of the fitted exponent a, for different values of the spin and N= 4,5,...,15.\n180.06025 0.06026 0.06027 0.06028 0.06029 0.0603-58-57.5-57-56.5-56-55.5\n0.016832 0.016836 0.01684 0.016844-247-246-245-244-243-242-241-240-239-238-237\nFigure 7: Examples of fitting for s= 3/2 andN= 4 (left) and N= 5 (right).\nRemarkably, we observe that it becomes independent of the value o f the coupling constant\nand of the spin, approaching a universal value of ∼1/2, compatible with a square root\nbehaviour.\nSimilarly, in table 3 we present the fitted values of the parameters bandc0, only for the\ncases= 3.\nN\n4 5 6 7 8 9 10 11 12 13 14 15\nb— — — 17.81 44.09 82.45 127.59 167.71 219.89 264.01 298.47 341.50\nc0— — — -7.58 -15.57 -24.23 -32.92 -41.24 -48.94 -55.95 -62.25 -67.85\nTable 3: Values of the fitted parameters bandc0, for spin s= 3 and N= 7,8,...,15.\nThese features, however, need a more careful analysis since it is e xtremely difficult and\ncomputationally challenging to study the data in the close vicinity of th e singularity r∗, as\nthe iterative procedure becomes extremely slow. The best way to o vercome this problem\nremains to find clever ways to solve the TBA equations (3.24) analytic ally, in order to have\na more quantitative analysis of these models. A first attempt in this d irection is provided in\nappendix B, with a simpler toy model.\n194 6 8 10 12 14 1600.10.20.30.40.50.60.70.80.91\n(a)s= 3/24 6 8 10 12 14 1600.10.20.30.40.50.60.70.80.91\n(b)s= 2\n6 8 10 12 14 1600.10.20.30.40.50.60.70.80.91\n(c)s= 5/26 8 10 12 14 1600.10.20.30.40.50.60.70.80.91\n(d)s= 3\nFigure 8: Fitted value of the exponent a, for different values of the spin.\n5 Conclusions\nFollowing the inverse scattering program, we have constructed ex actS-matrices of particles\nbelonging to spin smultiplets of quantum Uq(su2) group for any half and integer spins.\nThe scalar factors in front of the S-matrices are derived exactly to satisfy the unitarity and\ncrossing-symmetry in (2.32). The first two simple cases s= 1/2 and 1 match exactly with\nknown results of the quantum sine-Gordon and the sausage models . We have found many\nnewS-matrices in this way. For example, explicit expressions of new S-matrix elements for\ns= 3/2 are given in (2.36).\nWe have derived the TBA equations which can be associated with grap hs similar to but\n20different from Dynkin diagrams. Except for s= 1/2,1, these graphs are different from those\nof classical Lie algebras of finite or affine types. In our opinion, it is no t a coincidence that\nthe Hagedorn-like singularities occur for non-Dynkin graphs for sp ins≥3/2 and this fact\ndeserves, with no doubt, further investigations. We have shown h ow the pseudo-energies and\nthe free energies become complex if the temperature scale become s larger than the critical\nones. Although these exact results are based on a TBA with the sam e non-Dynkin graphs as\nshown in figs. 1 and 2 and a simplified toy kernel, we believe that the orig in of the transition\nto complex values should not be the kernel but the graphs which are of non-Dynkin types.\nThis is supported by the numerical analysis in which both the toy TBA a nd the TBA for\ns≥3/2 show qualitatively the same singularities. It will be interesting to und erstand\nthis from such mathematical considerations as cluster algebras an d hyperbolic algebras. In\nthe context of integrable QFTs, the fact that the pseudo-energ ies become complex implies\nthat the minimization of the free energy leading to the TBA equations fails and signals the\npresence of phase transitions. Unfortunately, we have no theor etical tools beyond the critical\nscales.\nThe numerical data suggest an interesting universality in which the c ritical exponents are\nquite close to 0 .5 independently of the spin sand the coupling constant γ= 1/N. If it\nis interpreted as 1 /2, the singular behaviour near the Hagedorn transitions is similar as a\nparticular deformation by the energy momentum T¯Twhere the free energy can be computed\nusing theBurgers equations. However, we want topoint out that o ur results arequalitatively\ndifferent fromthis T¯Tdeformationsince thephase shifts from(2.32) areregular for all v alues\nofsin the asymtptotic limit θ→ ∞while that of the T¯Tis singular.3\nThere are several unanswered questions, both either conceptu al or technical, which sug-\ngest some future investigations. Considering that our S-matrices are based on the spin s\nrepresentations of Uq(su2), we should understand why cases of s= 1/2,1 are so special to\nhave UV completeness while other values of scannot. Also, we can ask if exact S-matrices\nbased on other groups than su2can show similar singularities. Another interesting question\nis to study the phase transition in the limit of s→ ∞which may be applied to certain\ncondensed matter problems. On the numerical side, a deeper unde rstanding of the scaling\nfunction behaviour in the vicinity of the critical point r∗, maybe using the techniques sug-\ngested in [25] for the numerical integration of TBA, could shed more light on the phenomena\nappearing there.\n3Deformations by higher conserved charges ( T¯T)(s)which lead CDD factors of the “pole” type can also\nlead to the singularities even though the phase shifts are regular as pointed out in [25].\n21Acknowledgements\nWe thank Z. Bajnok, J. Balog, D. Bernard, O. Castro-Alvaredo, V . Dobrev, P. Dorey, B.\nDoyon, F. Essler, D. Fioravanti, S. Lukyanov, M. Mazzoni, S. Negr o, R. Nepomechie, L.\nPiroli, N. Reshetikhin, F. Sailis, M. Staudacher, S. van Tongeren, an d G. Takacs for valuable\ndiscussions, useful comments, and suggestions. This work was su pported by the National\nResearch Foundation of Korea (NRF) grant (NRF-2016R1D1A1B0 2007258) (CA), and by\nCommission IV (Theory) of I.N.F.N. under the grants GAST and DOT4 ( FR).\nA Explicit derivation of Bethe-Yang equations\nIt is important to find explicit expressions of the kernels of TBA equa tions. For this purpose,\nwe introduce another set of functions\nan(θ)≡i\n2πd\ndθlnen(θ) =γ\nπsinnπγ\ncosh2γθ−cosnπγ, (A.1)\nbn(θ)≡i\n2πd\ndθlngn(θ) =−γ\nπsinnπγ\ncosh2γθ+cosnπγ. (A.2)\nIn terms of these functions, the kernels can be derived from eqs.( 3.9-3.11) and (3.14-3.16)\n(n,m= 1,···,N−1)\nK0n=Kn0=−min(n,2s)−1/summationdisplay\ni=1a|n−2s|+2i−1, (A.3)\nK0N=KN0=−b2s, (A.4)\nKnm=Kmn=a|n−m|+an+m+2min(n,m)−1/summationdisplay\ni=1a|n−m|+2i, (A.5)\nKnN=KNn=bn−1+bn+1. (A.6)\nIt is more convenient to take Fourier transforms of the kernels. W ith the convention\nˆf(ω) =/integraldisplay∞\n−∞eiωθf(θ)dθ, f(θ) =/integraldisplay∞\n−∞e−iωθˆf(ω)dω\n2π, (A.7)\nthe Fourier transforms of an(θ) andbn(θ), withγ= 1/N, are given by\nˆan(ω) =sinh/parenleftbig\nπN−n\n2ω/parenrightbig\nsinh/parenleftbig\nπN\n2ω/parenrightbig,ˆbn(ω) =−sinh/parenleftbig\nπn\n2ω/parenrightbig\nsinh/parenleftbig\nπN\n2ω/parenrightbig. (A.8)\n22From these, we can obtain the Fourier transforms of the kernels a s follows:\nˆK00=sinh/parenleftbigN−2s\n2πω/parenrightbig\nsinh(sπω)\nsinh(πω) sinh/parenleftbigN\n2πω/parenrightbig, (A.9)\nˆKNN=sinh/parenleftbigN−2\n2πω/parenrightbig\nsinh/parenleftbigN\n2πω/parenrightbig, (A.10)\nˆK0N=ˆKN0=sinh(sπω)\nsinh/parenleftbigN\n2πω/parenrightbig, (A.11)\nˆKN−1,N=ˆKN,N−1=−sinh/parenleftbigN−2\n2πω/parenrightbig\nsinh/parenleftbigN\n2πω/parenrightbig, (A.12)\nˆK0n=ˆKn0=−sinh/parenleftbign\n2πω/parenrightbig\nsinh/parenleftbigN−2s\n2πω/parenrightbig\nsinh/parenleftbigN\n2πω/parenrightbig\nsinh/parenleftbig1\n2πω/parenrightbig,1≤n <2s, (A.13)\nˆK0n=ˆKn0=−sinh(sπω)sinh/parenleftbigN−n\n2πω/parenrightbig\nsinh/parenleftbigN\n2πω/parenrightbig\nsinh/parenleftbig1\n2πω/parenrightbig,2s≤n≤N−2, (A.14)\nˆKnN=ˆKNn=−2sinh/parenleftbign\n2πω/parenrightbig\ncosh/parenleftbig1\n2πω/parenrightbig\nsinh/parenleftbigN\n2πω/parenrightbig,1≤n≤N−2, (A.15)\nˆKnm=ˆKmn=sinh/parenleftbigN−n\n2πω/parenrightbig\nsinh/parenleftbigm\n2πω/parenrightbig\nsinhω\nsinh2/parenleftbig1\n2πω/parenrightbig\nsinh/parenleftbigN\n2πω/parenrightbig−δnm,1≤m≤n≤N−1.(A.16)\nThe kernel ˆK00can be obtained from (2.32). By defining the following kernels,\nˆp(ω)≡1\n2cosh/parenleftbig1\n2πω/parenrightbig, (A.17)\nˆKnm(ω)≡ˆKnm(ω)+δnm, (A.18)\none can easily check that the following functional relations are satis fied for all 1 ≤m≤N,\nˆKnm=δnm+ ˆp(ηn1ˆKn−1,m+ηn,N−1ˆKn+1,m)−ˆpδn,N−2δmN,1≤n≤N−1,(A.19)\nˆK0m=−ˆpˆK2s,m+ ˆpδ2s,N−1δmN, (A.20)\nˆKNm=δNm−ˆpˆKN−2,m, (A.21)\nˆK00= 1+ ˆp2ˆK2s,2s, (A.22)\nwhere we have used the short notation ηnm= 1−δnm, i.e.\nηnm=/braceleftigg\n0, n=m\n1, n/ne}ationslash=m.(A.23)\nInserting (A.19) into (3.20) for n= 1,···,N−1, we can find\nˆσn+ˆ˜σn= ˆp/parenleftig\nηn1ˆ˜σn−1+ηn,N−1ˆ˜σn+1+δn,N−2ˆσN+δn,2sˆσ0/parenrightig\n(A.24)\n23From (A.21) for n=N, we get\nˆσN+ˆ˜σN= ˆpˆ˜σN−2+δN−1,2sˆpˆσ0, (A.25)\nand from (A.20) and (A.22), we get\nˆσ0+ˆ˜σ0= ˆq+ ˆp(ˆ˜σ2s−δ2s,N−2ˆσN), (A.26)\nwhere ˆqis the formal Fourier transform of mcoshθ. Taking inverse Fourier transforms on\nthese equations, we get\nσ1(θ)+ ˜σ1(θ) =p⋆˜σ2(θ)+δ1,2sp⋆σ0(θ), (A.27)\nσn(θ)+ ˜σn(θ) =p⋆˜σn−1(θ)+p⋆˜σn+1(θ)+\n+δn,N−2p⋆σN(θ)+δn,2sp⋆σ0(θ), n= 2,...,N−2,(A.28)\nσN−1(θ)+ ˜σN−1(θ) =p⋆˜σN−2(θ)+δN−1,2sp⋆σ0(θ), (A.29)\nσN(θ)+ ˜σN(θ) =p⋆˜σN−2(θ)+δN−1,2sp⋆σ0(θ), (A.30)\nσ0(θ)+ ˜σ0(θ) =mcoshθ+p⋆˜σ2s(θ)−δ2s,N−2p⋆σN(θ). (A.31)\nHere we have used the universal kernel p(θ), defined in eq. (3.26). By minimizing the free\nenergywiththeseconstraints, wecanexpresstheTBAequations intermsofincidencematrix\nelements as eq. (3.24).\nB Simplified TBA equations with a toy kernel\nThe TBA system (3.24) is not solvable analytically and it is difficult to unde rstand how the\nTBA develops the singularities. To get an insight on this at technical le vel, we consider the\ntoy kernel4\np(θ) =1\n2δ(θ). (B.1)\nThe TBA equations become very simple, namely\nǫn(θ) =δn,0rcoshθ−1\n2N/summationdisplay\nm=0Inmlog/bracketleftbig\n1+e−ǫm(θ)/bracketrightbig\n, n= 0,1,···,N, (B.2)\n4The normalisation is chosen to match the normalisation of the integra ted universal kernel from the\nprevious sections.\n24and can be expressed as a set of algebraic equations\nxn(θ) =e−rcoshθδn,0N/productdisplay\nm=0[1+xm(θ)]Inm/2, x n(θ)≡e−ǫn(θ), r≡mR (B.3)\nfor each value of θ. Since these are not integral equations anymore, we can analyse it\nanalytically.\nk= 2sN r∗\n1≥4 0\n2≥5 <10−4\n34log4 = 1.38629436 ...\n35 1.38629436\n36 1.38629436\n37 1.38629436\n38 1.38629436\n45log8 = 2.07944154 ...\n46 2.07944154\n47 2.07944154\n48 2.07944154\n57 2.57478171\n58 2.57478171\nTable 4: r∗for various sandN. We observe that r∗depends only on sand it is independent of\nN. Fors= 1/2,1,r∗is zero, meaning that there is no phase transition.\nIn fact, these TBA equations whose graph is shown above in fig. 2 ar e exactly solvable by\nMathematica for N= 2s+1 with 2 s= 3,45in terms of a=e−rcoshθ,\nx[2s=3]\n0=−5\n2+1\n2a2−a−(1+a)2\n2a2√\n1−4a, (B.4)\nx[2s=4]\n0=−3\n2+1\n2a−a−(1+a)\n2a√\n1−8a. (B.5)\nAll other exact expressions for xn’s are also found but we will not put them here since they\nare much more complicated and not relevant in the further discussio ns.\nThese results show that the pseudo-energies can be real if\ne−rcoshθ≤1\n4for 2s= 3; e−rcoshθ≤1\n8for 2s= 4, (B.6)\n5Exact solutions for higher values of sare beyond the capacity of our Mathematica code.\n251.4 1.6 1.8 2 2.2 2.4-0.55-0.5-0.45-0.4-0.35-0.3-0.25-0.2-0.15-0.1-0.05\n2 2.5 3 3.5 4-0.3-0.25-0.2-0.15-0.1-0.050\nFigure 9: Toy TBA with s= 3/2,N= 4 (left) and s= 2,N= 5 (right).\nfor any value of θ. Therefore, the critical values r∗, which are the maximum values of rfor\nthem to remain all real, are found by considering θ= 0, namely,\nr∗=/braceleftigg\nlog4,for 2s= 3\nlog8,for 2s= 4.(B.7)\nFor other values of sandN, we can solve only numerically to find r∗. We list them for\ndifferent values of Nin table 4. It is interesting to notice that the critical values r∗where\nthe solutions turn into complex numbers, depend only on the spin sand not on the coupling\nconstant N= 1/γ. We do not understand this analytically but it is definitely due to the\nexceptionally simplified kernel. For s= 1/2,1 where no phase transition occurs, we observe\nthat the solutions are real for all ras expected.\nThe above exact solutions of the simplified TBA can be used to analyse the free energy\nusing (3.27). 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