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http://arxiv.org/abs/2406.18783v1	20240626230452	Psychological Profiling in Cybersecurity: A Look at LLMs and Psycholinguistic Features	[ "Jean Marie Tshimula", "D'Jeff K. Nkashama", "Jean Tshibangu Muabila", "René Manassé Galekwa", "Hugues Kanda", "Maximilien V. Dialufuma", "Mbuyi Mukendi Didier", "Kalala Kalonji", "Serge Mundele", "Patience Kinshie Lenye", "Tighana Wenge Basele", "Aristarque Ilunga", "Christian N. Mayemba", "Nathanaël M. Kasoro", "Selain K. Kasereka", "Hardy Mikese", "Pierre-Martin Tardif", "Marc Frappier", "Froduald Kabanza", "Belkacem Chikhaoui", "Shengrui Wang", "Ali Mulenda Sumbu", "Xavier Ndona", "Raoul Kienge-Kienge Intudi" ]	cs.CL	[ "cs.CL", "cs.LG" ]	1/f noise of a tiny tunnel magnetoresistance sensor originated from a wide distribution of bath correlation time Toshiki Yamaji July 1, 2024 ====================================================================================================================== § ABSTRACT The increasing sophistication of cyber threats necessitates innovative approaches to cybersecurity. In this paper, we explore the potential of psychological profiling techniques, particularly focusing on the utilization of Large Language Models (LLMs) and psycholinguistic features. We investigate the intersection of psychology and cybersecurity, discussing how LLMs can be employed to analyze textual data for identifying psychological traits of threat actors. We explore the incorporation of psycholinguistic features, such as linguistic patterns and emotional cues, into cybersecurity frameworks. Through case studies and experiments, we discuss the effectiveness of these methods in enhancing threat detection and mitigation strategies.Our research underscores the importance of integrating psychological perspectives into cybersecurity practices to bolster defense mechanisms against evolving threats. § INTRODUCTION Psychological profiling plays a crucial role in cybersecurity, particularly in understanding and identifying the traits and motives of cybercriminals. In computer science, cybersecurity aims to safeguard technology within computer systems, implementing security measures to prevent risks and threats that could harm the system. This field regulates security measures to thwart third-party invaders or intruders who engage in malicious activities such as stealing private, business, or organizational information for personal gain <cit.>. In the domain of cybercrime, understanding the identity and motives of intruders plays a key role in mitigating risks to information security <cit.>. Psychological profiling emerges as a valuable tool for understanding the psychological traits and characteristics of cybercriminals, which strengthens strategies against potential cyber threats and assists in the identification of intruders and their motives through an examination of behavior, nature, and thought process. Profiling in cybersecurity involves diverse criminological and criminal-law-based components, encompassing personal traits, criminal expertise, social attributes, and motivational factors. These elements help in understanding the predispositions, personality traits, demographics, socio-economic status, and motivations of cybercriminals, including those who are particularly elusive <cit.>. Cybercriminals frequently exhibit a range of psychological traits that strongly shape their behaviors and actions <cit.>. These individuals often possess a strong command of cyber technology, which they exploit for harmful purposes and various motives; common motives include financial gain, as seen in activities such as data theft and other forms of cyber fraud <cit.>. Many are driven by greed, pursuing financial rewards, while others seek power or revenge against certain groups or institutions. Some cybercriminals are thrill-seekers, relishing the risk involved in their illicit activities, or opportunists who take advantage of vulnerabilities for personal benefit <cit.>. There are also those who simply disregard legal and ethical standards, compromising their reputations within the cyber community. Traits of fearlessness, with little regard for potential consequences, and a lack of empathy are also prevalent. Moreover, some individuals demonstrate boldness, testing their hacking abilities against individuals and organizations. Collectively, these traits paint a complex picture of the motivations and behaviors driving cybercriminals in various scenarios <cit.>. Motivating factors behind cybercriminal personality traits include revenge and blackmailing. Understanding these traits can help minimize security risks and enable better analysis and resolution of cybercrimes <cit.>. In addition, integrating findings from Large Language Models (LLMs) and psycholinguistic tools, such as the Linguistic Inquiry and Word Count (LIWC) dictionary and the Medical Research Council (MRC) psycholinguistic database <cit.>, into psychological profiling can significantly enrich the understanding of cybercriminal behaviors and motivations. This holistic approach to psychological profiling can not only reveal the complex personalities of cybercriminals but also strengthen overall security measures, protecting both individuals and organizations from cyber threats. In this paper, we explore the intersection of psychology and cybersecurity, with a specific emphasis on the role of LLMs and psycholinguistic features in profiling cyber threats. The remainder of this work is organized as follows. Section <ref> discusses the fundamental role of psychological profiling in cybersecurity, outlining how it aids in understanding and mitigating the behaviors of cybercriminals. Section <ref> explores the application of LLMs in psychological profiling, highlighting their potential to decode complex patterns of cybercriminal activity. In Section <ref>, we examine the incorporation of psycholinguistic features into cybersecurity strategies, demonstrating how these tools can enhance the precision of psychological profiles. Section <ref> discusses different perspectives on psychological profiling in cybersecurity. Section <ref> addresses the ethical considerations and privacy implications inherent in the use of psychological profiling and data analysis in cybersecurity. Finally, Section <ref> discusses future directions for research in this area and Section <ref> concludes the paper with reflections on the evolving landscape of cybersecurity profiling. § PSYCHOLOGICAL PROFILING IN CYBERSECURITY Researchers and practitioners reveal a complex profile of cyber criminals, showcasing traits such as tech-savvy, well-networked, vengeful, goal-oriented, greedy, manipulative, risk-takers, opportunists, rule-breakers, fearless, emotionless, and daring <cit.>. More specifically, <cit.> identified a range of characteristics including smartness, creativity, and a need for control, shedding light on the multifaceted nature of individuals involved in cyber crimes, and uncovering motivating factors like monetary gain, thrill-seeking, and political beliefs that drive individuals towards engaging in cyber criminal activities. In addition to profiling traits, understanding the psychological effects of cybercrime remains essential. <cit.> indicated that exposure to cyber terrorism triggers heightened levels of stress and anxiety among individuals, akin to the psychological effects of conventional terrorism, emphasizing the pivotal role of perceived threats in shaping individuals' attitudes towards government surveillance, regulation, and military responses in the face of cyber threats. <cit.> underscored the significant influence of law enforcement's lack of cybercrime knowledge on low conviction rates and victim underreporting. The study revealed that victims often delay reporting cybercrimes due to embarrassment or a perception that they are better equipped to handle the situation themselves. This highlights the importance of training officers to increase their preparedness in dealing with cybercrime cases and engaging with victims. In a related vein, <cit.> explored the psychological impacts of hacking victimization and underlined the need for support organizations to address these issues. The study underscores the importance of raising awareness about the psychological effects of cybercrime and promoting support opportunities for victims. Its findings provide valuable insights for clinicians and support organizations, informing the development of treatment guidelines and interventions to address the negative psychological impacts of hacking. <cit.> investigated how limited experience and domain knowledge in cyberspace lead to the use of cognitive shortcuts and inappropriate heuristics, resulting in elevated levels of dread. In recent investigations, building upon prior research, <cit.> highlighted the importance of leveraging cybercriminals' cognitive biases to influence their behaviors during attacks. The study suggested that by using algorithms informed by cyberpsychology research, defenders can present low-risk, low-reward targets to steer hackers away from high-value assets. Studies show that attackers exhibit risk-averse behavior, preferring attacks on less secure machines to avoid the appearance of failure. Research on human subjects engaging in cybercriminal behavior revealed a strong relationship between key risk-taking and cybercriminal behaviors. <cit.> indicated that participants' exposure to fictional media, particularly crime-related television shows, can influence their attitudes towards criminal investigations and profiling techniques. The study revealed a correlation between media consumption habits and the perceived realism of investigative procedures portrayed in television episodes. Additionally, participants' beliefs about the role of criminal profilers and the importance of intuition in investigations were influenced by their media exposure. This underscores the nuanced relationship between media consumption and perceptions of criminal behavior and profiling accuracy. Expanding upon the evolving understanding of cybercriminal behavior, <cit.> highlighted the significance of intelligence, personality traits, and social skills in the effectiveness of cyber attacks. The study emphasized the role of environmental factors, such as family relationships and educational background, in shaping the behaviors of hackers. It suggested that a holistic approach, considering both individual characteristics and external influences, is crucial for developing a comprehensive psychological profile of cyber criminals. Additionally, the study noted the need for interdisciplinary collaboration between information technology and investigative psychology to combat cybercrime. Psychological profiling, rooted in behavioral analysis and psychological theory, aims to uncover patterns and traits indicative of malicious intent in cyber activities. This approach utilizes various aspects of human behavior, such as language use, decision-making processes, and emotional responses, to discern the psychological profiles of threat actors <cit.>. Leveraging techniques from psychology, including personality assessment and psycholinguistic analysis, enables the identification of anomalous behaviors and potential indicators of cyber threats. For instance, <cit.> emphasized the importance of profiling potential attackers in cybersecurity to enhance the accuracy of vulnerability severity scores using psychological and behavioral traits. Research investigated the influence of cultural and psychological factors on cyber-security behavior, utilizing the Big Five Framework to assess personality traits and their impact on user attitudes towards privacy and self-efficacy <cit.>. More specifically, <cit.> proposed machine learning models for psychological profiling of hackers based on the “Big Five” personality traits model (OCEAN - Openness, Conscientiousness, Extroversion, Agreeableness, Neuroticism) and their models achieved 88% accuracy in mapping personality clusters with different types of hackers (White Hat, Grey Hat, etc.), identifying cyber-criminal behaviors. <cit.> discovered that individuals attracted to hacking exhibit high scores on Machiavellianism and Psychopathy scales, with Grey Hat hackers showing opposition to authority, Black Hat hackers scoring high on thrill-seeking, and White Hat hackers displaying tendencies towards Narcissism. The Dark Triad traits significantly predict interest in different types of hacking, while thrill-seeking emerges as a key motivator for Black Hat hackers. Perceptions of apprehension for violating privacy laws negatively impact Grey Hat and Black Hat hacking. Moreover, <cit.> revealed that cybercriminals exhibit a range of behaviors and traits that deviate from societal norms, influenced by factors such as heredity, education, culture, and socio-economic status. Profiling methods focus on identifying key psychological features, modus operandi, and criminal motivations to aid in early detection and investigation of cybercrimes. The study emphasizes the significance of expert knowledge and advanced technologies in enhancing law enforcement efforts to combat cybercrime. Overall, the research underscores the evolving nature of criminal profiling in the digital era and the critical role it plays in addressing the growing threat of cybercriminal activities. In response to the escalating threat posed by cybercrimes, <cit.> highlighted the diverse motivations of hackers, including recreation, prestige, revenge, profit, and ideology, which influence their engagement in cyber activities. The study underscores the importance of not only teaching coding skills but also educating individuals about the risks and consequences of online actions to prevent cyber-crime involvement. Additionally, the research emphasizes the need to identify at-risk groups and individuals to target awareness campaigns and promote informed online behavior for future generations. Lastly, the study suggests that understanding social psychological theories can enhance communication with hacker communities and individuals, ultimately contributing to more effective cybersecurity practices. § LLMS IN PSYCHOLOGICAL PROFILING Large Language Models (LLMs), such as OpenAI's GPT series of models, Google's PaLM and Gemini, and Meta's LLaMA family of open-source models, have demonstrated remarkable capabilities in natural language understanding and generation tasks <cit.>. As these models continue to evolve and become more sophisticated, researchers and practitioners are exploring their potential applications beyond language tasks, venturing into the realm of psychological profiling (see Table <ref>). These models are utilized to profile individuals based on their language use patterns and communication styles, facilitating the early detection of potential threats <cit.>. The potential applications of LLM-based psychological profiling are vast and diverse <cit.>. In mental health settings, these techniques aid in the early detection of psychological disorders and the development of personalized treatment plans <cit.>. In human-AI interaction, understanding the perceived personalities of LLMs improves user engagement and trust, leading to more natural and effective interactions <cit.>. However, the application of LLMs to psychological profiling is not without challenges and ethical considerations. Existing personality models and assessment methods have been developed primarily for human subjects, and their suitability for evaluating artificial intelligence systems is questionable. Additionally, the fluid and context-dependent nature of LLM “personalities” raises concerns about the reliability and validity of traditional personality assessment techniques when applied to these models <cit.>. As researchers delve deeper into this emerging field, they must grapple with the complexities of transferring human-centric concepts like personality to artificial intelligence systems. LLMs are explored for psychological profiling tasks, such as detecting personality traits, values, and other non-cognitive characteristics <cit.>. In exploring the multifaceted landscape of psychological profiling with LLMs, researchers have embarked on various avenues to understand their potential applications. For instance, <cit.> focused on investigating the ability of LLMs to simulate human psychological behaviors using prompts to adopt different personas and respond to standardized measures of personality constructs to assess their psychometric properties. <cit.> repurposed standard psychometric inventories originally designed for assessing human psychological characteristics, such as personality traits, values, morality, and beliefs, to evaluate analogous traits in LLMs. <cit.> fine-tuned LLMs on psychometric test items related to the Big Five personality traits for evaluating personalities based on language. <cit.> introduced a method for administering personality tests on LLMs and shaping their generated text to mimic specific human personality profiles. Furthermore, <cit.> proposed PsychoBench, a framework for evaluating personality traits, interpersonal relationships, motivational tests, and emotional abilities to uncover complex psychological profiles within LLMs and their potential integration into human society as empathetic and personalized AI-driven solutions. <cit.> demonstrated that LLM agents conditioned on personality profiles can mimic human traits, with creative personas displaying more consistent behavior in both interactive and non-interactive conditions; the research highlights the importance of robust persona conditioning in shaping LLM behavior and emphasizes the asymmetry in linguistic alignment between different persona groups during interactions. <cit.> presented PsySafe, a framework designed to evaluate and improve the safety of multi-agent systems (MAS) by addressing the psychological aspects of agent behavior. PsySafe incorporates dark personality traits to assess and mitigate potential risks associated with agent behaviors in MAS; in addition, it includes identifying vulnerabilities, evaluating safety from psychological and behavioral perspectives, and implementing effective defense strategies. The findings yielded by PsySafe reveal several phenomena, including collective dangerous behaviors among agents, their self-reflection on engaging in such behaviors, and the correlation between psychological assessments and behavioral safety. While LLMs offer promising applications in psychological profiling, their language generation capabilities also raise concerns about potential misuse for cyber attacks and malicious activities <cit.>. Attack payloads and malware creation involve LLMs generating malicious code or new strains of malware through training on relevant data <cit.>. Automated hacking and vulnerability scanning tasks can be performed by LLMs, including generating code for automated hacking attacks, scanning software for vulnerabilities, or developing exploits <cit.>. In addition, LLMs can be used for social engineering and phishing purposes, leveraging their ability to mimic human language patterns to create convincing social engineering attacks, phishing emails, or disinformation campaigns <cit.>. Adversaries could potentially manipulate LLM outputs for malicious purposes using prompt injection techniques <cit.>. LLMs can generate highly personalized and persuasive phishing emails tailored to specific individuals within an organization, bypassing traditional detection systems. Studies show these AI-crafted attacks can be strikingly effective, with around 10% of recipients entering credentials on fake login portals <cit.>. The ability of LLMs to mimic human language patterns and adapt to different contexts makes them a powerful tool for deception and manipulation <cit.>. The 2023 Report of Voice of SecOps provides a comprehensive analysis of threats and stressors posed by LLMs, revealing that 51% of security professionals are likely to leave their job within 2024.[Generative AI and Cybersecurity: Bright Future or Business Battleground? Deep Instinct. (2023). Voice of SecOps Reports. Retrieved from <https://www.deepinstinct.com/voice-of-secops-reports>. Accessed on May 12, 2024.] The study surveyed over 650 senior security operations professionals in the U.S. to assess LLMs' impact on the cybersecurity industry. Findings indicate a 75% surge in attacks in 2022, with 85% attributing this increase to bad actors leveraging LLMs. Furthermore, 70% of respondents believe LLMs positively influence employee productivity and collaboration, while 63% perceive an enhancement in employee morale. Ransomware emerges as the greatest threat to organizational data security, with 46% of respondents acknowledging its severity and 62% indicating it as the top C-suite concern, a notable increase from 44% in 2022; the pressure to combat ransomware has prompted organizations to revise their data security strategies, with 47% now possessing a policy to pay the ransom, compared to 34% in the previous year. Moreover, the report reveals a 55% increase in stress levels among security professionals, primarily attributed to staffing and resource constraints, cited by 42% of respondents. § PSYCHOLINGUISTIC FEATURES Psycholinguistic features encompass a wide range of linguistic attributes and psychological constructs that reflect cognitive and emotional aspects of language use. Integrating psycholinguistic features into cybersecurity frameworks enhances the granularity of threat profiling techniques and enables a deeper understanding of cybercriminals' mental states and feelings <cit.>. Psycholinguistic features include sentiment analysis, linguistic complexity measures, lexical diversity metrics, and stylistic characteristics. Through advanced text analysis algorithms and machine learning algorithms, these features can be leveraged to identify anomalous patterns indicative of malicious intent. One of the powerful tools in psycholinguistic analysis is the Linguistic Inquiry and Word Count (LIWC) dictionary <cit.>. In the context of cyber attacks, LIWC has been used to detect deception in phishing emails by analyzing the psycholinguistic features that attackers employ to deceive end-users <cit.>. Research shows that phishers often use language conveying certainty (e.g. always, never), time pressure and work-related words to increase vulnerability of targets. Conversely, reward-related words like money or cash tend to decrease vulnerability as they are associated with scams. Beyond phishing, LIWC has been applied to study online predator behavior, analyze developer personalities, model social media rumors, and understand user reactions in crowdsourcing <cit.>. Building on the potential of LIWC for psycholinguistic analysis in cybersecurity, researchers explore its applications to understand attacker behavior and victim vulnerabilities. More precisely, <cit.> focused on analyzing the vulnerability factors of potential victims to cybergrooming using LIWC to quantify and understand the social-psychological traits that may make individuals more susceptible to online grooming; they reveal significant correlations between specific vulnerability dimensions and the likelihood of being targeted as a victim of cybergrooming. Interestingly, the research observed negative correlations between victims and certain family and community-related traits, challenging conventional beliefs about the key factors contributing to vulnerability in online contexts. <cit.> utilize LIWC and demonstrate that malicious insiders exhibit specific linguistic patterns in their written communications, including increased use of self-focused words, negative language, and cognitive process-related words compared to other team members; as insiders become more detached from the team, language similarity decreases over time. In a different angle, psycholinguistic features were utilized to examine the manipulative aspects of cybercrimes. More specifically, <cit.> investigated the psycholinguistic dimensions of social engineering within cybersecurity, employing activity theory to dissect the methods and techniques utilized by malicious actors. This research reveals the sophisticated tactics employed by social engineers to manipulate emotions, impede critical thinking, and exploit moral values to influence user behavior and extract sensitive information. <cit.> proposed a machine learning model for detecting sexual predation in chatrooms using psycholinguistic, content-based, and chat-based features, and show distinct characteristics that differentiate predators from non-predators. Particularly, <cit.> investigated the psychological traits and behaviors of individuals involved in self-reported criminal computer activities, emphasizing the role of extraversion in predicting such behavior and challenging stereotypes by shedding light on the complexities of personality factors in criminal/deviant computer behavior through the use of Likert-scale questionnaires and psychometric instruments. Furthermore, <cit.> conducted a study on phishing influence detection using a novel computational psycholinguistic analysis approach to identify influential sentences that could potentially lead to security breaches and hacking in online transactions and social media interactions, developing a language and domain-independent computational model based on Cialdini's principles of persuasion.[The 6 Principles of Persuasion: Tips from the leading expert on social influence, Douglas T. Kenrick. Posted Dec. 8, 2012. Retrieved from <https://www.psychologytoday.com/ca/blog/sex-murder-and-the-meaning-of-life/201212/the-6-principles-of-persuasion>. Accessed May 20, 2024.] <cit.> indicated that cyber offenders displayed similarities to the community sample on certain traits but exhibited differences from offline offenders, particularly in conscientiousness and openness to experience. Notably, cyber offenders showed lower scores on honesty-humility compared to the community sample, suggesting potential implications for intervention strategies targeting specific personality traits in this population. <cit.> emphasized the importance of understanding psycholinguistic features and psychology in cybersecurity to develop effective strategies and interventions. They explore the emotional responses triggered by cybersecurity breaches, focusing on the hacking of smart security cameras. The study identifies a 3-dimensional structure of emotional reactions, highlighting negative affectivity, proactive versus fight/flight action tendencies, and emotional intensity and valence. Personality characteristics, such as the Big Five traits and resilient/overcontrolled/undercontrolled types, were found to relate to these emotional dimensions. Recently, the application of sentiment analysis techniques has paved the way for building psychological profiles and detecting and understanding cyber threats. <cit.> utilized sentiment analysis to identify discussions around exploits, vulnerabilities, and attack planning on dark web forums even before these threats manifest in the real world, and to provide early warnings through the observation of changes in sentiment and semantic context. <cit.> proposed approaches to predict cyber-events by leveraging sentiment analysis on hacker forums and social media to analyze the sentiment expressed in online discussions and detect signals that may precede cyber attacks. <cit.> built user psychological profiles based on the sentiment analysis of their network browsing and email content, and demonstrate that this approach can proactively and accurately detect malicious insiders with extreme or negative emotional tendencies. Building upon recent studies and advancements, <cit.> developed a machine learning model called TrollHunter and collected a dataset of online trolling messages and found that troll messages exhibit more abusive language, lower cognitive complexity, and greater targeting of named entities and identities; the model achieved an 89% accuracy rate and F1 score in identifying trolling behavior. § DISCUSSION The integration of psychological profiling into cybersecurity practices offers a multifaceted approach to understanding and mitigating cyber threats. LLMs and psycholinguistic features provide deeper understanding into the behaviors, motivations, and emotional states of cybercriminals. This discussion section explores the potential benefits, and challenges of these techniques, drawing from the research findings presented earlier. §.§ Benefits of Psychological Profiling in Cybersecurity Psychological profiling in cybersecurity holds significant promise. Identifying psychological traits and patterns in cybercriminal behavior enables security professionals to anticipate and preemptively counteract potential threats. For instance, understanding the personality traits and motivations of different types of hackers (e.g., White Hat, Black Hat, Grey Hat) allows for more tailored security measures and interventions <cit.>. The use of LLMs enhances this profiling by analyzing large volumes of text data, identifying linguistic patterns that may indicate malicious intent. Psycholinguistic features, such as those derived from the LIWC dictionary, provide additional granularity. These features help in detecting subtle cues in language that might indicate deception, stress, or malicious intent. For example, certain linguistic markers can distinguish phishing emails from legitimate communications, thereby improving the accuracy of threat detection systems <cit.>. Moreover, the incorporation of psychological profiling can aid in the development of more personalized cybersecurity training programs. Understanding the psychological traits that make individuals more susceptible to cyber attacks allows organizations to design targeted awareness campaigns and training modules that address specific vulnerabilities. §.§ Challenges and Limitations Despite the promising applications, several challenges and limitations need to be addressed. One major challenge is the accuracy and reliability of psychological profiling techniques. While LLMs and psycholinguistic tools provide valuable insights, they come with inherent limitations. Implementing and maintaining these advanced profiling systems require a workforce equipped with specialized skills in artificial intelligence, cybersecurity, and psychological analysis. There is often a shortage of professionals with the necessary expertise to develop, deploy, and refine these tools. Addressing this skill gap is crucial for the effective utilization of psychological profiling in cybersecurity. The effectiveness of LLMs largely depends on the quality and diversity of the data they are trained on. Inaccurate models can result from poor-quality data, such as poisoned or contaminated datasets, or from non-representative data. Moreover, acquiring diverse and representative datasets is particularly challenging in the field of cybersecurity, where data sensitivity and proprietary information are significant concerns. Additionally, the use of these tools can lead to false positives and negatives, causing either unnecessary alarms or undetected threats. Thus, ensuring the robustness and validity of these models is vital for their successful deployment in real-world scenarios <cit.>. Another challenge lies in the dynamic and evolving nature of cybercriminal behavior. Cybercriminals continually adapt their tactics to evade detection, which means that profiling techniques must also evolve. Continuous updates and refinements to the models and algorithms are necessary to keep pace with these changes. The ethical implications of psychological profiling in cybersecurity cannot be overlooked. The use of personal data to create psychological profiles raises significant privacy concerns. It is essential to balance the benefits of enhanced security with the protection of individual privacy rights. Transparent policies and stringent data protection measures must be in place to ensure that the use of psychological profiling does not infringe on personal freedoms. § ETHICAL CONSIDERATIONS Ethical considerations are paramount when employing psychological profiling in cybersecurity. The potential for misuse of these technologies for surveillance, manipulation, or discrimination is a serious concern. For example, the ability of LLMs to generate persuasive phishing emails tailored to specific individuals poses a significant threat if used maliciously <cit.>. To mitigate these risks, it is crucial to establish ethical guidelines and regulatory frameworks that govern the use of psychological profiling tools. These guidelines should emphasize the importance of informed consent, data minimization, and transparency in the use of personal data. Additionally, there should be mechanisms for accountability and oversight to ensure that these technologies are used responsibly and ethically <cit.>. § FUTURE DIRECTIONS Future research should focus on improving the robustness of psychological profiling techniques. This includes developing more sophisticated models that can adapt to the evolving tactics of cybercriminals and integrating multimodal data sources (e.g., text, behavioral data, biometric data) to create more comprehensive profiles. Another promising direction is the exploration of collaborative approaches that combine human expertise with machine intelligence. Human analysts and AI systems can collaborate to achieve more effective and nuanced threat detection and mitigation strategies. Finally, ongoing efforts to address the ethical and privacy concerns associated with psychological profiling are essential. This includes developing new methods for anonymizing and protecting personal data while still enabling meaningful analysis, as well as fostering a culture of ethical awareness and responsibility among cybersecurity professionals. § CONCLUSION The integration of psychological profiling, LLMs, and psycholinguistic features into cybersecurity practices represents a significant advancement in the field. These techniques offer the potential to enhance threat detection and mitigation strategies by providing deeper understanding into the behaviors and motivations of cybercriminals. However, realizing this potential requires addressing the challenges and ethical considerations associated with these technologies. By doing so, we can create more robust and responsible cybersecurity frameworks that protect both organizations and individuals from evolving cyber threats. § ACKNOWLEDGMENTS The authors thank all Greprovad members for helpful discussions and comments on early drafts. acl_natbib
http://arxiv.org/abs/2406.18748v1	20240626202625	The localized phase of the Anderson model on the Bethe lattice	[ "Tommaso Rizzo", "Marco Tarzia" ]	cond-mat.dis-nn	[ "cond-mat.dis-nn", "cond-mat.stat-mech" ]	1 2 3 4 § ABSTRACT In this paper, we investigate the Anderson model on the Bethe lattice, focusing on the localized regime. Employing the cavity approach, we derive compact expressions for the inverse participation ratios (IPRs) that are equivalent to those obtained using the supersymmetric formalism and naturally facilitate a highly efficient computational scheme. This method yields numerical results with unprecedented accuracy, even very close to the localization threshold. Our approach allows for high-precision validation of all theoretical predictions from the analytical solution, including the finite jump of the IPRs at the transition. Additionally, we reveal a singular behavior of the IPRs near the critical point that has not been previously reported in the literature. This singular behavior is further confirmed by the numerical solution of the non-linear σ model on the Bethe lattice, which provides an effective description of Anderson localization. The localized phase of the Anderson model on the Bethe lattice Tommaso Rizzo1,2 and Marco Tarzia3,4 July 1, 2024 ============================================================== § INTRODUCTION Anderson localization (AL) <cit.> is one of the most spectacular phenomenon in condensed matter physics. It manifests as the suppression of wave propagation in a disordered medium above a critical value of the disorder strength (and for any finite disorder in low enough dimension) <cit.>. Over the past half-century the field has thrived, with recent experimental observations in diverse systems such as cold atomic gases <cit.>, kicked rotors <cit.>, and classical sound elastic waves <cit.> further highlighting the ubiquity and relevance of this phenomenon. On the theoretical side, the critical properties of AL are well established in low dimensions. According to the scaling hypothesis <cit.> d_L=2 is the lower critical dimension of the transition (for systems with orthogonal symmetry) <cit.>. The scaling arguments have been later supported and quantitatively confirmed by a renormalization group analysis in d = 2 + ϵ <cit.> of an effective field-theory description in terms of a non-linear σ model (NLσM) <cit.>. AL is also analytically tractable in the infinite-dimensional limit on the Bethe lattice (BL) <cit.>, an infinite tree (with no boundaries) in which each node as a fixed degree k+1. The hierarchical structure of the BL allows one to obtain a (complicated) non-linear integral self-consistent equation for the order parameter distribution function, which becomes asymptotically exact in the thermodynamic limit, and whose analysis yields the transition point and the critical behavior <cit.>. Despite these results have been firmly established already several years ago, the study of AL on the BL is still very active, and has continued to reveal new facets and intricacies. There are two main reasons for this. The first concerns the differences between the exotic critical behavior found on the BL and the one observed in finite dimensions and predicted by the scaling analysis. In particular, the diffusion coefficient (or the conductivity) vanishes exponentially at the critical disorder on the BL when the transition is approached from the metallic side <cit.>, while in finite-d such exponential behavior is replaced by a power law (with a d-dependent exponent ν (d-2), d being the spatial dimension and ν the critical exponent describing the divergence of the localization length at the critical disorder <cit.>). The other difference concerns the behavior of the inverse participation ratio (IPR). The IPR is defined as I_2 = ⟨∑_i=1^N \|ψ_α (i) \|^4 ⟩, and is essentially a measure of the inverse volume occupied by an eigenstate. On BLs of finite size N (, random-regular graphs in which every node has a fixed connectivity k+1 <cit.>, see below for a precise definition), I_2 ≃Λ/N in the metallic phase (with a disorder-dependent prefactor Λ which diverges exponentially for W → W_c^- <cit.>), exhibits a discontinuous jump at the transition, and stays of O(1) for W>W_c <cit.>. In contrast, in finite-dimensional systems the IPR vanishes as a power-law at the critical disorder with an exponent ν d <cit.>. Several works have addressed these apparent discrepancies. Both intuitive arguments and quantitative calculations <cit.> have provided strong indications of the fact that the BL limit is a singular point of AL and plays the role of the upper critical dimension of the problem, d_L = ∞, in agreement with previous conjectures <cit.>. The second reason for the remarkable resurgence of interest in AL on the BL and sparse random graphs can be attributed to its strong connection with many-body localization (MBL) <cit.>. MBL involves the localization of highly excited many-body eigenstates even in the presence of interactions, and has been a focal point of recent theoretical and experimental research <cit.>. Since the preliminary investigations, MBL was linked to a form of localization in the Fock space of Slater determinants <cit.> (see also Refs. <cit.>): In this representation, many-body configurations correspond to site orbitals on the graph, subject to (strongly correlated) diagonal disorder, while interactions serve as effective hoppings connecting them. Despite several simplifications in this analogy, it proves valuable for qualitatively understanding the problem <cit.>. In this context, a set of analytical <cit.> and numerical <cit.> explorations of the Anderson model on the BL has been conducted over the last decade. In the midst of such numerous investigations, the predominant research emphasis has leaned towards the delocalized side preceding the critical disorder, leaving the insulating regime relatively underexplored, with only a few notable exceptions <cit.>. Bridging this gap, in this work we perform a thorough investigation of the critical properties of AL on the infinite BL when the transition is approached from the localized phase. Possibly one reason the insulating phase has received comparatively less attention may be attributed to the inherent challenge posed by the fact that the order parameter distribution function (, the probability distribution of the local density of states, see below) exhibits power law tails, which are exceptionally difficult to sample accurately using conventional numerical methods. In fact, the first achievement of our work precisely consists in circumventing this problem: Using the cavity formalism, we derive compact and transparent expressions for the relevant observables (such as the IPR and the distribution of the wave-functions' amplitudes) that are equivalent to those obtained within the supersymmetric approach. These expressions lend themselves naturally to a highly efficient computational method allowing us to obtain results with unprecedented numerical even very close to the transition point. This enables us to precisely assess and validate all the predictions of the analytical solution <cit.> and recover the expected critical behavior <cit.>. In particular our results clearly show that that the IPR exhibit a finite jump at the localization transition, as predicted by the supersymmetric treatment <cit.>. This is particularly interesting, as the existence of such a finite jump has been questioned in some recent works <cit.>. The second noteworthy outcome of our investigation unveils a distinctive feature: the finite jump of the IPR at the critical point is followed by a square root singularity which, to the best of our knowledge, has never been reported in existing literature. Such singular behavior is further corroborated by the solution of the self-consistent equations found on the BL for the NLσM which provides an effective description of AL <cit.>. The analysis of the NLσM also helps to elucidate the highly non-trivial mathematical mechanism underlying this square root singularity. The paper is organized as follows: In Sec. <ref> we introduce the Anderson tight-binding model on the BL and briefly recall the definition of the key observables and the main features of its analytical solution; In Sec. <ref> we discuss the linearized self-consistent equations with respect to a small imaginary part which describe the localized phase and review the main features of their critical behavior; The main results of our work are contained in Sec. <ref>: We start by presenting our new approach to accurately solve the linearized equations, which allows one to retrieve the full probability distribution of the wave-functions' amplitudes in the localized phase; We discuss the resulting singular behavior of the IPR close to the transition point; We show that the solution of the effective NLσM fully support our findings; Finally, in Sec. <ref> we provide a summary of our results and a few perspectives for future investigations. In the Appendix sections <ref>–<ref> we present some technical details and supplementary information that complement the results discussed in the main text. § THE MODEL AND KNOWN RESULTS We consider the simplest model for AL, which consists in a non-interacting (spinless) quantum particle on a lattice in presence of a disordered potential: H = - ∑_⟨ i, j ⟩ t_ij( \| i ⟩⟨ j \| + \| j ⟩⟨ i \|) - ∑_i=1^N ϵ_i \| i ⟩⟨ i \| . The first term is a sum over all pairs of nearest neighbors sites and corresponds to the adjacency matrix of the considered lattice (t_ij is the hopping kinetic energy scale, which we take equal to 1 throughout). The second sum runs over all N sites of the lattice and corresponds to a diagonal random matrix containing the disordered potential. The on-site energies ϵ_i are independent and identically distributed random variables. It is custom to extract them accordingly to a uniform distribution in the interval [-W/2, W/2], W being the disorder strength. The model is defined on an infinite BL, which is formally described as an infinite random-regular graph (RRG) <cit.> in which each vertex has a fixed degree k+1, and can be thought as a tree wrapped onto itself and without boundaries. In fact it can be rigorously shown that RRGs of N nodes have locally a tree-like structure and loops whose typical length scales as ln N/ln k <cit.>. For concreteness in the following we mostly focus on the case k=2, but the same qualitative behavior is expected for any finite k strictly larger than 1. The order parameter associated to AL is the probability distribution of the local density of states (LDoS) <cit.>, defined as ρ_i (E) ≡∑_α \|ψ_α (i)\|^2 δ(E-E_α ) , where ψ_α and E_α are the eigenvectors and the eigenvalues of H. Physically ρ_i(E) is tightly related to the inverse lifetime of a particle of energy E created in i, and its typical value is proportional to the diffusion constant (or the dc conductivity). In the insulating phase the LDoS vanishes in the thermodynamnic limit for W>W_c, since the exponentially localized eigestates of energy E are typically very far from a given node i and do not contribute to the sum. Instead in the metallic phase the LDoS is finite with probability density P(ρ), since extended plane waves have typically amplitudes of order 1/N on all the nodes of the graph. Localization begins from the band edges <cit.>, therefore to see if all states are localized it is sufficient to look at the band center. Hence, for simplicity, we set E=0 throughout the rest of the paper. The distribution of the LDoS, as well as other properties of the spectral statistics of the model, are encoded in the statistics of the elements of the resolvent matrix <cit.>, defined as G_ij = ( iη𝕀 - H)^-1_ij , where 𝕀 is the identity matrix, H is the Hamiltonian (<ref>), and η is an infinitesimal imaginary regulator that softens the pole singularities in the denominator of G. On the BL the diagonal elements of G verify a set of self-consistent recursion relation <cit.>, which become asymptotically exact in the N →∞ limit <cit.>. Deriving these equations involves considering the resolvent matrices of modified Hamiltonians H^(i), where the node i has been removed from the lattice (, H^(i) is obtained by eliminating the i-th row and column from H). The crucial observation here is that, owing to the hierarchical structure of the BL, removing one node renders each of its neighbors uncorrelated from the others, as the lattice breaks into k+1 semi-infinite disconnected branches. Consequently, on any given site i one obtains (, by direct Gaussian integration <cit.> or by using the block matrix inversion formula, also called the Schur complement formula <cit.>): G_i → j =1/ϵ_i - iη - ∑_m ∈∂ i ∖ j t_mi^2 G_m → i , where G_i → j=( iη𝕀 - H^(j))^-1_ii are the so-called “cavity” Green's functions (, the diagonal element on node i of the resolvent of the Hamiltonian H^(j) obtained by removing the node j), ϵ_i is the on-site random energy taken from the uniform distribution, and ∂ i ∖ j denotes the set of all k + 1 neighbors of i except j. (Note that for each node one can define k + 1 cavity Green’s functions, each one satisfying a recursion relations of this kind when one of the k+1 neighbors of the node has been removed.) From the solution of these equations one can finally obtain the diagonal elements of the resolvent on the node i of the original problem as: G_ii =1/ϵ_i - iη - ∑_m ∈∂ i t_mi^2 G_m → i . Eq. (<ref>) should be in fact interpreted as a self-consistent integral equation for the probability distribution of the cavity Green's functions (in the N →∞ limit) P( G, G) = ⟨1/N(k+1)∑_i=1^N ∑_j ∈∂ iδ( G - G_i → j) × ×δ( G - G_i → j) ⟩ , where the average is performed over the disorder distribution. Such integral self-consistent equation can be solved numerically using population dynamics algorithms: The probability distribution of the cavity Green's functions is approximated by the empirical distribution of a large pool of Ω complex elements ( G_α, G_α), P( G, G) ≃ω^-1∑_α=1^Ωδ ( G - G_α) δ ( G - G_α); At each iteration step k instances ( G_α, G_α) are extracted from the pool and a value of ϵ is taken at random from the uniform distribution; A new instance of ( G_α, G_α) is generated using Eq. (<ref>) and inserted in a random position of the pool until the process converges to a stationary distribution (convergence can be monitored for instance by checking that some moments of P( G, G) reach a stationary value). Once the fixed stationary distribution of the cavity Green's function is found, one can implement a similar procedure to obtain the probability distribution P( G, G) of the Green's function of the original problem from Eq. (<ref>) (see also Refs. <cit.> for more details). It is easy to show that the LDoS on a given node i of the lattice, Eq. (<ref>), is proportional to the imaginary part of the Green's function (in the η→ 0^+ limit): ρ_i = 1/πlim_η→ 0^+ Im G_ii , from which the average density of states (DoS) at E=0 is simply given by ρ = 1/N∑_i δ (E_α) = 1/N∑_i ρ_i = 1/π G_ii . Similarly, the generalized inverse participation ratios, defined as I_p = ⟨∑_α∑_i \|ψ_α (i)\|^2 pδ (E_α) ⟩/⟨∑_αδ (E_α) ⟩ , are associated to the p-th moments of the Green's functions in the limit η→ 0^+: \|G_ii\|^p = \|∑_α \|ψ_α (i) \|^2 1/ iη - E_α\|^p ≈∑_α \|ψ_α (i)\|^2 p1/(η^2+E_α^2)^p/2 = ∑_α \|ψ_α (i)\|^2 pδ(E_α) 1/η^p-1∫ _-∞^+∞1/(1 + x^2)^p/2 x . Averaging over all sites, from Eqs. (<ref>), (<ref>), and (<ref>) one obtains a simple spectral representation of the generalized IPRs (for p>1): I_p = √(π) Γ ( p/2)/Γ( p-1/2)lim_η→ 0^+η^p-1⟨ \|G_ii\|^p ⟩/⟨ Im G_ii⟩ . In the metallic phase P( G, G) (and consequently P( G, G)) converges to a stable non-singular distribution. Hence, ⟨ \|G\|^p ⟩ is finite and from Eq. (<ref>) one immediately sees that all the generalized IPR vanish for η→ 0^+. In the insulating phase, instead, P( G, G) (and consequently P( G, G)) is singular in the η→ 0 limit: The (marginal) probability distribution of G has a maximum in the region ImG ∼η and power-law tails P( Im G) ∼√(η)/( Im G)^3/2 with a cutoff at η^-1. Hence the main contribution to the moments comes from the cutoff, ⟨ ( Im G)^p ⟩∝η^1-p (for p ≥ 1/2). The generalized IPR are all of O(1) for W ≥ W_c, and have a finite jump at the transition. (The average DoS, instead, is continuous across the transition.) The normalization integral is dominated by the region Im G ∼η, and the typical value of LDoS is of order η. This behavior reflects the fact that in the localized phase wave-functions are exponentially localized on few O(1) sites where ρ_i takes very large values, while the typical value of the LDoS is exponentially small and vanishes in the thermodynamic limit for η→ 0^+. Quest'ultimo paragrafo sugli elementi fuori diagonale di potrebbe anche togliere. For completeness, it is worth mentioning that the distribution of the eigenfunctions' amplitudes at different points can also be determined from the statistics of the Green's functions via the following relation which holds for η small (again, we specify to the case E=0): \|G_ii\|^p\|G_jj\|^q\|G_ij\|^l ≈∑_α\|ψ_α(i)\|^2 p+l\|ψ_α(j)\|^2 q+l/(η^2+E_α^2)^p/2+q/2+l/2 . The correlation function of wave-functions' amplitudes is tightly related to \|G_ij\|^2, which corresponds to p=q=0 and l=2 (which is equivalent to the case p=q=1 and l=0). § THE SELF-CONSISTENT EQUATIONS IN THE LINEARIZED REGIME AND THE CRITICAL BEHAVIOR As discussed above, in the localized phase the imaginary part of the Green's functions goes to zero linearly with η. It is then convenient to write in full generality G_i → j = g_i → j+ η i ĝ_i → j . When η is small the cavity equation can be linearized and can be rewritten as (we set t_ij=1 on all edges ⟨ i,j ⟩ throughout): g_i → j = 1/ϵ_i - ∑_m ∈∂ i ∖ j g_m → i , ĝ_i → j = g_i → j^2( 1+ ∑_m ∈∂ i ∖ jĝ_m → i) . The above equations have two important features: i) The equation for the real part does not depend on the imaginary part, as the g_i → j's obey the equation corresponding to η=0; ii) The equation for the imaginary part is linear and thus it does not depend on η. The critical disorder W_c is found by studying the stability of the linearized equations (<ref>) and (<ref>) <cit.>. After some manipulations (see App. <ref>) one finds that a solution of the linearized equations of the form P(g,ĝ) ≃f(g)/ĝ^̂1̂ ̂+̂ ̂β̂ (for ĝ≫ 1) only exists if the function f(g) satisfies the following integral equation: f(g) = ∫ K_β (g,g_1) f(g_1) g_1 , which defines a linear β-dependent integral operator with the (non-symmetric) kernel <cit.> K_β (g,g_1) = k \|g\|^2 β∫ϵ p(ϵ) g̃P̃ (g̃) δ( g - 1/ϵ - g_1 - g̃) . Here p (ϵ) is the uniform box distribution of width W of the random energies and P̃ (g̃) is the probability distribution of the sum of the real part of k-1 cavity Green’s functions, Eq. (<ref>). In order for the localized phase to be stable the largest eigenvalue λ_β of the integral operator must be smaller than 1. It is possible to show (see App. <ref>) that, due to a symmetry of the problem <cit.>, for each left eigenvector ϕ(g) of the integral operator the function \|g_1\|^-2 βϕ(1/g_1) is also a right eigenvector (with the same eigenvalue) of the integral operator with β→ 1 - β. Hence the spectrum of (<ref>), and in particular its largest eigenvalue, must be symmetric around β=1/2, as schematically illustrated in Fig. <ref>. The condition that Eq. (<ref>) admits a solution fixes the value of β (the solution with β>1/2 must be picked since in the strong disorder limit one has that β→ 1). The critical point is identified by the point where the solution no longer exists (, the largest eigenvalue of the integral operator becomes larger than one for any β). Due to the symmetry β→ 1 - β, at the transition point one has that β=1/2<cit.>. One can estimate the largest eigenvalue numerically by suitably discretizing the Kernel (<ref>) on a finite grid. This has been recently been done for k=2 with great accuracy in Ref. <cit.> (see also Ref. <cit.>). One finds that the largest eigenvalue for W close to W_c and for β close to 1/2 behaves as: λ_β≃ 1 - c_1 (W-W_c) + c_2 (β -1/2)^2 , with the numerical coefficients given in Ref. <cit.>: W_c ≃ 18.17 , c_1 ≃ 0.0308 , c_2 ≃ 3.18 . For W ≳ W_c we thus have β≃1/2 + √(c_1/c_2)√(W - W_c) . La discussione da qui fino alla fine della sezione si puo' anche togliere. For completeness, it is worth mentioning that the same integral operator (<ref>) with the kernel K_β=1/2 (g,g_1) defined in (<ref>) also controls the long distance behavior the two-point correlation function. In particular, in the localized phase the L dependence of \|G_i,i+L\|^2 is obtained by applying L times the integral operator (<ref>) for β=1/2 divided by the branching ratio k (see App. <ref> and Ref. <cit.> for a detailed explanation). Hence, at large L the behavior of the two-point correlation function is dominated by the largest eigenvalue of the operator, yielding \|G_i,i+L\|^2 ≃ (λ_1/2/k)^L, with λ_1/2→ 1 for W → W_c^+. Yet, the spectrum of (<ref>) is continuous, resulting in a power-law sub-leading correction to the exponential decay <cit.>, yielding: C(L) ∝( λ_1/2/k)^L L^-3/2 = k^-L e^-L/ξ_ loc/L^3/2 . The localization length diverges at the critical point as: ξ_ loc = - [ ln (λ_1/2) ]^-1≃1/c_1 (W-W_c) , where the numerical value of c_1 for k=2 has been computed in Ref. <cit.> and is given in Eq. (<ref>). As mentioned above, the power law prefactor Finally, the properties of the largest eigenvalue λ_β of (<ref>) also control the critical behavior on the metallic side of the transition. In fact, as discussed in Refs. <cit.>, if one performs the analytic continuation of the solution of λ_β=1 below W_c one finds that β acquires an imaginary part: β = 1/2± i√(c_1/c_2)√(W_c - W) . Such imaginary part controls in turn the critical behavior of the correlation volume Λ of typical eigenstates, which is predicted to diverge exponentially as <cit.>: Λ∝exp[ π√(c_2/c_1)/√(W_c - W) ] . The physical interpretation of Λ is the following: For W ≲ W_c typical wave-functions can be thought as the result of the hybridization of many (, an extensive number) of resonant localized peaks very far away from each other: typical eigenstates have O(N/Λ) bumps localized in a small region of the BL where the amplitude is of order Λ/N (to ensure normalization), separated by regions of radius lnΛ where the amplitude is very small <cit.>. In the delocalized phase the imaginary part of the Green's functions remains finite even as the imaginary regulator η goes to zero. As shown in Ref. <cit.>, this implies that the correlation function behaves as: C(L) ∝ηΛk^-L/L^3/2 . § CRITICAL BEHAVIOR OF THE INVERSE PARTICIPATION RATIO In this section we introduce suitable variables whose typical values are related to the (generalized) IPR's, and describe an algorithm that allows one to compute the I_p's with very high numerical accuracy, arbitrarily close to the critical point. The generalized IPR's are related via Eq. (<ref>) to the p-th moment of \|G_ii\| which is broadly distributed, according to Eq. (<ref>). The power-law tails of its probability distribution would lead to divergent expressions for ⟨ \|G_ii\|^p ⟩. In practice this does not occur because the power-law behavior is cut off at large values of the imaginary part at η^-1, which corresponds to the limit of validity of the linearized equations. Yet, in order to compute \|G_ii\|^p we need to take into account the region of large values of the imaginary part of the Green's functions and not the region of typical finite values. It seems therefore that the linearized equations are not useful. Luckily enough, this is not the case. To see this we define: M_ii≡1/G_ii = ϵ_i - i η - ∑_m ∈∂ i G_m → i = m_ii - iη m̂_ii , from which one immediately obtains that ⟨ \|G_ii\|^p ⟩ = ∫ Q(m,m̂) 1/(m^2+m̂^2 η^2)^p/2 m m̂ . Similarly, using the fact that ĝ_ii = m̂_ii/(m_ii^2 + η^2 m̂_ii^2), ⟨ Im G_ii⟩ is expressed as: ⟨ Im G_ii⟩ = ∫ Q(m,m̂) ηm̂/m^2+m̂^2 η^2 m m̂ , Given that m̂ is strictly positive we can make the change of variables m = ηm̂ x that leads to ⟨ \|G_ii\|^p ⟩ = ∫ Q(ηm̂ x ,m̂) (η m̂)^1-p/(1 + x^2)^p/2 x m̂ , ⟨ Im G_ii⟩ = ∫ Q(ηm̂ x ,m̂) 1/1 + x^2 x m̂ . In the η→ 0 limit we can approximate Q(ηm̂ x,m̂) ≈ Q(0,m̂) and perform the integration over x explicitly. From Eq. (<ref>) we immediately obtain that the average DoS (<ref>) is given by: ρ = ∫ Q(0 ,m̂) m̂ . Plugging Eqs. (<ref>) and (<ref>) into Eq. (<ref>), one finally obtains: I_p = ρ^-1∫ Q(0 ,m̂) m̂^1-p m̂ . A similar expression has been derived in Refs. <cit.> in the supermatrix NLσM framework. We will discuss this connection in Sec. <ref>. To sum up, although the moments of the local Green's functions are controlled by the fact that \|G_ii\| is O(1/η) with probability O(η), they can be computed in terms of the typical values of M_ii, whose real part that is typically O(1) and whose imaginary part that is typically O(η). The fact that in the localized phase one can use the linearized equations to compute the relevant observables, such as the (generalized) IPR, facilitates the adoption of highly efficient computational methods that strongly reduces the effect of the finite size of the population compared to the delocalized phase. §.§ An efficient computational scheme for Q(0,m̂) Here we introduce a modification of the population dynamics algorithm which allows us performing the extrapolation of Q(m,m̂) to m=0 very efficiently, thereby allowing one to evaluate Eqs. (<ref>) and (<ref>) with arbitrary accuracy. In fact, from Eq. (<ref>) we have that m_ii = ϵ_i - ∑_m ∈∂ i g_m → i. Hence, the probability that m_ii=0 is equal to the probability that ϵ_i = ∑_m ∈∂ i g_m → i. This occurs with probability density 1/W if \|∑_m ∈∂ i g_m → i\|<W/2, and with zero probability otherwise. Based on this observation, we thus proceed in the following way: For a given value of W > W_c, we implement the standard population dynamics algorithm described in Sec. <ref> and obtain the stationary probability distribution of the cavity Green's function P(g,ĝ) in the linearized regime, corresponding to the solution of Eqs. (<ref>) and (<ref>); We extract k+1 elements (g_α,ĝ_α) from the population and compute m and m̂ from Eq. (<ref>). We define S = ∑_α=1^k+1 g_α; If (and only if) \|S\|<W/2 we add m̂^1-p/W to the numerator and 1/W to the denominator of I_p; We repeat this process several times and divide the numerator and the denominator by the total number of attempts; We renew the elements of the pool of the cavity Green's function by performing a few steps of the standard population dynamics algorithm and repeat the whole process several times until the desired accuracy on I_p is reached. It is worth to mention that the algorithm described here, which is schematically summarized in App. <ref>, can be straightforwardly extended to the computation of generic two-points correlation functions. §.§ Numerical results for the generalized IPRs Below we present the numerical results obtained applying the procedure described above. (All the results presented in this paper are obtained with pools of Ω = 2^28 elements.) We start by focusing on the full probability distribution of m̂ when m is identically equal to zero (divided by ρ to normalize it to 1): Q(0,m̂)/ρ. These probability distributions are plotted in the left panel of Fig. <ref> for several values of the disorder close to the critical point, W_c ≈ 18.17 <cit.>. The figure shows the appearance of the power-law tails at large m̂ with a disorder-dependent exponent β, as expressed in Eq. (<ref>). The values of β extracted from the fit of the tails is reported in the right panel of Fig. <ref>. The dashed line represents the prediction of Eq. (<ref>) obtained from the direct diagonalization of the integral operator (<ref>) close to the critical point performed in <cit.>, which is in excellent agreement with the numerical results. In the left panel of Fig. <ref> we explicitly check that the IPR measured from exact diagonalizations of RRGs of N nodes (see App. <ref> for more details) converges in the large N limit to the values obtained using the “improved” population dynamics scheme described in Sec. <ref>. Yet, upon decreasing W towards the critical point, the finite-size corrections to the asymptotic value becomes stronger and one needs to diagonalize larger systems in order to see the convergence. As a consequence, a precise estimation of the IPR sufficiently close to W_c from exact diagonalizations of finite-size samples is practically out of reach. Concretely, with currently available resources one cannot get reliable results for W ≲ 22. (The finite N corrections of the IPR to the N →∞ value will be studied in detail in a forthcoming work.) In the right panel of Fig. <ref> we show that the IPR obtained using standard population dynamics for the (non-linearized) self-consistent cavity equations (<ref>) in presence of a small but finite imaginary regulator converges in the small η limit to the value obtained directly at η=0 from Eq. (<ref>) using the algorithm described in Sec. <ref>. However, as W gets closer to W_c the finite-η corrections become stronger and one needs to consider smaller and smaller η to see convergence: The data are well fitted by I_2 (η) ≃ I_2(η=0) + a_ηη^b_η (dashed lines), with an exponent b_η decreasing with W and approaching zero at W_c as b_η∝ (W-W_c)^κ. Since upon decreasing η the probability distribution of the imaginary part of the Green's functions becomes broader and broader, obtaining accurate estimations for its moments, which are controlled by the tails, becomes increasingly hard. In practice, using the standard population dynamics algorithm one can measure η\|G_ii\|^2 precisely enough only for η≳ 10^-7. For these reasons, computing the η→ 0 limit of the IPR close enough to W_c with the standard approach is essentially unfeasible. Finally, we specifically focus on the critical behavior of the (generalized) IPR close to W_c. In Fig. <ref>(left) we plot I_p (for η=0 and N→∞) as a function of W≥ W_c, for p=1.4, p=2, and p=4, showing that I_p jumps to a finite value at W_c, as predicted by the analytic solution <cit.>. The behavior of I_p for W ≳ W_c is well described by: I_p ≃ I_p^ (c) + a_p √(W - W_c) . Specifically, for p=2 we find I_2^ (c)≃ 0.304 and a_2 ≃ 0.094. To support this claim, in the right panel of Fig. <ref> we perform a parametric plot of I_2 - I_2^(c) as a function of β-1/2, showing that close enough to the localization transition the data are well described by a linear relation. The confirmation of the jump in the IPR's at the localization transition, as predicted by the supersymmetric analysis <cit.>, is an important result, particularly because it has been challenged in recent studies <cit.> (see also <cit.>). Moreover, the square root singularity identified in Fig. <ref>(left)—standing out as one of the most significant contributions of our work—has not been previously documented in the literature. Exploring the possible connection between this distinctive behavior and the recently emphasized transverse length's singular behavior <cit.>—which governs the exponential decay of wave-functions along typical branches of the tree—would offer intriguing insights on the geometric structure of Anderson localized eigenstates on the BL. To conclude this section, it is worth mentioning that by extending the analysis to higher values of the connectivity of the BL (not shown), we observe that the amplitude of the jump of the IPR at W_c grows with k and appears to approach 1 in the infinite connectivity limit, as predicted in <cit.>. §.§ The Distribution function of the eigenfunctions' amplitudes Comparison of Eq. (<ref>) with Eq. (<ref>) and averaging immediately leads to: ⟨∑_α \|ψ_α(i)\|^2 pδ(E_α) ⟩ = ∫ Q(0,m̂) m̂^1-p m̂ . This implies that the moments of the wave-functions' amplitudes can be directly expressed in terms of the distribution of the imaginary part of G_ii^-1 on the scale η. More precisely introducing the distribution T(u) as in <cit.>: T(u) ≡1/ρ⟨∑_αδ ( u-\|ψ_α (i) \|^2 ) δ(E_α)⟩ , Eq. (<ref>) leads to: T(u) = ρ^-1 Q(0,1/u) 1/u^3 . In Refs. <cit.> the authors obtained the following expressions for the T(u) and I_p analogous to Eqs. (<ref>) and (<ref>). T(u) = ^2 F_l (u)/ u^2 , I_p = p (p-1) ∫ u^p-2 F_l(u) u , in terms of a function F_l(u). By comparison one easily sees that F_l(u) is related to our Q(0,m̂) through Q(0,1/u) = ρ u^3 ^2 F_l (u)/ u^2 . The numerical results for the distributions T(u) are shown in the left panel of Fig. <ref> for several values of the disorder across the localized phase. Since Q(0,m̂) goes to zero as 1/m̂^1+β for large m̂ we have T(u) ∝1/u^2-β . From the above expression one has that ∫ T(u) u is divergent for small values of u. On the other hand this is not consistent with the fact that ∫ T(u) u = N exactly by definition. Ref. <cit.> argues that the matching between the finite N result and thermodynamic limit expression (<ref>) occurs because the integral must be truncated at a some value u_N such that ∫_u_N^∞ T(u) u = N and this leads naturally to u_N ∝1/N^1/1-β . This phenomenon is clearly illustrated in the middle and right panels of Fig. <ref> for W=34 (similar results, not shown, are found for other values of W within the localized phase). In the middle panel we plot the probability distributions of the wave-functions' amplitudes computed from exact diagonalizations of finite RRGs of N nodes (see App. <ref>). For u>u_N these distributions coincide with the one obtained using the cavity approach on the infinite BL, and feature a power-law behavior given in Eq. (<ref>) with β≈ 0.784. For u ≃ u_N the probability distributions on finite graphs exhibit a crossover to a different behavior at small amplitudes described by an integrable square root singularity, T(u) ∝ 1/√(u). The position of the crossover moves to smaller values when the system size is increased, in agreement with the arguments given above. This is shown in the right panel, which indicates that the dependence of the crossover u_N upon the system size is very well described by Eq. (<ref>). §.§ Singular behavior of the IPR within the NLσM formulation In order to confirm the singular behavior of the IPR described in Sec. <ref> and reported in Fig. <ref> and to understand its origin, in this section we consider an effective field-theoretical description of the localization transition first introduced in Ref. <cit.>, in which AL was mapped onto a non-linear σ model with non-compact symmetry. The NLσM representation is obtained as the n →∞ limit of an n-orbital generalization of the problem, which can be viewed as describing an electron hoping between metallic granules containing n orbitals and located at the nodes of the same Bethe lattice. For n=1 the Anderson tight-binding model (with random hopping t_ij) is recovered. Its n-orbital generalization is expected to exhibit the same gross features and the same critical behavior, with the advantage that analytical calculations are usually somewhat simpler. The NLσM on an infinite BL was solved via the supersymmetry approach in Refs. <cit.>. Such solution is expressed in terms of the following self-consistent integral equation for an order parameter function ψ(t), which is essentially akin to the Laplace transform of the probability distribution of the imaginary part of the Green's functions (with the change of variable t= ln s, s being the variable of the Laplace transform): ψ(t) = ∫_- ∞^+∞ t^' L_γ (t - t^') d(t^') ψ^k (t^') , d(t) = exp( -2 e^t ) , L_γ (t) = e^t/2ℓ_γ (t) , ℓ_γ(t) = ( γ/2 π)^1/2 e^- γcosh t [ sinhγcosh t + + ( coshγ - sinhγ/2 γ)] . C'e' una cosa che non capisco sull'espansione attorno a γ_c. Se ci mettiamo vicino al punto critico, γ = γ_c - δγ, ed espandiamo l'equazione (36), abbiamo: ψ_c(t) + δψ_1 (t) + δψ_2 (t)= ∫_- ∞^+∞ t^' [ Γ_c (t - t^') + δΓ (t - t^') ] [ ψ_c(t^') + δψ_1 (t^') + δψ_2 (t^') ]^k , dove ho definito Γ(t - t^') ≡ L_γ (t - t^') d(t^'), con δΓ proporzionale a δγ. Se assumiamo che δψ_1 ∝√(δγ) e δψ_2 ∝δγ abbiamo: δψ_1 (t) = k ∫_- ∞^+∞ t^' Γ_c (t - t^') (ψ_c(t^'))^k-1δψ_1 (t^') 0 = ∫_- ∞^+∞ t^' δΓ (t - t^') ψ_c(t^')^k + k ∫_- ∞^+∞ t^' Γ_c (t - t^') (ψ_c(t^'))^k-1δψ_2 (t^') + k (k-1) ∫_- ∞^+∞ t^' Γ_c (t - t^') (ψ_c(t^'))^k-2 (δψ_1 (t^'))^2 . La prima equazione ci dice che δψ_1 e' un autovettore dell'operatore linearizzato (che, a parte il k davanti, e' lo stesso della fuzione di correlazione a due punti) con autovalore 1. Possiamo riscrivere la seconda equazione in termini dell'operatore critico: k ∫_- ∞^+∞ t^' Γ_c (t - t^') (ψ_c(t^'))^k-1( δψ_2 (t^') + (k-1) (δψ_1 (t^'))^2/ψ_c(t^')) = - ∫_- ∞^+∞ t^' δΓ (t - t^') ψ_c(t^')^k . La cosa che non capisco e' perche' il fatto che questo operatore ha un autovalore massimo uguale a uno e uno spettro continuo sia incompatibile col fatto che δψ_1 ∝√(δγ) e δψ_2 ∝δγ. Non e' incompatibile, e' pero' non banale. I write: E(t) ≡ψ(t)- ∫_- ∞^+∞ t^' L_γ (t - t^') d(t^') ψ^k (t^') we expand the equation we have 0=∫ dt' A(t,t')δψ(t')+ d E/d γδγ + ∫ dt'dt” B(t,t',t”) δψ(t')δψ(t”) Where A(t,t') is the linear operator and B(t,t',t”) is a another regular operator. Now I write δψ(t) in the base of the eigenvectors of A δψ(t) = ∑_q a_q ψ_q(t) And I multiply the equation times the left eigenvector and integrate, I obtain λ_q a_q + δγ∫ dt e^-t Z(t) ψ_q(t) d E/d γδγ where the properties of ψ_q(t) are given by Zirnbauer. a_q = δγ∫ dt e^-t Z(t) ψ_q(t) d E/d γ/λ_q Now if there is only one eigenvalue λ_0 that goes to zero at the critical point the equation becomes an equation for a_0: 0=δγ∫ dt e^-t Z(t)d E/d γ + a_0^2∫ dt dt'dt” e^-t Z(t) B(t,t',t”) ψ_0(t') ψ_0(t”) from which I obtain δψ(t) ≈ a_0 ψ_0 , _O=O(δγ^1/2) But the spectrum is not continuous therefore this argument cannot be simply made. One has to study carefully the projection of d E/d γ over the ψ_q(t). In the NLσM formulation the parameter γ is a dimensionless coupling constant, which plays the role of t_ij/W. The solution of this equation vanishes at t →∞, due to the fact that d(t →∞) = 0, and goes to a constant for t → - ∞ where d(t → - ∞) = 1. In fact one can show <cit.> that the solution decreases monotonically from 1 to 0 as t varies from -∞ to +∞ and has a sharp kink in a region where it decreases rapidly. In the localized phase (, small values of γ), the kink is located somewhere near t=0. For γ larger than a critical value γ_c, instead, the kink is unstable and runs away to minus infinity. Thus the existence of a non-trivial solution of Eq. (<ref>) characterizes the localized phase, while a trivial solution (ψ(t)=0 for t>-∞) corresponds to the metallic phase. To find the limit of stability of the insulating phase one can consider the linearized equation which describes the effect of infinitesimal perturbations on the solution ψ(t) = 1 for large negative t. This analysis leads to the study of the spectral properties of the kernel L_γ, whose largest eigenvalue λ_γ (β) is given by <cit.>: λ_γ (β) = ∫_- ∞^+∞ t e^(1/2 - β) t ℓ_γ (t) , with β∈ [0,1]. λ_γ (β) shares the very same properties (discussed in Sec. <ref>) of the largest eigenvalue of the integral operator defined by the Kernel (<ref>) which emerges in the Anderson problem when studying the stability of the linearized solution of the cavity equations in the localized phase with respect to a small imaginary part of the Green's functions. In particular λ_γ (β) is symmetric for β→ 1 - β and is thus minimal for β=1/2. Close to the transition β behaves as β≃ 1/2 + cst√(γ_c - γ). The fact that β=1/2 at the critical point then yields a closed equation for γ_c <cit.>. When the transition is approached from the localized side, γ≲γ_c, the solution of Eqs. (<ref>) assumes the asymptotic form ψ(t) ≃ 1 - c e^β t , (for t ≪ -1) , where c is a γ-dependent constant of order unity. The left exponential tail of ψ(t) corresponds in fact to the power-law tails of Q(0,m̂) at large m̂ (see Eq. (<ref>) and Fig. <ref>(left)). We have solved Eqs. (<ref>) numerically for k=2 by iteration for several values of γ≤γ_c. For k=2 the critical point is located at γ_c ≃ 0.06803. In practice we discretized the integral over t^' on a finite mesh of constant spacing of N_ bin points in the interval [t_ min, t_ max]. The boundaries of the interval are chosen in such a way that ψ(t) = 1 for t < t_ min and ψ (t) = 0 for t> t_ max within the numerical accuracy. Furthermore, in the interval t ∈ [t_ min, t_ tail] the function ψ(t) is set to be equal to Eq. (<ref>), with β obtained from the solution of Eq. (<ref>). The constant c is fixed in a self-consistent way, by imposing the continuity of the logarithmic derivative of ψ(t) for t=t_ tail. Below we show the results obtained for N_ bin = 6144, t_ min = -64, t_ tail = -44, and t_ max = 16. Given the solution ψ(t) of Eq. (<ref>), the IPR is obtained as <cit.>: I_2 = 2 ∫_- ∞^+∞ t e^t exp( -2 e^t ) ψ^k+1 (t) . The numerical results for I_2 are reported in the left panel of Fig. <ref>, showing that, as for the Anderson model, the IPR has a finite jump at the transition followed by a square root singularity, as in Eq. (<ref>). To understand the origin of such behavior, in the middle panel of Fig. <ref> we plot the difference between the solution found at γ≲γ_c and solution found right at the critical point, ψ_c (t): δψ (t;γ) ≡ψ(t; γ) - ψ_c (t). Close to γ_c one has: I_2(γ) - I_2^(c)≃ 2 (k+1) ∫_- ∞^+∞ t e^t exp( -2 e^t ) ψ_c^k (t) δψ (t;γ) . The function δψ (t;γ) is the largest in correspondence of the kink, which is located approximately around t=0 and whose position moves to the left as γ is increased towards γ_c. Due to the term e^t exp( -2 e^t ) in the equation above the IPR is also dominated by the region around the kink. Yet, as shown in the figure, the δψ (t;γ)'s obtained at different γ collapse on the same function when divided by √(γ_c - γ), implying that I_2(γ) - I_2^(c)∝√(γ_c - γ) close to γ_c. A further confirmation comes from the inspection of the γ dependence of the prefactor c of the left exponential tail of ψ(t), Eq. (<ref>). As shown in the right panel of Fig. <ref>, the prefactor behaves as c(γ) ≃ c_c + c_0 √(γ_c - γ) (with c_c and c_0 of O(1)), implying that for t ≪ -1 one has: δψ (t;γ) ≃√(γ_c - γ) e^1/2 t( δ_1 t + δ_2 + O(t √(γ_c - γ)) ) , with δ_1 and δ_2 of order 1. The analysis of the NLσM thus provides a clear mathematical explanation of the origin of the square root singularity of the IPR: Although the main contribution to I_2 comes from the region around the kink, the matching with the tails at t ≪ -1 produces a scaling regime close to W_c. In this regime, the β dependence of the tails also imparts its influence on the bulk of the distributions. As explained above, ψ(t) is essentially the Laplace transform of the probability distribution of the imaginary part of the Green's function (see Eq. (<ref>)), with the change of variable t = ln s. Hence, the region around the kink corresponds to the values of m̂ of order 1, while the exponential tails at t ≪ -1 of ψ(t), Eq. (<ref>), correspond to the power-law tails of Q(0,m̂) for m̂≫ 1 with exponent 1+β. Drawing inspiration from our examination of the NLσM, below we endeavored to apply a similar scaling analysis to the Anderson model. In order to do so, we define δQ̃(m̂;W) as the difference between the order parameter distribution function Q(0,m̂)/ρ found at disorder W>W_c and right at the critical point W=W_c: δQ̃(m̂;W) ≡Q(0,m̂;W)/ρ(W) - Q(0,m̂;W_c)/ρ(W_c) , in terms of which one has: I_2(W) - I_2^(c) = ∫δQ̃(m̂;W)/m̂ m̂ . In Fig. <ref> we show that, when divided by the square root distance from the critical point (δ W)^1/2 = √(W-W_c), the δQ̃ (m̂;W)'s computed for different disorder levels tend to collapse on the same scaling function when W approaches W_c. The right panel of Fig.<ref> highlights this data collapse particularly in the region m̂≳ 1, which gives the dominant contribution to the integral (<ref>). This implies that the square root singularity of the IPR observed in Fig. <ref> is due to the square root dependence of δQ̃ in the bulk, and in particular at small m̂. In a specular way, the collapse implies that at large m̂ and near W_c, the tails of the order parameter distribution function behave as: Q(0,m̂) ≃c_c + c_0 √(W-W_c)/m̂^1/2 + √(c_1/c_2)√(W - W_c) , (which is the analog of Eq. (<ref>)) where c_1 and c_2 are given in Eq. (<ref>) for k=2 <cit.>. The mechanism by which the square root dependence of the prefactor of the tails is directly inherited from the square root dependence of the exponent of the tails is not obvious, and is certainly an interesting question for future investigations. § TWO-POINTS CORRELATION FUNCTION Finally, we focus on the critical behavior of the two-point function defined in Eq. (<ref>). It is easy to show that, thanks to the tree-like structure of the BL, the off-diagonal elements of the resolvent on two nodes at distance L along a branch of the tree can be expressed in terms of the product of the diagonal elements of the (cavity) Green's functions along the branch: G_i,i+L= t G_i → i+1 t G_i+1 → i+2 ⋯ t G_i+L-1 → i+L G_i+L,i+L . The moments of G_i,i+L can also be computed with high accuracy using a procedure analogous to the one described in Sec. <ref> to evaluate the moments of G_ii. To this aim we introduce: M_i,i+L≡1/G_i,i+L = m_i,i+L + i η m̂_i,i+L . Following the same steps as above, one can show that lim_η→ 0^+η^p-1 \|G_i,i+L\|^p = ∫_-∞^+∞ x 1/(1 + x^2)^p/2 ×∫_-∞^+∞m̂_i,i+L Q_L(0,m̂_i,i+L) 1/\|m̂_i,i+L\|^p-1 , where Q_L( M_i,i+L) is the probability distribution of M_i,i+L. This relation has been obtained in the η→ 0 limit by approximating Q_L(ηm̂ x,m̂) ≈ Q_L(0,m̂). As detailed in App. <ref>, in order to evaluate Q_L(0,m̂_i,i+L) within the population dynamics algorithm one proceed in a way similar as the one described in Sec. <ref>. The results of this procedure are illustrated in Fig. <ref>. (Note that the value of the correlation function in L=0 is proportional to the IPR, which is of O(1) in the whole localized phase including the critical point.) The expected asymptotic behavior of the two-point function at large L obtained from the analytic solution on the infinite BL is given by Eq. (<ref>). In the left panel of Fig. <ref> we plot lim_η→ 0 k^L L^3/2 (η \|G_i,i+L\|^2) as a function of L for several values of the disorder in the localized phase and for k=2. The plot shows an exponential decay at large L, in agreement with Eq. (<ref>). The localization length can be thus defined from: ξ_ loc^-1 = lim_L →∞Θ(L) , Θ(L) ≡ - 1/L ln ( lim_η→ 0 k^L L^3/2η \|G_i,i+L\|^2 ) . In the middle panel of Fig. <ref> we plot Θ(L) as a function of L for several values of W. The plateau reached by Θ(L) at large L provides an estimation of ξ_ loc^-1. The values of ξ_ loc^-1 obtained in this way are reported in the right panel of Fig. <ref>, showing that ξ_ loc is described by Eq. (<ref>), with c_1 ≃ 0.0307 ± 0.0002 and W_c ≃ 18.17 ± 0.05, in perfect agreement with the critical behavior obtained by diagonalizing explicitly the integral operator (<ref>) that governs the linear stability of the cavity equations in the localized phase for k=2 <cit.>. § CONCLUSIONS In this paper we have analyzed the localized phase of the Anderson model on the infinite BL. We have put forward an improved population dynamics scheme to compute the moments of the imaginary part of the Green's function directly in the limit η=0 with unprecedented accuracy even very close to the critical point. This approach allows one to validate the critical behavior predicted by the supersymmetric analysis <cit.> with very high accuracy. It also unveils a remarkable feature that has not been reported in the previous literature: The finite jump of the IPR at the transition is followed by a square root singularity, whose existence is also confirmed by the analysis of the effective NLσM formulation of the problem on the BL. It would be interesting to interpret this result in terms of the geometric structure of localized eigenstates on the BL, and understand whether the singular behavior of the IPR is related to the one of the transverse localization length which controls the exponential decay of the wave-functions on typical branches <cit.>. Ultimately, delving into the loop corrections to the BL solution of AL and broadening the analysis of Ref. <cit.> to encompass the insulating phase presents a highly intriguing prospect. Given that a comprehensive understanding of the loop corrections hinges on mastering very precisely the BL solution, the current investigation serves as a pivotal stride forward, laying the foundation for further exploration in this direction. We warmly thank Y. Fyodorov, G. Lemarié and A. D. Mirlin for illuminating discussions. § STABILITY OF THE LINEARIZED EQUATION The linearized cavity equations (<ref>) and (<ref>) must be interpreted as a self-consistent integral equation for the probability distribution P(g,ĝ): P(g,ĝ) = ∫ϵ p(ϵ) ∏_i=1^k [ g_i ĝ_i P(g,ĝ) ] δ(g - 1/ϵ - ∑_i=1^k g_i) δ(ĝ - g^2 ( 1 + ∑_i=1^k ĝ_i ) ) . This equation is more conveniently written performing the Laplace transform with respect to ĝ <cit.> (note ĝ takes only strictly positive values): P̂(g,s)=∫ϵ p(ϵ) ∏_i^k [ P̂(g_i,s g^2) g_i] δ(g - 1/ϵ - ∑_i=1^k g_i) e^- s g^2 . Following Ref. <cit.> we identify the localization transition as the point where the above equation ceases to have a solution. To do so we focus on the region s ≪ 1, corresponding to the tail of the probability of the imaginary part, ĝ≫ 1. At small values of s we assume that: P̂(g,s) ≈ P_0(g) + f (g) s^β , where P_0(g) ≡P̂(g,0) is the probability distribution of the real part of the Green's function, corresponding to the solution of Eq. (<ref>) with η=0. Note that the ansatz (<ref>) corresponds to Eq. (<ref>) of the main text. Plugging the above small-s form into the equation (<ref>) we obtain the following linear equation for the function f (g): f (g)=k ∫ϵ p(ϵ) ∏_i=2^k[ P_0(g_i) g_i ] δ(g - 1/ϵ - ∑_i=1^k g_i) \|g\|^2β f (g_1) g_1 We now introduce the probability distribution of the sum of the real part of k-1 cavity Green's functions: P̃ (g̃) ≡∫∏_i=1^k-1[ P_0 (g_i) g_i ] δ( g̃ - ∑_i=1^k-1 g_i ) . (Note that for k=2 one has that P̃ (x) = P_0 (x) <cit.>.) Inserting this identity into Eq. (<ref>) we obtain: f (g)=k ∫ϵ p(ϵ) g̃P̃ ( g̃ ) δ(g - 1/ϵ - g_1 - g̃) \|g\|^2β f (g_1) g_1 , which coincides with Eqs. (<ref>) and (<ref>) of the main text. The condition that the above homogeneous equation admits a solution fixes the value of β. This is only possible if the largest eigenvalue λ_β of the integral operator is smaller than 1. The critical point is identified by the point where no solution exists. It is easy to check that the probability distribution of the real part of the Green's function P_0(g) is an eigenvector of the integral operator for β=0, corresponding to the largest eigenvalue k (see Fig. <ref>). As explained in the main text, it can be shown that β=1/2 at the critical point. In fact, since the integral operator above is non-symmetric, for each eigenvalue, there will be a right and left eigenvector. After inegrating over the δ-function, using the fact that δ (w(g̃)) = δ(g̃_0)/\|w^'(g̃_0)\|, with g̃_0 = ϵ - g_1 - 1/g and \|w^'(g̃_0)\| = g^2, one gets: λ_β ψ_β (g) = k \|g\|^2(β-1)∫ϵ p(ϵ) P̃( ϵ -g_1 - 1/g) ψ_β (g_1) g_1 , λ_β ϕ_β (g_1) = k ∫ϵ p(ϵ) P̃( ϵ -g_1 - 1/g) \|g\|^2(β-1)ϕ_β (g) g . From the second equation, defining ψ_1-β (g_1) = \|g_1\|^-2 βϕ(1/g_1) and changing variable g → 1/g in the left hand side, one sees that ψ_1-β (g_1) is a right eigenvector of the integral operator (<ref>) for β→ 1 - β with the same eigenvalue λ_β. Hence the spectrum of (<ref>), and in particular its largest eigenvalue, must be symmetric around β=1/2, as schematically illustrated in Fig. <ref>. As mentioned in the main text, the largest eigenvalue of the integral operator (<ref>) for β=1/2 also determines the long distance behavior of the two-point correlation function. The argument goes as follows. Setting η = 0 in Eq. (<ref>), the cavity recursion relation for the imaginary part of the cavity Green's function can be written as: G_i → j = ∑_m ∈∂ i / j G_m → i/( ϵ _i + ∑_m ∈∂ i / j G_m → i)^2 + ( ∑_m ∈∂ i / j G_m → i)^2 = \| G_i → j\|^2 ∑_m ∈∂ i / j G_m → i . Considering a node of the BL labeled as i_n (in absence of one of its neighbors labeled as i_n+1), the cavity recursion equation for G_i_n → i_n+1 can be telescoped in the following way in terms of the imaginary parts of the Green's function on the (k+1) k^n-1 nodes (labeled as i_i) at distance n from i_n <cit.>: G_i_n → i_n+1 = ∑_paths P P:i_n → i_1∏_i_m ∈ P\| G_i_m → i_m+1\|^2 G_i_1 → i_2 , where P are all the (k+1) k^n-1∼ k^n directed paths of length n of the BL originating from the node i_n and ending on the nodes i_1. As explained in the main text, the correlation function of wave-functions' amplitudes on two nodes i and j of the BL is encoded in \|G_i,j\|^2. Thanks to the tree-like structure of the BL, it is easy to show that the off-diagonal elements of the resolvent on two nodes at distance n along a branch of the tree can be expressed in terms of the product of the diagonal elements of the (cavity) Green's functions along the branch: G_i_1,i_n= G_i_1 → i_2 G_i_2 → i_3 ⋯ G_i_n-1→ i_n G_i_n,i_n . Hence, the products of \| G_i_m → i_m+1\|^2 appearing in Eq. (<ref>) is proportional to the two-point correlation function between nodes i_1 and i_n. One thus obtains that the typical value of the imaginary part of the cavity Green's function on node i_n G_i_n → i_n+1^ typ∝ k^n C(n) G_i_1 → i_2^ typ In the localized phase, W>W_c, for η=0 the typical value of the imaginary part of the Green's function decreases under iteration. § ANALYSIS OF THE CRITICAL BEHAVIOR OF THE TWO-POINTS CORRELATION FUNCTION As explained in the main text, the off-diagonal elements of the resolvent on two nodes at distance L along a branch of the BL can be expressed in terms of the diagonal elements of the (cavity) Green's functions as in Eq. (<ref>). As a consequence, the generic moment of the correlation \|G_i,i+L\|^p can be written in terms of the following linear operator K_p (G,G_1) ≡∫ϵ p(ϵ) δ( G - 1/ϵ - iη - G_1 - G̃) \|G\|^p P̃ (G̃) G̃ , where P̃ (G̃) is the distribution of the sum of k-1 cavity Green's functions P̃ (G̃) ≡∫∏_i=1^k-1[ P(G_i) G_i ] δ(G̃ - ∑_i=1^k-1 G_i ) . The expression above reduces to Eq. (<ref>) for p=2. With this definition the two-point correlation function can be written as: \| G_i,i+L\|^p = ∫ P(G_2) G_2 ϵ p(ϵ) G̃P̃ (G̃) 1/\|ϵ - iη - G - G_2 - G̃\|^p K_p^L-1 (G,G_1) P(G_1) G G_1 , where K_p^L(G,G_1) is the L-th power of the operator K_p. Similarly to the integral operator associated to the linear stability of the cavity equation with respect to the imaginary part discussed in the previous section, we note that the operator (<ref>) is not symmetric, therefore for each eigenvalue there will be a left and right eigenvector, respectively ϕ_λ(G) and ψ_λ (G). On the other hand the symmetry of the problem implies that if in the expression for \| G_i,i+L\|^p we replace P(G_1) with a generic positive A(G_1) and P(G) with a generic positive B(G), the result must be symmetric with respect to the exchange A ↔ B. This implies that we can express the left eigenvector as a function of the right one: ϕ_λ(G) ∝∫ϵ p(ϵ) G̃P̃ (G̃) ψ_λ (G_1) /\|ϵ- iη -G - G_1 - G̃ \|^p G_1 . Similarly we have the following orthonormality relationships ∫ϵ p(ϵ) G̃P̃ (G̃) ψ_λ (G_1) ψ_λ' (G)/\|ϵ- iη - G - G_1 - G̃ \|^p G G_1 ∝δ_λλ' . From which we can define: ϕ_λ(G) = A_λ^-1∫ϵ p(ϵ) G̃P̃ (G̃) ψ_λ (G_1) /\|ϵ- iη -G - G_1 - G̃ \|^p G_1 . with A_λ≡∫ϵ p(ϵ) G̃P̃ (G̃) ψ_λ (G_1) ψ_λ (G)/\|ϵ- iη - G - G_1 - G̃ \|^p G G_1 . It follows that we have: \| G_i,i+L\|^p = ∫_λλ^L-1 ( ∫ϵ p(ϵ) G̃P̃ (G̃) P (G_1) ψ_λ (G)/\|ϵ- iη - G - G_1 - G̃ \|^p G G_1 )^2 /( ∫ϵ p(ϵ) G̃P̃ (G̃) ψ_λ (G_1) ψ_λ (G)/\|ϵ- iη - G - G_1 - G̃ \|^p G G_1 ) Both integrals above are of the form ∫ f(M)1/\|M\|^p dM ≈η^1-p√(π)Γ((p-1)/2)/Γ(p/2)∫ f(0,m̂)m̂^1-p d m̂ so that: \| G_i,i+L\|^p = η^1-p√(π)Γ((p-1)/2)/Γ(p/2)∫_λλ^L-1 ( ∫ f_1 (0,m̂)m̂^1-p d m̂)^2 /( ∫ f_2(0,m̂)m̂^1-p d m̂) As usual, in the localized phase we will be interested in considering the small η limit in which the cavity equations can be linearized with respect the imaginary part. Performing the Laplace transform with respect to the imaginary part we obtain the following expression <cit.>: K_p (g,s\|g_1,s_1) ≡∫ϵ p(ϵ) g̃P̃ (g̃ , s_1) δ(g - 1/ϵ - g_1 - g̃) δ(s_1-s g^2 ) \| g \|^p e^-s_1 . We note that the function e^-s_1P̃ (g̃,s_1) tends to a constant P̃ ( g̃ ,0) for s_1 going to zero and tends to zero for s_1 going to infinity. As a consequence for s_1 going to zero the eigenvectors of the operator take the form g_λ(g)s^α. To proceed in a systematic way we perform the change of variable τ=ln s, by writing [USE t INSTEAD OF τ] h(s) =∫ B(s,s') g(s') s' , h(τ) =∫ B(τ,τ') s'/τ' g(τ') τ' , h(τ) ( s/τ)^1/2 =∫ ( s/τ)^1/2 B(τ,τ') ( s'/τ')^1/2( ( s'/τ')^1/2 g(τ') ) τ' . From Eqs. (<ref>) and (<ref>) we thus obtain that the eigenvector of the original problem can be written in terms of the eigenvector of the new operator B_p(g,τ \| g_1, τ_1) ≡ e^τ/2 K_p(g,τ\|g_1,τ_1)e^τ_1/2 = ∫ϵ p(ϵ) g̃P̃ (g̃ , e^τ_1) δ(g - 1/ϵ - g_1 - g̃) δ(τ_1-τ-ln(g^2) ) \|g \|^p-1 exp( -e^τ_1) . In the limit τ_1 → -∞ (, s → 0) we have that e^-e^τ_1P̃ (g̃,e^τ_1) →P̃ (g̃,0) and the operator B_p becomes invariant under translation of τ and τ_1. Therefore at large τ the operator is diagonal in momentum space: ∫ B_p (g,τ\|g_1,τ_1) f(g_1) e^ iμτ_1 g_1 τ_1 ≈ e^ iμτ∫ϵ p(ϵ) g̃P̃ (g̃, 0) δ(g - 1/ϵ - g_1 - g̃) \| g \|^p-1+ 2 iμ f(g_1) g_1 , where the function f(g) must thus be a solution of the eigenvalue equation: λ_p,μ f(g) = ∫ϵ p(ϵ) g̃P̃ (g̃, 0) δ(g - 1/ϵ - g_1 - g̃) \| g \|^p-1+ 2 iμ f(g_1) g_1 . For p=2 we recognize the operator (divided by a factor k) controlling the critical point for β=1/2, Eqs. (<ref>), (<ref>), and (<ref>), studied in Ref. <cit.>. Thus, according to Eq. (<ref>), the largest eigenvalue of this operator behaves as 1/k - c_1/k (W-W_c) close to W_c, with correction of order μ^2. For small values of μ it is convenient to study the eigenvector in two different region. For large positive or negative τ of order μ^-1 we have: τ = x/μ , exp( -e^τ_1 ) P̃ (g̃, e^τ_1) →P̃ (g̃, 0) θ(-x) , CHECK B(g,x\|g_1,x_1) →∫ϵ p(ϵ) g̃P̃ (g̃, 0) δ(g - 1/ϵ - g_1 - g̃) \| g \|^p-1 δ(x_1-x) θ(-x) . CHECK Since the variables x and g are decoupled, the eigenvector takes the form: ψ_λ (g,x/μ) → f(g) sin (x) θ(-x) , λ_μ= λ_0+ c_2 μ^2 , with some constant that is given by (write the formula). Note that the fact that the cos x term is absent follows from the continuity of the solution in zero. In the region of finite τ where exp(-e^τ) P̃(g̃,e^τ) is different from a step function, the solution must match the large τ behavior sinμτ≈μτ and the eigenvector is given by ψ_λ(g,τ) = μ ψ(g,τ) where ψ(g,τ) is the solution of the equation λ_0 ψ(g,τ) = ∫ϵ p(ϵ) g̃P̃ (g̃ , e^τ_1) δ(g - 1/ϵ - g_1 - g̃) δ(τ-τ_1 -2ln(g) ) \| g \|^p-1exp( -e^τ_1) ψ(g_1,τ_1) g_1 τ_1 , with the condition that: ψ(g,τ) → f (g) τ for τ→ -∞ . § ALGORITHM TO EVALUATE EQ. (<REF>) The algorithm implemented to evaluate Eq. (<ref>) and compute Q(0,m̂) is schematically summarized as follows: xx x̄x x̄x x̄x x̄x x̄x x̄x x̄xxxx Algorithm computing I_p and Q(0,m̂) begin Initialize population of Ω elements (g_α, ĝ_α)_α = 1, …, Ω Iterate population using Eqs. (<ref>) and (<ref>) until convergence to a stationary distribution begin I_p = 0; Q(0,m̂)=0 for r=1 to N_ avg do a=0; b=0 for i=1 to N_ est do Sample k+1 elements from the pool (g_α_j, ĝ_α_j), j=1,…,k+1 Extract a random energy ϵ from the box distribution of width W Compute S = ∑_j=1^k+1 g_α_j if \|S\| < W/2 then Compute m̂ from Eq. (<ref>): m̂ = 1 + ∑_j=1^k+1ĝ_α_j a = a+m̂^1-p/W b=b+1/W Add m̂ to Q(0,m̂) end if end for a=a/N_ est b=b/N_ est I_p= I_p+ a/b Renew all the elements of the population end for I_p= I_p/N_ avg Normalize Q(0,m̂) end end § EXACT DIAGONALIZATIONS OF THE ANDERSON MODEL ON RRGS OF N NODES We preform exact diagonalizations of the Anderson tight-binding model (<ref>) on finite Bethe lattices of fixed connectivity k+1=3. Finite BLs are in fact random-regular graphs of N nodes, a class of random lattices that have locally a tree-like structure but do not have boundaries. More precisely, a (k+1)-RRG is a lattice chosen uniformly at random among all possible graphs of N vertices where each of the sites has fixed degree k+1. The properties of such random graphs have been extensively studied (see Ref. <cit.> for a review). A RRG can be essentially thought as a finite portion of a tree wrapped onto itself. It is known in particular that for large number of vertices any finite portion of a RRG is a tree with a probability going to one as N →∞: RRGs have loops of all size but short loops are rare and their typical length is of order ln N/ln k <cit.>. Thanks to the sparse nature of the graph, exact diagonalizations can be efficiently performed using the Arnoldi method, which provides a few eigenvalues and eigenvectors around E=0. In practice we consider the 64 nearest eigenstates to zero energy. When comparing the results obtained from exact diagonalizations with the analytic predictions obtained at E=0, a suitable approach is taken to minimize corrections arising from the small deviation of eigenvalues from precisely zero energy. This is achieved through the following procedure: We start by noticing that the eigenvectors of H are also eigenvectors of H + γ𝕀 with all eigenvalues shifted by γ, E_α→ E_α + γ. Thus an eigenvector of H of energy E_α is an eigenvector of zero energy of an Anderson model (<ref>) with all random energies shifted by -E_α (note that since we consider only a finite number of eigenvectors, the E_α's are of order 1/N, and thus only a few random energies ϵ_i - E_α will fall outside the box of width W after the shift). Since H = ∑_i ϵ_i converges to a normal distribution with zero mean and variance N W^2/12, shifting the trace of H by N E_α must be reweighted by a factor e^- 6 N E_α^2/W^2. As a result, to obtain the averages at zero energy of a generic observable which depends on the wave-functions' amplitudes, we use the following expression: O({ψ_α (i) } ) = ∑_α e^- 6 N E_α^2/W^2 O({ψ_α (i) }) /∑_α e^- 6 N E_α^2/W^2 . § COMPUTATION OF THE TWO-POINT FUNCTION In order to evaluate the moments of the correlation function using Eq. (<ref>) we start by defining M_i,i+L (defined in Eq. (<ref>)) as the product of two random variables defined as follows: M_i,i+L = H_i,i+L M_i+L,i+L , H_i,i+L = 1/G_i → i+1 G_i+1 → i+2⋯ G_i+L-1 → i+L= h_i,i+L + i η ĥ_i,i+L , M_i+L,i+L = ϵ_i+L - iη - ∑_m ∈∂ i+L G_m → i+L = m_i+L,i+L - i η m̂_i+L,i+L . Note that H_i,i+L and M_i+L,i+L are correlated since one of the neighbors of i+L is the node i+L-1 and the Green's function G_i+L-1 → i+L enters in the sum on the right hand side of the last equation. In the η→ 0 limit we have that: m_i,i+L = h_i,i+L m_i+L,i+L , m̂_i,i+L = ĥ_i,i+L m_i+L,i+L - h_i,i+Lm̂_i+L,i+L . Since m_i+L,i+L = ϵ_i+L - ∑_m ∈∂ i+L g_m → i+L, one has that the probability that m_i,i+L=0 is equal to the probability that ϵ_i = ∑_m ∈∂ i+L g_m → i+L. This occurs with probability density 1/W if \|∑_m ∈∂ i+L g_m → i+L\|<W/2, and with zero probability otherwise. Based on this observation, we implement the following algorithm to compute the correlation function η^p-1 \|G_i,i+L\|^p between two nodes at distance L: We apply the standard population dynamics method to obtain a stationary probability distribution of the cavity Green's function P(g,ĝ) in the linearized regime, corresponding to the solution of Eqs. (<ref>) and (<ref>); We compute H_i,i+L on a chain of length L; We compute S = ∑_m ∈∂ i+L g_m → i+L on the last node of the chain; If and only if \|S\|<W/2 we add \| h_i,i+Lm̂_i+L,i+L\|^1-p/W to the value of the correlation; We repeat this procedure N_ est times and divide the result by the total number of attempts. We renew the pool of the cavity Green's function by performing a few steps of the population dynamics algorithm and repeat the whole process N_ avg times until the desired accuracy on I_p is reached. The algorithm is schematically summarized below: xx x̄x x̄x x̄x x̄x x̄x x̄x x̄xxxx Algorithm computing η^p-1 \|G_i,i+L\|^p begin Initialize population of Ω elements (g_α, ĝ_α)_α = 1, …, Ω Iterate population using Eqs. (<ref>) and (<ref>) until convergence to a stationary distribution begin η^p-1 \|G(L)\|^p = 0 for r=1 to N_ avg do a=0 for i=1 to N_ est do Sample one element from the pool G_1 = (g, ĝ) Initialize H = h + η iĥ = 1/G_1 for m=1 to L-1 do Sample k-1 elements from the pool G_j= (g_α_j, ĝ_α_j), j=1,…,k-1 Sample a random energy ϵ_m from the box distribution Compute the cavity Green's function G_m = 1/(ϵ_m - G_1 - ∑_j G_j) Update H = h + η iĥ = H/G_m end for Sample k+1 elements from the pool G_j = (g_α_j, ĝ_α_j), j=1,…,k+1 Extract a random energy ϵ from the box distribution of width W Compute S = ∑_j=1^k+1 g_α_j if \|S\| < W/2 then Compute m̂ = 1 + ∑_j=1^k+1ĝ_α_j a = a+ \| h m̂\|^1-p/W end if end for a=a/N_ est η^p-1 \|G(L)\|^p = η^p-1 \|G(L)\|^p + a Renew all the elements of the population end for η^p-1 \|G(L)\|^p = η^p-1 \|G(L)\|^p/ N_ avg end end
http://arxiv.org/abs/2406.18309v1	20240626125007	Automated Immunophenotyping Assessment for Diagnosing Childhood Acute Leukemia using Set-Transformers	[ "Elpiniki Maria Lygizou", "Michael Reiter", "Margarita Maurer-Granofszky", "Michael Dworzak", "Radu Grosu" ]	cs.LG	[ "cs.LG", "q-bio.QM" ]	Automated Immunophenotyping Assessment for Diagnosing Childhood Acute Leukemia using Set-Transformers This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 101034277. Elpiniki Maria Lygizou1, Michael Reiter1, Margarita Maurer-Granofszky2, Michael Dworzak2, Radu Grosu1 1TU Wien elpiniki.lygizou@tuwien.ac.at, rei@cvl.tuwien.ac.at, radu.grosu@tuwien.ac.at 2St. Anna Children's Cancer Research Institute margarita.maurer@ccri.at, michael.dworzak@ccri.at Received 7 March 2024 / Accepted 23 May 2024 =========================================================================================================================================================================================================================================================================================================================================== § ABSTRACT Acute Leukemia is the most common hematologic malignancy in children and adolescents. A key methodology in the diagnostic evaluation of this malignancy is immunophenotyping based on Multiparameter Flow Cytometry (FCM). However, this approach is manual, and thus time-consuming and subjective. To alleviate this situation, we propose in this paper the FCM-Former, a machine learning, self-attention based FCM-diagnostic tool, automating the immunophenotyping assessment in Childhood Acute Leukemia. The FCM-Former is trained in a supervised manner, by directly using flow cytometric data. Our FCM-Former achieves an accuracy of 96.5% assigning lineage to each sample among 960 cases of either acute B-cell, T-cell lymphoblastic, and acute myeloid leukemia (B-ALL, T-ALL, AML). To the best of our knowledge, the FCM-Former is the first work that automates the immunophenotyping assessment with FCM data in diagnosing pediatric Acute Leukemia. immunophenotyping, multiparameter flow cytometry, set-transformers, self-attention § INTRODUCTION Acute Leukemias are a heterogeneous group of hematologic malignancies (cancers), which progress rapidly. Hence, their prompt detection is crucial for a successful treatment. These diseases are primarily categorized based on the lineage of the affected cells, referring to the type of precursor cells (early-form cells), that begin to multiply at an accelerated rate. Immunophenotyping is an essential part of the precise diagnosis and classification of acute leukemia <cit.>, although it is manual, time-consuming, and subjective, by relying on the experience and knowledge of domain experts. A reliable tool for immunophenotyping is flow cytometry, a laser-based biophysical technique which provides a quick and comprehensive multi-parameter analysis of individual cells or particles. In order to automate the immunophenotyping assessment in the diagnosis of Childhood Acute Leukemia, we introduce in this paper the FCM-Former, a machine learning and self-attention-based classification algorithm for FCM data. The FCM-Former is based on the Set-Transformer architecture of <cit.>, and it is designed to work directly with the FCM data obtained from CCRI in Vienna, without any additional pre-processing. This direct approach, also allows the FCM-Former to take advantage of the high dimensionality of the FCM data. The main goal of the FCM-Former, is to accurately classify the malignancy into one of three lineages: B-ALL, T-ALL and AML. To the best of our knowledge, the FCM-Former is the first work that automates the immunophenotyping assessment with FCM data in diagnosing pediatric Acute Leukemia. Our work thus represents an important step towards applying advanced machine learning techniques, to critical healthcare challenges. In particular, in the accurate and timely diagnosis of pediatric acute leukemia. The rest of this paper is structured as follows. In Section 2 we review foundational concepts for this work. In Section 3, we discuss the related work and previous approaches. In Section 4, we describe the experimental methodology, leading to Section 5, where we present our experimental results. Finally, we conclude in Section 6, by summarizing our key insights, and proposing directions for future work. § BACKGROUND §.§ Multiparameter Flow Cytometry Multiparameter flow cytometry (FCM) serves as a robust and powerful tool for both analytical and preparative applications <cit.>,<cit.>. In FCM, a blood or bone-marrow sample of patients is stained with a specific combination of fluorochrome-labelled antibodies (markers) uniquely binding to antigens, intracellular or on cell's surface. The resulting data are a set of measurements (feature vectors) of the physical (size, granularity), and biological (multiple surface/intracellular markers) properties of every single cell (an event). This set (sample) thus characterizes the phenotype of a hole cell population. §.§ Transformers In this section, we delve into the core principles of the Transformer <cit.> and Set-Transformer <cit.> models, focusing on the self-attention mechanisms behind them. Transformers. These are advanced deep learning models, primarily developed for natural language processing (NLP). Their unique architecture, is characterized by a self-attention mechanism, allowing them to focus on complex relationships within data, and capture meaningful patterns in large-scale data. This leads to an effective context understanding. The multi-headed self-attention mechanism, as introduced in <cit.>, is defined for a given set of n query vectors Q (n corresponds to the set's size consists of n elements) each with dimension d_q, Q ∈ ℝ^n_q × d_q, key matrices K ∈ ℝ^n_v × d_q and value matrices V ∈ ℝ^n_v × d_v, where d_q = d_v = d for the sake of simplicity. It can be described as a function according to the formula below: attn(Q,K,V) := softmax(QK^T/√(d))V (1) If Q and K are derived from the same set of inputs (as in self-attention), the QK^T multiplication is quadratic in the set size, which prevents the direct application to FCM data. Set-transformers (ST). This derivative of the original transformer architecture, is designed to operate on set-structured data, where ordering is irrelevant to the input information. The associated model adapts the transformer's architecture to handle input data, that lacks a clear sequential or grid-like structure, similar to how the FCM data are in our application. The building block of Set Transformers reduces the O(n^2) complexity of self-attention to O(nm) by incorporating inducing points into the standard multi-head self-attention block of Formula (1), where n is the input dimension and m is the number of learnable parameters (inducing points). The process begins by projecting Q, K, V onto h different d_h^q, d_h^q, d_h^v,-dimensional vectors, where d_h^q = d_h^v = d/h, such that: Multihead(Q, K, V ) := concat(O_1,..., O_h)W^O (2) where O_j = attn(QW_j^Q, KW_j^K, VW_j^V) (3) where W_j^Q, W_j^K, W_j^V are projection operators of dimensions ℝ^d_q × d_h^q, ℝ^d_q × d_h^q and ℝ^d_v × d_h^v, respectively, and W^O is a linear operator of dimension d × d relating O_1,… O_h to each other. Furthermore, given a set S of d-dimension vectors, we initialize m d-dimensional inducing points I ∈ ℝ^m × d. Then, the Multihead Set-Transformer Attention Block (MSAB) is computed by the following formulas: MSAB(I,S) := LayerNorm(X + rFF(X)) (4) where X = LayerNorm(I + Multihead(I, S, S)) (5) where rFF denotes a row-wise feedforward layer, and LayerNorm is layer normalization as described in <cit.>. Finally, the Set-Transformer Attention Block (STAB) is defined as follows: STAB(S) := MSAB(S,MSAB(I,S)) (6) § RELATED WORK Manual analysis of FCM data, which plays a crucial role in various medical and biological fields, typically involves representing and transforming the high-dimensional space of raw data, into 2-D plots for human interpretation. This technique, while making the data more comprehensible, can lead to a loss of information. However, machine learning methods, can utilize the full data space, and tackle this shortcoming. Automated FCM data analysis, primarily focuses on identifying and classifying distinct or specific cell populations. Initial methods in this domain pooled events from different samples, employing classifiers based on single-event pairs and labels <cit.>,<cit.>,<cit.>, but were limited to fixed decision regions. This approach was less effective in discerning relational positioning among cell populations, a key factor in detecting rare or abnormal cells, particularly in Minimal Residual Disease (MRD) detection <cit.>. Subsequent developments in FCM analysis therefore shifted towards processing a hole sample in a unified manner. Techniques such as Gaussian Mixture Models (GMM) <cit.>, and Convolutional Neural Networks (CNNs) <cit.> emerged, which were applied to multiple 2-D projections of the data space. These methods addressed some limitations of earlier approaches, particularly in maintaining relational context among cell populations. They are less suited for tabular data analysis, which characterizes FCM data. In the realm of automated immunophenotyping statistical methods have been proposed employing distance-based analysis in the space of principal components calculated on a database of FCM reference samples <cit.>. While these methods rely on strict standardization in the data acquisition process (flow cytometer settings, FCM panels, etc.), our claim is to process data in diverse conditions without the need for such rigid standardization by using machine learning models being able to identify and relate structures in the data space and thus deal to a larger degree with data distortions. More recently, attention-based models have gained prominence in automated FCM analysis <cit.>,<cit.>,<cit.>,<cit.>. These models, using attention mechanisms, emulate the human logic of manual FCM data analysis, but keep and leverage the high-dimension data space information of the FCM data. They enable event-level classification by learning the importance of various cell populations within a sample. They are well known for their SOTA performance in tasks such as automated MRD detection <cit.>, and recently, in adults acute leukemia diagnosis <cit.>. However, these methods often assume a fixed set of features during training and inference, which can be a limitation given the high variability of FCM data features even within a single dataset. Some recent studies have explored combining features from different samples using techniques like nearest neighbor imputation <cit.>,<cit.>, but the efficacy of these approaches is still under scrutiny due to potential inaccuracies in imputed values affecting downstream analysis <cit.>. A late work <cit.> attempts to address these issues by employing a feature-agnostic, attention-based method with promising results. § EXPERIMENTAL SETUP §.§ Data Immunophenotyping for diagnosing childhood acute leukemia typically engaged the use of multiple tubes per sample, each with a different combination of markers. FCM data are presented in matrix format, where each sample comprises multiple diagnostic tubes. Each tube holds thousands of events, corresponding to feature vectors of individual cells. Our model utilizes a fixed number of features, and consists of 18 markers and 4 physical properties, as measured by the forward and side scatter of the laser light of the flow cytometer, thereby standardizing the input. Our training fixed-feature list is as follows: FSC-A, FSC-W, FSC-H, SSC-A, CD45, CD71, CD34, CD19, (i)CD79A, (i)CD3, (i)CD22, CD10, CD5, CD7, CD13, CD117, CD33, SY41, LZ, (i)MPO, CD64 and CD65. FCM-Former thus involves the aggregation of the features present across three datasets, by following the guidelines presented in <cit.>, <cit.>, with missing values imputed as zeros. In our work, we ensured that our model training was not biased, by the presence or absence of markers related with lineage-specific markers, which are typically either used or excluded by experts, following the analysis' conclusion of the initial tubes. We represent a single sample by a matrix E ∈ℝ^N × m, where N denotes the number of cells (events) in the sample, and m denotes the number of features per cell (which was 22 in our case, as listed above). N is equal to t × K, where t is the number of tubes (typically 8-13) and K the number of cells in every tube (typically 10^4 - 10^5, the exact value varies for every tube and every sample). For every index n ∈1,...,N, E_n ∈ℝ^m is a quantitative representation of physical and biological properties of every cell. §.§ Datasets We evaluate FCM-Former on samples of blood or bone marrow of pediatric patients with B-ALL, T-ALL or AML. The data set consisting of 960 samples was collected at CCRI from 2011 to 2022, with a BD LSR II flow cytometer or BD FACSSymphony A3 and FACSDiva Software (all Becton Dickinson, San Jose, CA). The samples were stained using a multi-color approach, based on a CD45-Backbone. Markers against lymphoid lineages in each tube allowed defining potential control cells. Immunphenotyping was essentially performed as proposed in <cit.>. Sampling and research were approved by local Ethics Committees, and informed consent was obtained from patients, their parents, or legal guardians, according to the Declaration of Helsinki. For all samples ground truth information was acquired by manual immunophenotyping assessment, conducted by CCRI experts. §.§ Model An overview of the FCM-Former architecture is depicted in Figure 1. Our model incorporates an encoder coupled with a linear classification layer. We use an ST encoder as presented in Section 2. Inspired by Vision-Transformers (ViT) <cit.>, our model is augmented by an additional class token, a learnable feature vector, into the encoder's input. At the output of the ST encoder, the trained class token is retrieved and then fed into a linear classification layer. We treat our problem as a single-label classification, and use a cross-entropy loss for supervised training. FCM-Former processes a single sample of FCM data in a single forward pass. Unlike typical transformer-based approaches that incorporate an embedding step, our model is applied directly to FCM samples, specifically bypassing any form of positional embedding. We set the number of induced points to m=16, hidden dimension d=32 ,and the number of attention heads to 4, for all three layers. We train our model for 200 epochs and use an early stop after 50 epochs if there is no improvement of the accuracy on the validation set. Throughout all the experiments, we use the cosine-annealing learning rate scheduler with an initial learning rate of 0.001, lowering to a minimum of 0.0002 over 10 iterations for fine-tuning purposes. The Adam optimizer is applied across these experiments while batch processing is not part of our experimental setup. All training processes are executed using an NVIDIA GeForce RTX 3090. The resulting model is comparatively lightweight with 31,572 parameters. The accuracy and the ROC-AUC are used as evaluation metrics. § RESULTS Here we present the results of the conducted experiments, evaluated on accuracy and roc-auc metrics. To ensure the robustness and generalizability of our results, we implemented a 5-fold cross-validation technique. For all experiments, the data are divided into 660 training samples, 100 validation samples, and 200 test samples. The model demonstrates exceptional proficiency in identifying the lineage of Childhood Acute Leukemia, achieving a peak accuracy of 0.965 and peak roc-auc value 0.9708 on the test datasets. The average accuracy of the model on test datasets across all folds is 0.9408 ± 0.0217 and the average roc-auc respectively is 0.9638 ± 0.0063. We additionally experimented with implementing a cross-attention mechanism in our model and trained it accordingly, using as a query Q the learnable vector of the class token, and K, V the linear projections of the input set, as in self-attention. However, the outcomes of this cross-attention mechanism under-performed compared to self-attention, indicating that cross-attention constrains the model's ability to effectively attend to the most relevant parts and relationships within the entire input dataspace, rather than enhancing it. Furthermore, our model is adaptable to variability across different clinical centers and devices. It facilitates straightforward retraining on new FCM data, with diverse features, highlighting its scalability and potential for integration into clinical routine. We identified major causes for misclassification. Cross-lineage marker expression contributed significantly to errors and was the most common cause. Some misclassifications revealed inherent biological complexity, as seen in cases of mixed phenotype acute leukemias (MPAL). Additionally, cases with minimal blast percentages (less than 5%) underscored the impact of low cellularity on accurate classification. Poor sample quality and the resulting compromised data quality may also pose challenges to precise classification. These insights highlight the significance of detailed marker analysis and acknowledging biological heterogeneity in improving machine learning models for the classification of leukemia. § CONCLUSIONS We proposed FCM-Former, a new and automated method for immunophenotyping to diagnose childhood acute leukemia. We trained FCM-Former in a supervised manner and showed that is capable of generalizing to new, unseen data. To the best of our knowledge, FCM-Former is the first attempt to automate the diagnosis of pediatric acute leukemia using FCM data. FCM-Former employs self-attention mechanisms, enabling it to attend to all cells in the sample at once, taking advantage of the whole high-dimension data-space, and avoiding the information loss encountered in the traditional process of manual immunophenotyping assessment. 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http://arxiv.org/abs/2406.17724v1	20240625171335	Spatiotemporal statistical features of velocity responses to traffic congestions in a local motorway network	[ "Shanshan Wang", "Michael Schreckenberg", "Thomas Guhr" ]	physics.soc-ph	[ "physics.soc-ph", "physics.data-an" ]	Re-examination of the role of displacement and photon catalysis operation in continuous variable measurement device-independent quantum key distribution Arvind July 1, 2024 ======================================================================================================================================================== Abstract. The causal connection between congestions and velocity changes at different locations induces various statistical features, which we identify and measure in detail. We carry out an empirical analysis of large-scale traffic data on a local motorway network around the Breitscheid intersection in the North Rhine-Westphalia, Germany. We put forward a response function which measures the velocity change at a certain location versus time conditioned on a congestion at another location. We use a novel definition of the corresponding congestion indicator to ensure causality. We find that the response of velocities to the congestion exhibits phase changes in time. A negative response at smaller time lags transforms into positive one at larger time lags, implying a certain traffic mechanism. The response decays as a power law with the distance. We also identify a scaling property leading to a collapse of the response functions on one curve. Keywords: response function, traffic congestion, power law, scale invariance 1pt 1pt § INTRODUCTION The traffic flow on road networks <cit.> consists of free flow and congested flow. According to the three-phases traffic theory <cit.>, the congested flow further contains synchronized flow and wide moving jams. Extensive studies on the dynamic behavior of traffic flow has been devoted to modeling and simulations in past decades <cit.>. At present, neither models nor simulations fully capture the realistic traffic situations, which may be affected by commuting, weather, seasons, road construction, traffic accidents, big city events, etc. A huge amount of traffic data collected by the global positioning system tracking devices, inductive loop detectors, video recording devices, etc. <cit.> is available, making the empirical studies possible <cit.>. Along with theoretical studies, data-driven analyses to explore traffic flow dynamics <cit.>, traffic patterns <cit.>, traffic congestion <cit.>, traffic flow prediction <cit.> and the resilience after traffic jams <cit.> are called for and pose a variety of challenges. Due to the non-stationary in the time series of traffic observables, including traffic flows and velocities, a traffic network can be viewed as a complex system, where traffic observables are correlated in various ways in time and space. Temporal correlation matrices together with the technique of k-means clustering have been used for identifying different quasi-stationary states in traffic systems <cit.>. The states, manifesting themselves in correlation structures, carry certain traffic patterns, related to non-stationary. In contrast to a financial market <cit.>, the presence of spatial information <cit.> renders a traffic network more complicated. The correlations between time series measured at arbitrarily labeled locations or positions induce a topology which has to be mapped on the real topology, i.e. of the road map. This has led to the identification of collective and sub-collective traffic behavior in the motorway network of North Rhine-Westphalia <cit.>. The propagation of effects among road sections via a traffic network takes time, inducing temporal shift of correlation structures. Non-synchronized time series from different road sections bring about cross-correlations with a time lag or lead. A recent study <cit.> discloses a spectral transition in the symmetrized matrix of time-lagged correlations. Importantly, the spectral transition is associated with the duration of traffic congestion. In addition to correlations, response functions as a novel concept are introduced to explore the interaction of road sections with non-synchronized time series <cit.>. They measure the time-dependent response of some observable, conditioned on events which are encoded in indicator functions. The response function has been used in financial markets to study how the trading price changes conditioned on a buy or a sell <cit.>. It consists of a response variable and a triggered event. In traffic, the latter could be traffic congestion <cit.>, traffic accidents <cit.>, road construction <cit.>, the presence of trucks <cit.>, etc. Our previous study on the response of velocities to heavy congestion <cit.> was conducted with five neighbouring sections on a motorway. The response function measures the average velocity changes versus time on a motorway section conditioned on the heavy congestion occurring on a different section at an earlier time. Despite the fact that a remarkable response with phase transitions shows up, it is difficult to build a causal relation between the velocity change and the heavy congestion, as congestion may occur simultaneously on other sections in addition to the section as the trigger. Furthermore, the degree of velocity changes on a section depends largely on its specific traffic environment, for instance, a section with bottleneck, a must-pass section for commuting, or a section on a bridge. In our previous study <cit.>, we ignore the effect of traffic environment on the responses, as the considered sections are rather close to each other. When studying the responses among sections distributed on a motorway network in two dimensions, the environments for the sections may vary largely over the network, which has to be taken into account. We extend the study of the considered motorway network from one to two dimensions. First, we introduce a conditional indicator, which rules out the possibility of synchronized congestion on multiple sections and guarantees the response only caused by the section as a trigger. Second, we employ an alternative definition of response functions, which removes the effect of random noise on velocity changes. Third, by considering many sections distributed on a two-dimensional motorway network, we are able to explore the spatial features of responses, in addition to the temporal features. This paper is organized as follows. In Sec. <ref>, we provide some basic information and concepts for this study, including the considered local motorway network, the used traffic data, the method to aggregate velocities across multiple lanes, and the network distances. In Sec. <ref>, we introduce the response functions for this study as well as the indicator functions that ensure causality. In Sec. <ref>, we analyze our empirical results and find phase changes in time as well as power laws and scaling invariance in space. We conclude in Sec. <ref>. § DATA DESCRIPTION In Sec. <ref>, we introduce a local motorway network considered in this study and the information of traffic data. In Sec. <ref>, we describe the method of averaging velocities across multiple lanes on a motorway section. In Sec. <ref>, we brief the network distance and its computation method. §.§ The studied motorway network and traffic data In this study, we focus on a local motorway network near Breitscheid in North Rhine-Westphalia (NRW), Germany, which is part of the large-scale NRW motorway network. The considered local network is mainly composed of motorway A52 connecting the densely populated cities of Düsseldorf and Essen, motorway A3 connecting the cities of Duisburg and Mettmann, motorway A40 connecting the cities of Duisburg and Essen, and motorway A524 connecting the city of Duisburg with other motorways, as displayed in Fig. <ref>a. The intersection of motorways A3 and A52 is at Breitscheid and carries heavy traffic flow of commuters during rush hours on workdays. We select a section on motorway A52 as close to this intersection as possible to study the effect of its congestion on other sections nearby. As seen in Fig. <ref>, this section toward the north-east (NE) is our section j locating in the center of the network and playing the role of congestion. The other sections either toward north-east or toward south-west (SW) are the sections i response to it. Within the network distance of 15 km from section j, we have 68 sections i in total. Our traffic data is accumulated with inductive loop detectors on the motorway network. It includes the information on traffic flow and on velocity with a resolution of one minute for each lane on each motorway section. The data used in this study comprises 179 workdays selected during the period from Dec. 1, 2016 to Nov. 30, 2017. On each considered workday, our central section j contains the velocity of at least one minute lower than or equal to 10 km/h from 5:00 to 22:59, which guarantees the presence of heavy congestion of section j during this period. Moreover, the used data for each section from 5:00 to 22:59 on the 179 workdays has high quality with more than 96.7% non-missing values. We fill the missing values in the data with the linear interpolation of neighboring, non-missing values <cit.>. §.§ Velocities on individual sections One motorway section has one or more lanes, leading to one or multiple velocities per minute. The later case requires aggregation of the velocity across multiple lanes, such that for one section there is one velocity per minute. For the velocity aggregation, we use the flow-weighted velocity here, which is different from the density-weighted velocity that we used in our previous studies <cit.>. The traffic flow is the number of vehicles passing through a road section per unit time, while the density is the number of vehicles passing through per unit distance. The former is a time-dependent observable obtained from data directly, while the latter is a space-dependent quantity that has to be worked out via flow and velocity. In contrast to density-weighted velocity, the flow-weighted velocity better reflects the velocity per individual vehicle. Let the traffic flow and velocity at time t on lane m of section i be denoted q_i,m(t) and v_i,m(t), respectively. We define the flow-weighted velocity v_i(t) for section i across multiple lanes as the sum of the velocity times the flow on each lane divided by the total flow on this section, v_i(t)=∑_m q_i,m (t)v_i,m(t)/∑_m q_i,m(t) . Distinguishing car flows and truck flows, we further extend the above equation to v_i(t)=∑_m (q_i,m^(c) (t)v_i,m^(c)(t)+q_i,m^(t) (t)v_i,m^(t)(t))/∑_m (q_i,m^(c)(t)+q_i,m^(t)(t)) , where the superscripts (c) and (t) indicate the quantities for cars and for trucks, respectively. For convenience, we refer to the flow-weighted velocity as velocity in the following. As an example, Fig. <ref>a shows the time evolution of the velocity on the central section j averaged over 179 workdays. Two valleys are visible for morning and afternoon rush hours, where the valley during afternoon rush hours is much deeper and wider. Setting the critical velocity at 10 km/h, the time period with lower velocity is the congested phase, otherwise is the non-congested phase, depicted in Fig. <ref>b. As expected, many congestions, in particular short congestions, occur during rush hours, but less or even none exist during non-rush hours. §.§ Network distances A network distance is the distance of the shortest path on the network between two locations. As for our local motorway network, the locations at the ends of a path are the motorway sections. The path along the motorway network is composed of many short motorway pieces connecting two close locations. Each short piece is similar to a straight line and its distance is approximately a straight-line Euclidean or a geodetic distance. Therefore, a network distance of a path is the sum of distances of all short pieces along the path. In this way, we obtain network distances between any two sections with the help of the Java application Osmosis and the Python packages OSMnx and NetworkX. Exchanging the origin and the destination of two given sections i and j, the distances in the unit of kilometers change very little. In view of this, the network distance from i to jis equal to the network distance from j to i, i.e. l_ij = l_ji. Figure <ref> visualizes the sections j within different ranges of network distances to the central section j. The shortest path between two sections is a curve rather than a perfect straight-line. Thus, the sections i within each distance range are not located in a ring or a circle centered around the central section j. As an example, Fig. <ref> visualizes the sections within different distance ranges and the covered areas of the motorway network within a given distance range l from the central section j. § RESPONSE FUNCTIONS To study the response to congestion, we define an indicator function for a given critical velocity v_c as ε_j(t)={[ 1, if v_i(t)<v_c ,; 0, if v_i(t)≥ v_c , ]. where ε_i(t)=1 for congested traffic and ε_i(t)=0 for non-congested traffic. The three-phases theory <cit.> provides a possible interpretation for congested traffic. In a local motorway network, simultaneous congestions may occur on multiple sections, obscuring the causality between congestion and velocity changes on different sections. To unambiguously disclose this causality, it is essential to capture the effect of congestion on one section without the interference of simultaneous congestions on others. We define an indicator of congestion on section j under the condition that there are no congestions on other sections k. Furthermore, we do that with spatial resolution by only including network distances l_ij smaller than a given distance threshold l_ω. Hence we introduce ω_j(t\|l_ω)=ε_j(t)∏_l_kj≤ l_ω, k≠ j(1-ε_k(t)) . A simultaneous congestion on any section k with k≠ j implying ω_j(t\|l_ω)=0. In this way, it removes the contribution of congestion from multiple sections to the velocity change under consideration. When the congestion is absent in any section at time t except for section j, the conditional indicator is ω_j(t\|l_ω)=1 and only the congestion on section j contributes to the velocity change. A velocity change is also termed a velocity increment between times t and t+τ on section i, Δ v_i(t,τ)=v_i(t+τ)-v_i(t) , where τ is referred to as time lag. The velocity increment varies largely at different traffic environments. We define the response function of velocities to the conditional indicators given distance l_ω as, R_ij(τ\|l_ω)=⟨Δ v_i(t,τ)ω_j(t\|l_ω)⟩-⟨Δ v_i(t,τ)⟩⟨ω_j(t\|l_ω)⟩ . The average ⟨⋯⟩ is on the times t. The response function (<ref>) depends on the chosen critical velocity v_c. It measures, on average, how large the velocity on section i relatively changes from time t to t+τ, if a congestion is only on section j at time t. From a formal mathematical viewpoint, the response function is a time-lagged covariance. As one of the time series is an indicator, we prefer the term response functions. If R_ij(τ \|l_ω)>0, the two observables move in the same direction, i.e., the increase (or decrease) of the velocity change is accompanied by the increase (or decrease) of the conditional indicator. In contrast, the two quantities move in opposite directions when R_ij(τ \|l_ω)<0. The second term in Eq. (<ref>) is the unconnected part, hence the response vanishes if there is no mutual dependence between the congestions and the velocity change. It depends on the studied system if one finds it convenient to include this unconnected part. The effect of congestion propagates both in time and in space via the neighbouring sections <cit.>. A section geographically close to the congested section suffers more influences from the congestion than a section far away <cit.>. As the network distance l_ij between the impacted section i and the congested section j plays an important role in the congestion propagation, we incorporate the spatial information into the response function (<ref>). To assess the spatial characteristics in a more general way, we average over all impacted sections i in the region defined by l_ij<l, ⟨ R_ij(τ,l\|l_ω)⟩_i=∑_iR_ij(τ \|l_ω)Θ (l-l_ij)/∑_iΘ (l-l_ij) , where the step function Θ (l-l_ij)={[ 1, if l≥ l_ij; 0, if l<l_ij ]. extracts all sections i that satisfy the condition of distances. The average in Eq. (<ref>) captures the response within the specified range, and washes out the noise in the individual response functions. § EMPIRICAL RESULTS AND DISCUSSION To empirically work out the response functions, we first apply Eq. (<ref>) to the time series of each workday and then average the response values for each given τ over different workdays to obtain R_ij(τ\|l_ω). Averaging R_ij(τ \|l_ω) over different distance ranges l by Eq. (<ref>) finally results in ⟨ R_ij(τ,l\|l_ω)⟩_i. In the following, we first discuss the response behavior with respect to the time evolution in Sec. <ref>. We then analyze the transitions of response phases in Sec. <ref>. We also explore how the response changes with the increase of distance ranges in Sec. <ref>. We further inspect the feature of the scale invariance in the response function in terms of distance ranges in Sec. <ref>. §.§ Time-dependent response behavior According to Fig. <ref>, we select three typical time periods, i.e. morning rush hours from 6:00 to 10:59, afternoon rush hours from 15:00 to 19:59, and non-rush hours from 10:00 to 14:59. Each time period contains 300 minutes with a time step of 1 minute. Considering a motorway network centered around section j within the largest reachable network distance l_ω=15 km for conditional indicator ω_j(t), we work out the averaged responses of velocities on sections i to the congestion on section j, shown in Fig. <ref>b, within different distance ranges l running from 2 km to 15 km at an increment of 1 km. Here the range within l=1 km only contains one section i and the averaging of results is unable to eliminate the individuality carried by section i paired with section j. We therefore ignore this case. A strength difference in individuality is visible in Fig. <ref>d. In spite of it, the basic characteristics of response curves are similar. Figure <ref>b depicts the overall characteristics of responses and correspondingly Fig. <ref>e zooms in the negative responses at small τ. Within each l, the averaged response ⟨ R_ij(τ,l \|l_ω)⟩_i depending on time lag τ drops down to be negative and then raises up to be positive. The negative value persists for more than 10 minutes until the positive value shows up. Such behavior emerges from the both morning and afternoon rush hours. In contrast, the response is too weak to be observed during non-rush hours. Usually the congested phases dominate most of time during rush hours, while non-congested phases are prevalent most of time during non-rush hours. The comparison between rush and non-rush hours turns out that the presence of remarkable responses is stimulated by the congested phase rather than the non-congested phase. Essentially, the response function is a covariance function which reveals the collective motion of two quantities, e.g. Δ v_i(t,τ) and ω_j(t) in our study. The case of ω_j(t)=0 is complicated. It corresponds to the non-congested phase on section j and the congested phase on both section j and any section i. Therefore the contribution to response with ω_j(t)=0 is difficult to be distinguished. Differently, the case of ω_j(t)=1 only corresponds to the congested phase on section j accompanied with non-congested phases on all sections i. The resulting response is causally related to the congestion on section j to some extent. For the negative response at small τ, when the binary conditional indicator ω_j(t)=1, the velocity changes Δ v_i(t,τ) relative to the average velocity change caused by noise information become negative, implying the velocity on section i decreases due to the congestion on section j. On the other hand, for the positive response at large τ, the Δ v_i(t,τ) relative to its average become positive when ω_j(t)=1, suggesting the velocity increases on section i conditioned on the congestion on section j. For comparison, we also work out the responses with regard to different types of indicators, given in a uniform formula by R_ij(τ\|l_ω)=⟨Δ v_i(t,τ)η_j(t\|l_ω)⟩-⟨Δ v_i(t,τ)⟩⟨η_j(t\|l_ω)⟩ . When the indicator η_j(t\|l_ω)=ω_j(t\|l_ω), we arrive at the response function (<ref>) with respect to the congestions only occurs on section j. When η_j(t\|l_ω)=ε_j(t), there are responses to the congestion on section j regardless of the simultaneous congestions on other sections i, see Fig. <ref>a. Obviously, this response is stronger than the response to the congestion only on section j comparing Figs. <ref>a and b, since the former contains a part of responses to other sections i. In other words, the latter exactly excludes the response components caused by other sections i apart from section j, so as to preserve the causality between each section pair. For an extreme scenario, every section in the considered local motorway network is congested. In this scenario, the conditional indicator is defined as ω̃_j(t\|l_ω)=∏_l_kj≤ l_ωε_k(t) . Setting η_j(t\|l_ω)=ω̃_j(t\|l_ω) in Eq. (<ref>) yields the response to the congestion on every section. It is, however, zero response for each τ shown in Fig. <ref>c, as the aforementioned scenario is rather impossible to reach unless the whole local motorway network is broken down. From our empirical data, we have not met this scenario so far. §.§ Transitions of response phases To explore the traffic dynamics from the perspective of velocity responses, we analyze different regimes. First, negative and positive responses are separated by the critical point τ_c (0<τ_c<30 min), at which the response vanishes, ⟨ R_ij(τ,l \|l_ω)⟩_i \|_τ=τ_c=0 . In our previous study <cit.>, we refer to the response occurring before τ_c as transient response (or response phase 1) and after τ_c as long-term response (or response phase 2). The phase 1 (phase 2) with the negative (positive) response reveals the lowering (raising) of the velocity on section i caused by the congestion on section j. Furthermore, the response has a minimum at τ_min (0<τ_min<30 min) and a maximum at τ_max (30 min <τ_max<240 min), where its derivative vanishes, ∂/∂τ⟨ R_ij(τ,l \|l_ω)⟩_i \|_τ=τ_min or τ_max=0 . The extremal points may be viewed as indicating transitions, reflecting competitions between the vehicle deceleration and acceleration on the impacted sections i. The three critical points separate the response in time and space into four regions, yielding a phase portrait for each rush hours, as depicted in Fig. <ref>. For 0<τ≤τ_min, the congestion causes a high possibility of vehicle deceleration, resulting in the decrease of the velocity on section i. Vehicles decelerate to a minimal value at around 5 or 4 minutes for different distance ranges. In comparison to the initial velocity, the velocity on section i changes negatively. The magnitude of velocity changes decays with distance ranges. For τ_min<τ≤τ_c, with the congestion relief, vehicle acceleration occupies the most of time, leading to the increase of the velocity from a negative value to the initial value. Roughly speaking, the larger the distance ranges, and the more quickly the vehicles recover to their initial velocities. The persistent acceleration during τ_c<τ≤τ_max further drives the velocity to a positive value. The vehicle acceleration for a long time attracts more traffic flow, which further reverses the change of velocity and leads to a reduction in velocity during τ_max<τ≤240 min. §.§ Power laws Figures <ref>b,e and <ref> reveal that the response is not only time-dependent but also distance-dependent. Fixing a specific time lag τ, the dependence of responses on the distance range l, as shown in Fig. <ref>a, behave as a power law ⟨ R_ij(τ,l \|l_ω)⟩_i=α(τ \|l_ω) l^β(τ \|l_ω) , where α(τ \|l_ω) is the l-independent part and β(τ \|l_ω) the exponent. Both depend on τ and l_ω. We determine them by fitting to the empirical result, as shown in Fig. <ref>a. Around the critical point τ_c, such a fit is not possible with statistical significance. This region has to be excluded. The results for β(τ \|l_ω) are shown in Figs. <ref>b and c for morning and afternoon rush hours during workdays. As seen, the exponent β(τ \|l_ω) depends on the time lag τ considered. Importantly, there is a jump occurring in the region around τ_c. §.§ Scale invariance Guided by our naked eyes, we phenomenologically describe the collapses of curves in Fig. <ref>b by shifting horizontally and stretching vertically. To obtain well curve collapses, the responses before and after the critical point τ_c are rescaled by different methods, r(τ̃)={[ ⟨ R_ij(τ,l\|l_ω)⟩_i/\|min(⟨ R_ij(τ,l\|l_ω)⟩_i)\| , with τ̃=τ, if τ<τ_c ,; [0.5cm] ⟨ R_ij(τ-τ_c,l\|l_ω)⟩_i/\|max(⟨ R_ij(τ,l\|l_ω)⟩_i)\| , with τ̃=τ-τ_c, if τ≥τ_c . ]. For time lags τ<τ_c, the response is rescaled only by dividing the magnitude of the minimal response. After τ_c, the response is not only shifted left by τ_c, but also divided by the magnitude of the maximal response. The rescaled responses r(τ̃) versus the time lag τ̃ in Fig. <ref> show that all curves are very close to each other and roughly collapse to a single curve. This phenomenon indicates a potential presence of scaling invariance. Furthermore, the difference in rescaling methods before and after τ_c suggest distinguishable traffic dynamics for different response phases. To validate and refine our findings, we explore the behavior of scaling invariance employing the power law (<ref>). It is known <cit.> that the only solution of the scaling-invariant criterion is a power law. We sketch the reasoning for the response function ⟨ R_ij(τ,l \|l_ω)⟩_i in Appendix <ref>. Assuming that the function ⟨ R_ij(τ,l \|l_ω)⟩_i in terms of distance ranges l is invariant under all rescalings, we have <cit.> ⟨ R_ij(τ,l \|l_ω)⟩_i=μ(λ,τ\|l_ω) ⟨ R_ij(τ,λ l\|l_ω)⟩_i , where λ is a scaling factor. According to Eqs. (<ref>) and (<ref>), μ(λ,τ\|l_ω) is a function in terms of λ and τ, μ(λ,τ\|l_ω)=λ^-β(τ \|l_ω) . We reformulate Eq. (<ref>) as ⟨ R_ij(τ,l \|l_ω)⟩_i=λ^-β(τ \|l_ω)⟨ R_ij(τ,λ l \|l_ω)⟩_i . Setting l=1 and λ=l in the above equation yields ⟨ R_ij(τ,1 \|l_ω )⟩_i=l^-β(τ \|l_ω)⟨ R_ij(τ,l \|l_ω)⟩_i , which means for a given τ, the responses for different distance ranges l are rescaled to the response within l=1 by multiplying l^-β(τ \|l_ω). In other words, at a given τ, the points of responses for different distance ranges l overlap with each other. For different τ, the connection of all overlapping points performs a collapse curve. Therefore, if the scaling invariance exists, the time-dependent curves of responses rescaled by multiplying l^-β(τ \|l_ω) should collapse to a single curve. Figure <ref>b displays the empirical results of l^-β(τ \|l_ω)⟨ R_ij(τ,l \|l_ω)⟩_i during morning and afternoon rush hours. Deviating from the critical point τ_c, all curves within different distance ranges l basically overlap with each other. As the region around τ_c does not allow a power-law analysis, we eliminate the effects from the critical point τ_c by fitting β(τ \|l_ω) to an exponential function β(τ \|l_ω) = aexp(bτ)+cexp(dτ) , where a, b, c and d are fit parameters. The exponential function (<ref>) well describes the dependence of β(τ \|l_ω) on τ, as displayed in Fig. <ref>c, and in particular fills suitable values to substitute for the distorted β(τ \|l_ω) around τ_c. For distinguishing, we refer to the fitted β(τ \|l_ω) as β̃(τ \|l_ω). With β̃(τ \|l_ω), the rescaled responses almost collapse to the same curve regardless of the critical point τ_c, as shown in Fig. <ref>c. The overlap of all curves during afternoon rush hours looks much better than during afternoon rush hours. One possible reason owes to fitting errors either in the power law (<ref>) or in the exponential function (<ref>). However, the most possible reason lies in the proportion of congestions during each rush hours. A higher proportion of congestions leads to the better statistic for responses, and further to the better collapses of rescaled responses. In Fig. <ref>, a higher proportion of congestion exactly occupies the afternoon rush hours than the morning rush hours, corresponding to the better curve collapses for afternoon rush hours. Our empirical results, therefore, corroborate the assumption of scale invariance in the response function ⟨ R_ij(τ,l \|l_ω)⟩_i in terms of l give each τ. The values of exponent β(τ \|l_ω ) before and after τ_c (see Sec. <ref>) differ the scaling behavior for the response phases separated by τ_c. § CONCLUSIONS To study the causality between the congestion and velocity changes, we introduced a new response function with a conditional indicator. The conditional indicator rules out the synchronization of congestion occurring on multiple motorway sections. The response function quantifies the causal connection between the impacted sections and the congested section. From a formal mathematical viewpoint, it is a (time-lagged) covariance. When the two quantities move towards the same direction, a positive response shows up and the velocity increases due to the congestion. Conversely, a negative response appears and the velocity decreases compared with the initial velocity. We found a phase change from negative responses at small time lags to positive responses at large time lags, separated by the critical point τ_c at which the response vanishes. The points τ_min and τ_max correspond to the minimal and the maximal response, respectively, where the minimal responses occur at around the time lag of 4 or 5 minutes. These points distinguish the vehicle deceleration from the vehicle acceleration. The latter leads to a velocity change relative to its average recovering from a negative value to a positive one. Therefore the acceleration prompts the change of response phases distinguished by the critical point τ_c. The three points separate the response phases into four regions with different traffic dynamics. Furthermore, we also found the distance-dependent response at a fixed time lag τ decays as a power law in terms of the distance ranges within which the responses are averaged. We notice that a power law does not necessarily imply heavy tails, which depend on the exponent. Here, we focused on the scale invariance in response curves which we confirmed empirically. § ACKNOWLEDGMENTS We are grateful to Sebastian Gartzke for fruitful discussions. We thank Strassen.NRW for providing the empirical traffic data. § AUTHOR CONTRIBUTIONS T.G. and M.S. proposed the research. S.W. and T.G. developed the methods of analysis. S.W. performed all the calculations. All authors contributed equally to analyzing the results, writing and reviewing the paper. tocsectionReferences 10 Hansen1959 Walter G Hansen. How accessibility shapes land use. J. Am. I. Planners, 25(2):73–76, 1959. 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Setting l=1 in the above equation gives μ(λ)=ℛ(1)/ℛ(λ) Therefore Eq. (<ref>) becomes ℛ(λ)ℛ(l)=ℛ(1)ℛ(λ l) . By differentiating both sides with regard to λ, we have ℛ'(λ)ℛ(l)=lℛ(1)ℛ'(λ l) . Here ℛ'(·) represents the derivative of ℛ(·) regarding its argument inside the bracket. Letting λ=1 gives rise to ℛ'(1)ℛ(l)=lℛ(1)ℛ'(l)= lℛ(1)dℛ(l)/dl . We rewrite Eq. (<ref>) dℛ(l)/ℛ(l)=ℛ'(1)/ℛ(1)dl/l . Integrating both sides results in lnℛ(l)=ℛ'(1)/ℛ(1)ln l +c , where c is a constant. Let l=1 such that we are able to obtain c=lnℛ(1). This leads to ℛ(l)=ℛ(1)l^β , where β=ℛ'(1)/ℛ(1). Therefore, the scaling invariance results in the power-law response function in terms of distances. In other words, the power-law response function in terms of distances is the only function that meets the scaling invariant criterion (<ref>). § FIGURES FOR DIFFERENT TIME PERIODS
http://arxiv.org/abs/2406.18065v1	20240626045019	On Calibration of Speech Classification Models: Insights from Energy-Based Model Investigations	[ "Yaqian Hao", "Chenguang Hu", "Yingying Gao", "Shilei Zhang", "Junlan Feng" ]	eess.AS	[ "eess.AS", "cs.SD" ]	Large Language Models for Cuffless Blood Pressure Measurement From Wearable Biosignals Marc Cheong ====================================================================================== § ABSTRACT For speech classification tasks, deep learning models often achieve high accuracy but exhibit shortcomings in calibration, manifesting as classifiers exhibiting overconfidence. The significance of calibration lies in its critical role in guaranteeing the reliability of decision-making within deep learning systems. This study explores the effectiveness of Energy-Based Models (EBMs) in calibrating confidence for speech classification tasks by training a joint EBM integrating a discriminative and a generative model, thereby enhancing the classifier’s calibration and mitigating overconfidence. Experimental evaluations conducted on three speech classification tasks specifically: age, emotion, and language recognition. Our findings highlight the competitive performance of EBMs in calibrating the speech classification models. This research emphasizes the potential of EBMs in speech classification tasks, demonstrating their ability to enhance calibration without sacrificing accuracy. [Corresponding Author.] [Equal Contribution.] § INTRODUCTION Despite the impressive performance of deep learning models in speech classification <cit.>, issues such as overconfidence, calibration errors, and uncertainty estimation may hinder their reliability and generalization in real-world scenarios <cit.>. Confidence calibration in these models poses a significant challenge <cit.>. For example, in speech emotion recognition (SER) systems, the inherent uncertainty in modeling emotions affects the trustworthiness of the model's predictions <cit.>. Overconfidence and underconfidence can indicate suboptimal calibration, leading to false positives or missed opportunities <cit.>. The current state of research in speech classification models often overlooks the issue of confidence calibration, resulting in a lack of reliable methods and leading to uncertainty in predictions. The current state of confidence calibration in speech classification underscores a lack of reliable methods, fostering uncertainty and mistrust in model predictions. Undoubtedly, ensuring a well-calibrated confidence measure in a classification model is crucial for accurate predictions. Therefore, developing methodologies to adjust the predictions of a speech classification model is essential, balancing calibration with performance. Existing techniques, like Temperature Scaling and Vector Scaling, typically rescale the posterior distributions of classifier predictions <cit.>. However, these methods need post-processing adjustments and require a persistent development set with enough samples. Alternatively, adjusting calibration during the model training process, such as using confidence regularization, offers another approach <cit.>. Recently, the effectiveness of EBMs in achieving enhanced model calibration has been demonstrated <cit.>, wherein the joint training process incorporates both discriminative and generative models. The EBMs characterize the relationship between density of input data and model energy, enabling predictions based on energy minimization. While this flexibility is advantageous, the training process of energy models involves intricate adjustments, making it a challenging endeavor <cit.>. Following <cit.>'s work, <cit.> has intricately improved the training process of EBMs, substantially boosting both training efficiency and ultimate performance. Despite their effectiveness in computer vision, EBMs' potential for calibrating speech classification models remains untapped. In this paper, we explore the effectiveness of EBMs in enhancing the calibration of speech classification models. Through experiments on three distinct speech classification tasks, we compare EBMs with traditional softmax-based models. Results reveal that EBMs achieve an average reduction of 7.787% in Expected Calibration Error (ECE) across the tasks, indicating improved calibration. Additionally, Negative Log-Likelihood (NLL) shows an average reduction of 0.172, indicating enhanced model fitting to observed data and more accurate probability predictions. Furthermore, we compared EBMs with other calibration methods such as Temperature Scaling and Logistic Scaling, and the results demonstrate that EBMs exhibit a significantly greater reduction in overconfidence compared to these post-processing methods. The key contributions of this paper are summarized as follows: * We introduce joint EBMs in speech classification tasks to improve calibration by modeling the energy function with a deep neural network, maintaining accuracy while enhancing reliability. This joint energy model optimizes not only for classification tasks but also learns the underlying probability distribution within the data, resulting in improved calibration observed through the model's cautious decision-making when encountering inputs deviating from the training data distribution. * We assess the performance of EBMs across three speech tasks and datasets, specifically targeting language, emotion, and age recognition. This evaluation demonstrates that EBMs can significantly reduce ECE without compromising model accuracy, and mitigate overconfidence issues in speech classification models. * We conduct comparative analyses with other calibration methods, and explore model training dynamics and confidence distributions to address model overconfidence. Specifically, our results show that EBMs outperform other post-processing methods in achieving effective calibration without requiring additional auxiliary datasets. § METHOD §.§ Energy-based Models The fundamental principle underlying an EBM is to construct a function E(𝐱):R^D → R that maps each point in the input space to a singular, non-probabilistic scalar referred to as the energy <cit.>. This scalar, denoted by E(𝐱), is a key component in the Gibbs distribution, allowing the derivation of a probability density p(𝐱). The relationship is formalized as follows: p_θ (𝐱) = exp(-E_θ(𝐱)/T) /Z(θ), where E_θ(𝐱) representing the energy, is a nonlinear regression function parameterized by θ, T refers to the temperature parameter, and Z(θ) signifies the normalizing constant, also known as the partition function: Z(θ) = ∫_𝐱exp(-E_θ (𝐱) /T) dx. §.§ Energy-based Classifier The EBM exhibits an intrinsic association with contemporary machine learning, particularly discriminative neural classifier <cit.> f_θ(𝐱) :ℝ^D →ℝ^K. This classifier assigns logits to each class for a given input 𝐱 using the softmax function: p_θ(y \| 𝐱) = exp(f_θ(𝐱)[y]/T)/∑_i^K exp(f_θ(𝐱)[i]/T), where f_θ(𝐱)[y] denotes the logit associated with the y-th class label within the output of f_θ(𝐱). This connection allows us to view the discriminative classifier f(𝐱) as an energy function in the EBM framework E_θ(𝐱, y) = -f_θ(𝐱)[y]/T. Consequently, the Helmholtz free energy function E(𝐱; f) for a given data point 𝐱∈ℝ^D can be represented as the negative logarithm of the partition function: E_θ(𝐱; f) = -T ·log∑_i^K exp(f_θ(𝐱)[i]/T). Additionally, the logits from f(𝐱) enable the definition of an EBM for the joint distribution of data points 𝐱 and labels y: p_θ(𝐱, y) = exp(-E_θ(x,y))/∫_x ∑_i^K exp(-E_θ(x,y)dx)=exp(f_θ(𝐱)[y]/T)/Z(θ). Marginalizing over y provides an unnormalized density model for 𝐱: p_θ(𝐱) = ∑_y p_θ(𝐱, y) = ∑_y exp(f_θ(𝐱)[y]/T)/Z(θ), which is precisely the definition of EBM. This reinterpretation underscores the intrinsic compatibility between the softmax classifier and the EBM, offering a unified perspective on their shared principles. §.§ Optimization We employ a joint model, integrating an energy-based classifier and a generative model, wherein the EBM is trained to learn the energy function that best captures the data distribution <cit.>. The objective function is as following: log p_θ(𝐱, y) = log p_θ(y\|𝐱) + log p_θ(𝐱), which represents the logarithm of joint distribution of data and labels. The conditional distribution p_θ(y\|𝐱) signifies the softmax classification model, while p_θ(𝐱) captures the marginal data distribution. The loss function is then aligned with the logarithm of the likelihood, as explained in the following: log p_θ(𝐱,y) =log p_θ(y\|𝐱) + log p_θ(𝐱) = logexp(f_θ(𝐱)[y]/T)/∑_i exp(f_θ(𝐱)[i]/T) +log∑_yexp(f_θ(𝐱)[y])/T/Z_θ. The derivative of the first term in the Eq.(<ref>) is relatively straightforward, representing the loss function for training the classifier. The derivative of the second term is: ∇_θlog p(𝐱) = ∇_θlog∑_y exp(f_θ(𝐱)[y]/T) - ∇_θlog Z(θ) = ∇_θlog∑_y exp( f_θ(𝐱)[y]/T)-𝔼_x ∼ p_θ(x)[log∑_y exp( f_θ(𝐱)[y]/T)] =-∇_θ E_θ(𝐱) + 𝔼_x ∼ p_θ(x)[∇_θ E_θ(𝐱) ]. By employing a one-sample Monte Carlo estimate ∇_θlog Z_θ∼ -∇_θ E_θ(x̃), where x̃ is sampled from the EBM's distribution p_θ(x). §.§ SGLD-Based Training Method According to Eq. (<ref>), we utilize Langevin MCMC for sampling from p_θ(𝐱) to train the EBMs <cit.>. Stochastic Gradient Langevin Dynamics (SGLD) is a dynamic optimization algorithm that combines the principles of stochastic gradient descent with Langevin dynamics. To initiate the Langevin sampling process, we begin by drawing an initial sample x_0 from a straightforward prior distribution. Subsequently, we simulate an overdamped Langevin diffusion process for K steps, employing a positive step size ϵ > 0. The iteration for each step k = 0, 1, …, K - 1 is expressed as: x_k+1 = x_k + ϵ^2/2∇_x_klog p_θ(x_k) + ϵ z_k = x_k - ϵ^2/2∇_x_k E_θ(x_k) + ϵ z_k, where ∇_x_klog p_θ(x_k) represents the gradient of the log probability with respect to x_k, and z_k is a random noise term. Notably, as ϵ→ 0 and K →∞, the final sample x_K converges to a distribution that matches p_θ(x) under certain regularity conditions. §.§ Evaluation Metrics Expected Calibration Error in Classification. A calibrated classifier aligns confidence with accuracy <cit.>. ECE quantifies calibration by binning predictions and measuring the difference between expected confidence and accuracy. Mathematically, ECE is expressed as: ECE = ∑_b=1^B\| B_b \|/N\|acc(B_b) - conf(B_b) \|, where B is the number of bins, B_b represents the b-th bin,\|B_b \|is the number of samples in binB_b,Nis the total number of samples,acc(B_b)is the average accuracy in binB_b, andconf(B_b)is the average confidence in binB_b. Negative Log-Likelihood in Classification. NLL is a key metric for assessing a classification model's calibration. It measures the agreement between predicted probabilities and actual labels by computing the logarithm of the predicted probability assigned to the true label for each sample: NLL = -1/N∑_i=1^Nlog(P(ŷ_̂î)\|x_i), where N is the total number of samples, ŷ_̂î represents the true label of the i-th sample, and P(ŷ_̂î) denotes the predicted probability associated with the true label. A lower NLL indicates better calibration, signifying that the model's predicted probabilities closely match the actual outcomes. In this study, we concentrate on calibration performance, aiming to demonstrate that incorporating EBMs improves confidence calibration without impacting the model’s classification effectiveness. Consequently, we use accuracy as the primary metric to affirm the preservation of core classification capabilities. § EXPERIMENTS §.§ Datasets Multiple datasets will be used in these experiments, with the speech sampled at 16 kHz. The duration of training data for each task and the data split of those datasets is listed in Table <ref>. AP17-OLR <cit.>, CASIA <cit.> and VoxCeleb-Enrichment <cit.> datasets are used in our experiments for language, emotion and the age group classification respectively. AP17-OLR consists of 10 different languages. The test set contains three subsets with different durations (1 second, 3 second, and full length). For speech emotion classification task, we conduct out experiments on CASIA with six emotion categories (i.e., angry, surprise, sad, fear, happy, and neutral). For the age group classification, we used the VoxCeleb Enrichment dataset to train the model. VoxCeleb Enrichment were extracted from YouTube videos, the audio clips were recorded in a variety of acoustic environments. The audios were divided into four age groups . §.§ Experimental Settings The input features are 32-dimensional Mel Filter-Banks extracted using the librosa package <cit.> with a window length of 25ms and a shift of 10ms with Hamming window. Mean and variance normalization is applied during instance normalization on Mel Filter-Banks features. A 192-frame segment (320-frame for age classification) is randomly chunked from each utterance. All our experiments are based on the Wide-ResNet architecture <cit.>, featuring a width of 5, depth of 28, payload learning rate of 0.2, 50 SGLD sampling steps, and a buffer size of 10,000 . Frame-level feature extraction is based on ResNet topology with 3 groups of residual blocks. Then the frame-level features are fed into the average pooling layer to get utterance-level embeddings, and the final classification layer dimensions are 10, 6, and 4, respectively, for language, emotion, and age group classification. We optimize the model with stochastic gradient descent (SGD) <cit.> optimizer, the learning rate warms up to 0.1 during the first 1000 steps, and reduce the learning rate at epoch [40, 80, 120] with a decay rate of 0.2. Both softmax-based models and EBMs stick to the same training settings. §.§ Performance analysis and discussion Table <ref> summarizes the performance of softmax and energy-based classifiers across three classification tasks. The energy-based classifier outperform in the age classification task, achieving a higher accuracy of 74.96% and significantly improving calibration with a reduction in ECE from 16.332% to 3.208%. While the emotion classification task showed a minor accuracy drop, the EBMs exhibited substantial calibration improvement. Likewise, in the language classification task, the energy model demonstrated superior calibration with a notable reduction in ECE. These results highlight the efficacy of energy-based classifiers in enhancing calibration across diverse tasks, suggesting their potential for reliable predictions with well-calibrated uncertainty estimates. Reliability diagrams. To evaluate calibration performance, reliability diagrams are utilized, visually representing the consistency between predicted probabilities and actual outcomes in Figures 1. It's evident that EBMs exhibit superior calibration, displaying smaller gaps and significantly lower ECE compared to softmax-based models across three classification scenarios. Particularly, softmax-based models in three speech tasks consistently demonstrate overconfidence, as they tend to assign excessively high probabilities to predicted classes, a common issue observed in deep learning models <cit.>. Notably, the reliability diagram for language classification using EBMs closely follows the diagonal, achieving an ECE of only 1.0%, indicating nearly perfect calibration. These findings underscore that EBMs can significantly alleviate the issue of overconfidence in speech classification models, thereby achieving better calibration performance. Comparative evaluation of other calibration methods. We conduct a comparative analysis between EBMs and two post-processing calibration methods, namely Temperature Scaling and Logistic Scaling, across three speech classification tasks. The results are summarized in Table <ref> alongside ECEs. It is evident that these two post-processing calibrators provide limited improvement in model calibration. This restricted effectiveness may stem from potential disparities between the auxiliary data and the target distribution, resulting in suboptimal calibration adjustments. In contrast, EBMs consistently demonstrate superior performance, resulting in significant reductions in the ECE, without the requirement of supplementary training data. Why softmax-based models are poorly calibrated? As illustrated in Figure <ref>, softmax models prioritize optimizing accuracy over achieving minimizing NLL. This observation aligns with the findings in <cit.>, which suggest that modern neural networks can overfit to NLL without overfitting accuracy. This phenomenon indicates that while softmax models may achieve high classification accuracy, they may not necessarily provide well-calibrated probability estimates. In other words, the pursuit of higher accuracy values can sometimes come at the cost of the model's ability to accurately reflect confidence in its predictions, thereby compromising calibration quality. In contrast, EBMs excel in reducing NLL while maintaining high accuracy. Despite slower convergence during EBM training, they achieve lower NLL and higher confidence levels. Confidence distribution. We analyze the problem of model overconfidence by visualizing the confidence distribution in Figure 3. It demonstrates a significant prevalence of excessively high confidence levels for incorrect predictions across three speech tasks, resulting in unreliable confidence estimates for softmax-based models. For example, within the confidence range of 0.9-1, the softmax-based model for age recognition yields 300 misclassified samples, whereas EBMs show only 10 misclassifications. It is noteworthy that EBMs exhibit a reduction in the confidence range of incorrect predictions across the three speech tasks, with accurate predictions predominantly falling within higher confidence intervals, leading to reliable confidence. § CONCLUSIONS In this study, we explored the effectiveness of joint EBMs in calibrating speech classification tasks. Our results show that joint EBMs optimize both the classifier and the generative model to enhance calibration by gaining profound insights into the data distribution while also serving as a regularization mechanism, effectively mitigating overfitting tendencies. These findings demonstrate that EBMs can notably generate well-calibrated predictions without compromising accuracy across diverse speech classification tasks. IEEEtran
http://arxiv.org/abs/2406.18273v1	20240626115629	Lift-and-Project Integrality Gaps for Santa Claus	[ "Etienne Bamas" ]	cs.DS	[ "cs.DS", "cs.CC" ]	"\nCAS: Confidence Assessments of classification algorithms for Semantic segmentation of EO data\n (...TRUNCATED)
http://arxiv.org/abs/2406.17664v1	20240625155311	Magnetic Force Microscopy: High Quality Factor Two-Pass Mode	["Christopher Habenschaden","Sibylle Sievers","Alexander Klasen","Andrea Cerreta","Hans Werner Schum(...TRUNCATED)	cond-mat.mes-hall	[ "cond-mat.mes-hall" ]	"\n\nxxx\n\nMagnetic Force Microscopy: High Quality Factor Two-Pass Mode]Magnetic Force Microscopy: (...TRUNCATED)
http://arxiv.org/abs/2406.18368v1	20240626141257	Future singularity in anisotropic universe	[ "Taishi Katsuragawa", "Shin'ichi Nojiri", "Sergei D. Odintsov" ]	gr-qc	[ "gr-qc", "astro-ph.CO", "hep-th" ]	"\n\n=5000\n\n\n\nKEK-TH-2632\nKEK-Cosmo-0348\n\n\ntaishi@ccnu.edu.cn\n\nInstitute of Astrophysics, (...TRUNCATED)
http://arxiv.org/abs/2406.19347v1	20240627172122	Thermal Dynamics of Heat Pipes with Sub-Critical Nanopores	[ "Sumith Yesudasan" ]	cond-mat.mes-hall	[ "cond-mat.mes-hall" ]	"\n\n\n\n\nsyesudasan@newhaven.edu\nDepartment of Mechanical and Industrial Engineering, University (...TRUNCATED)
http://arxiv.org/abs/2406.18770v1	20240626214250	ADO-LLM: Analog Design Bayesian Optimization with In-Context Learning of Large Language Models	[ "Yuxuan Yin", "Yu Wang", "Boxun Xu", "Peng Li" ]	cs.LG	[ "cs.LG" ]	"\n\n\n\n\n\n\nGPT4AIGChip: Towards Next-Generation AI Accelerator Design Automation via Large Langu(...TRUNCATED)

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