Daily Papers

Graph neural networks that model 3D data, such as point clouds or atoms, are typically desired to be SO(3) equivariant, i.e., equivariant to 3D rotations. Unfortunately equivariant convolutions, which are a fundamental operation for equivariant networks, increase significantly in computational complexity as higher-order tensors are used. In this paper, we address this issue by reducing the SO(3) convolutions or tensor products to mathematically equivalent convolutions in SO(2) . This is accomplished by aligning the node embeddings' primary axis with the edge vectors, which sparsifies the tensor product and reduces the computational complexity from O(L^6) to O(L^3), where L is the degree of the representation. We demonstrate the potential implications of this improvement by proposing the Equivariant Spherical Channel Network (eSCN), a graph neural network utilizing our novel approach to equivariant convolutions, which achieves state-of-the-art results on the large-scale OC-20 and OC-22 datasets.

Sparse autoencoders (SAEs) have lately been used to uncover interpretable latent features in large language models. Here, we explore their potential for decomposing latent representations in complex and high-dimensional biological data, where the underlying variables are often unknown. On simulated data we show that generative hidden variables can be captured in learned representations in the form of superpositions. The degree to which they are learned depends on the completeness of the representations. Superpositions, however, are not identifiable if these generative variables are unknown. SAEs can to some extent recover these variables, yielding interpretable features. Applied to single-cell multi-omics data, we show that an SAE can uncover key biological processes such as carbon dioxide transport and ion homeostasis, which are crucial for red blood cell differentiation and immune function. Our findings highlight how SAEs can be used in advancing interpretability in biological and other scientific domains.

Hyperdimensional computing (HDC) is a biologically-inspired framework which represents symbols with high-dimensional vectors, and uses vector operations to manipulate them. The ensemble of a particular vector space and a prescribed set of vector operations (including one addition-like for "bundling" and one outer-product-like for "binding") form a *vector symbolic architecture* (VSA). While VSAs have been employed in numerous applications and have been studied empirically, many theoretical questions about VSAs remain open. We analyze the *representation capacities* of four common VSAs: MAP-I, MAP-B, and two VSAs based on sparse binary vectors. "Representation capacity' here refers to bounds on the dimensions of the VSA vectors required to perform certain symbolic tasks, such as testing for set membership i in S and estimating set intersection sizes |X cap Y| for two sets of symbols X and Y, to a given degree of accuracy. We also analyze the ability of a novel variant of a Hopfield network (a simple model of associative memory) to perform some of the same tasks that are typically asked of VSAs. In addition to providing new bounds on VSA capacities, our analyses establish and leverage connections between VSAs, "sketching" (dimensionality reduction) algorithms, and Bloom filters.

Aiming at better representing multivariate relationships, this paper investigates a motif dimensional framework for higher-order graph learning. The graph learning effectiveness can be improved through OFFER. The proposed framework mainly aims at accelerating and improving higher-order graph learning results. We apply the acceleration procedure from the dimensional of network motifs. Specifically, the refined degree for nodes and edges are conducted in two stages: (1) employ motif degree of nodes to refine the adjacency matrix of the network; and (2) employ motif degree of edges to refine the transition probability matrix in the learning process. In order to assess the efficiency of the proposed framework, four popular network representation algorithms are modified and examined. By evaluating the performance of OFFER, both link prediction results and clustering results demonstrate that the graph representation learning algorithms enhanced with OFFER consistently outperform the original algorithms with higher efficiency.

The utility of a learned neural representation depends on how well its geometry supports performance in downstream tasks. This geometry depends on the structure of the inputs, the structure of the target outputs, and the architecture of the network. By studying the learning dynamics of networks with one hidden layer, we discovered that the network's activation function has an unexpectedly strong impact on the representational geometry: Tanh networks tend to learn representations that reflect the structure of the target outputs, while ReLU networks retain more information about the structure of the raw inputs. This difference is consistently observed across a broad class of parameterized tasks in which we modulated the degree of alignment between the geometry of the task inputs and that of the task labels. We analyzed the learning dynamics in weight space and show how the differences between the networks with Tanh and ReLU nonlinearities arise from the asymmetric asymptotic behavior of ReLU, which leads feature neurons to specialize for different regions of input space. By contrast, feature neurons in Tanh networks tend to inherit the task label structure. Consequently, when the target outputs are low dimensional, Tanh networks generate neural representations that are more disentangled than those obtained with a ReLU nonlinearity. Our findings shed light on the interplay between input-output geometry, nonlinearity, and learned representations in neural networks.

This study investigates how fake news uses a thumbnail for a news article with a focus on whether a news article's thumbnail represents the news content correctly. A news article shared with an irrelevant thumbnail can mislead readers into having a wrong impression of the issue, especially in social media environments where users are less likely to click the link and consume the entire content. We propose to capture the degree of semantic incongruity in the multimodal relation by using the pretrained CLIP representation. From a source-level analysis, we found that fake news employs a more incongruous image to the main content than general news. Going further, we attempted to detect news articles with image-text incongruity. Evaluation experiments suggest that CLIP-based methods can successfully detect news articles in which the thumbnail is semantically irrelevant to news text. This study contributes to the research by providing a novel view on tackling online fake news and misinformation. Code and datasets are available at https://github.com/ssu-humane/fake-news-thumbnail.

We propose procedures for evaluating and strengthening contextual embedding alignment and show that they are useful in analyzing and improving multilingual BERT. In particular, after our proposed alignment procedure, BERT exhibits significantly improved zero-shot performance on XNLI compared to the base model, remarkably matching pseudo-fully-supervised translate-train models for Bulgarian and Greek. Further, to measure the degree of alignment, we introduce a contextual version of word retrieval and show that it correlates well with downstream zero-shot transfer. Using this word retrieval task, we also analyze BERT and find that it exhibits systematic deficiencies, e.g. worse alignment for open-class parts-of-speech and word pairs written in different scripts, that are corrected by the alignment procedure. These results support contextual alignment as a useful concept for understanding large multilingual pre-trained models.

The expanse of information available over the internet makes it difficult to identify whether a specific work is a replica or a duplication of a protected work, especially if we talk about visual representations. Strategies are planned to identify the utilization of the copyrighted image in a report. Still, we want to resolve the issue of involving a copyrighted image in a video and a calculation that could recognize the degree of similarity of the copyrighted picture utilized in the video, even for the pieces of the video that are not featured a lot and in the end perform characterization errands on those edges. Machine learning (ML) and artificial intelligence (AI) are vital to address this problem. Numerous associations have been creating different calculations to screen the identification of copyrighted work. This work means concentrating on those calculations, recognizing designs inside the information, and fabricating a more reasonable model for copyrighted image classification and detection. We have used different algorithms like- Image Processing, Convolutional Neural Networks (CNN), Image hashing, etc. Keywords- Copyright, Artificial Intelligence(AI), Copyrighted Image, Convolutional Neural Network(CNN), Image processing, Degree of similarity, Image Hashing.

Neural Radiance Fields (NeRFs) have achieved great success in the past few years. However, most current methods still require intensive resources due to ray marching-based rendering. To construct urban-level radiance fields efficiently, we design Deformable Neural Mesh Primitive~(DNMP), and propose to parameterize the entire scene with such primitives. The DNMP is a flexible and compact neural variant of classic mesh representation, which enjoys both the efficiency of rasterization-based rendering and the powerful neural representation capability for photo-realistic image synthesis. Specifically, a DNMP consists of a set of connected deformable mesh vertices with paired vertex features to parameterize the geometry and radiance information of a local area. To constrain the degree of freedom for optimization and lower the storage budgets, we enforce the shape of each primitive to be decoded from a relatively low-dimensional latent space. The rendering colors are decoded from the vertex features (interpolated with rasterization) by a view-dependent MLP. The DNMP provides a new paradigm for urban-level scene representation with appealing properties: (1) High-quality rendering. Our method achieves leading performance for novel view synthesis in urban scenarios. (2) Low computational costs. Our representation enables fast rendering (2.07ms/1k pixels) and low peak memory usage (110MB/1k pixels). We also present a lightweight version that can run 33times faster than vanilla NeRFs, and comparable to the highly-optimized Instant-NGP (0.61 vs 0.71ms/1k pixels). Project page: https://dnmp.github.io/{https://dnmp.github.io/}.

Compositionality, the notion that the meaning of an expression is constructed from the meaning of its parts and syntactic rules, permits the infinite productivity of human language. For the first time, artificial language models (LMs) are able to match human performance in a number of compositional generalization tasks. However, much remains to be understood about the representational mechanisms underlying these abilities. We take a high-level geometric approach to this problem by relating the degree of compositionality in a dataset to the intrinsic dimensionality of its representations under an LM, a measure of feature complexity. We find not only that the degree of dataset compositionality is reflected in representations' intrinsic dimensionality, but that the relationship between compositionality and geometric complexity arises due to learned linguistic features over training. Finally, our analyses reveal a striking contrast between linear and nonlinear dimensionality, showing that they respectively encode formal and semantic aspects of linguistic composition.

Neural text-to-speech (TTS) has achieved human-like synthetic speech for single-speaker, single-language synthesis. Multilingual TTS systems are limited to resource-rich languages due to the lack of large paired text and studio-quality audio data. In most cases, TTS systems are built using a single speaker's voice. However, there is growing interest in developing systems that can synthesize voices for new speakers using only a few seconds of their speech. This paper presents ZMM-TTS, a multilingual and multispeaker framework utilizing quantized latent speech representations from a large-scale, pre-trained, self-supervised model. Our paper is the first to incorporate the representations from text-based and speech-based self-supervised learning models into multilingual speech synthesis tasks. We conducted comprehensive subjective and objective evaluations through a series of experiments. Our model has been proven effective in terms of speech naturalness and similarity for both seen and unseen speakers in six high-resource languages. We also tested the efficiency of our method on two hypothetical low-resource languages. The results are promising, indicating that our proposed approach can synthesize audio that is intelligible and has a high degree of similarity to the target speaker's voice, even without any training data for the new, unseen language.

Despite achieving remarkable performance on various vision-language tasks, Transformer-based Vision-Language Models (VLMs) suffer from redundancy in inputs and parameters, significantly hampering their efficiency in real-world applications. Moreover, the degree of redundancy in token representations and model parameters, such as attention heads, varies significantly for different inputs. In light of the challenges, we propose SmartTrim, an adaptive acceleration framework for VLMs, which adjusts the computational overhead per instance. Specifically, we integrate lightweight modules into the original backbone to identify and prune redundant token representations and attention heads within each layer. Furthermore, we devise a self-distillation strategy to enhance the consistency between the predictions of the pruned model and its fully-capacity counterpart. Experimental results across various vision-language tasks consistently demonstrate that SmartTrim accelerates the original model by 2-3 times with minimal performance degradation, highlighting the effectiveness and efficiency compared to previous approaches. Code will be available at https://github.com/kugwzk/SmartTrim.

Equivariant Transformers such as Equiformer have demonstrated the efficacy of applying Transformers to the domain of 3D atomistic systems. However, they are still limited to small degrees of equivariant representations due to their computational complexity. In this paper, we investigate whether these architectures can scale well to higher degrees. Starting from Equiformer, we first replace SO(3) convolutions with eSCN convolutions to efficiently incorporate higher-degree tensors. Then, to better leverage the power of higher degrees, we propose three architectural improvements -- attention re-normalization, separable S^2 activation and separable layer normalization. Putting this all together, we propose EquiformerV2, which outperforms previous state-of-the-art methods on the large-scale OC20 dataset by up to 12% on forces, 4% on energies, offers better speed-accuracy trade-offs, and 2times reduction in DFT calculations needed for computing adsorption energies.

Releasing open-source large language models (LLMs) presents a dual-use risk since bad actors can easily fine-tune these models for harmful purposes. Even without the open release of weights, weight stealing and fine-tuning APIs make closed models vulnerable to harmful fine-tuning attacks (HFAs). While safety measures like preventing jailbreaks and improving safety guardrails are important, such measures can easily be reversed through fine-tuning. In this work, we propose Representation Noising (RepNoise), a defence mechanism that is effective even when attackers have access to the weights and the defender no longer has any control. RepNoise works by removing information about harmful representations such that it is difficult to recover them during fine-tuning. Importantly, our defence is also able to generalize across different subsets of harm that have not been seen during the defence process. Our method does not degrade the general capability of LLMs and retains the ability to train the model on harmless tasks. We provide empirical evidence that the effectiveness of our defence lies in its "depth": the degree to which information about harmful representations is removed across all layers of the LLM.

We propose the Multi-Track Music Machine (MMM), a generative system based on the Transformer architecture that is capable of generating multi-track music. In contrast to previous work, which represents musical material as a single time-ordered sequence, where the musical events corresponding to different tracks are interleaved, we create a time-ordered sequence of musical events for each track and concatenate several tracks into a single sequence. This takes advantage of the Transformer's attention-mechanism, which can adeptly handle long-term dependencies. We explore how various representations can offer the user a high degree of control at generation time, providing an interactive demo that accommodates track-level and bar-level inpainting, and offers control over track instrumentation and note density.

Uniformity plays a crucial role in the assessment of learned representations, contributing to a deeper comprehension of self-supervised learning. The seminal work by Wang2020UnderstandingCR introduced a uniformity metric that quantitatively measures the collapse degree of learned representations. Directly optimizing this metric together with alignment proves to be effective in preventing constant collapse. However, we present both theoretical and empirical evidence revealing that this metric lacks sensitivity to dimensional collapse, highlighting its limitations. To address this limitation and design a more effective uniformity metric, this paper identifies five fundamental properties, some of which the existing uniformity metric fails to meet. We subsequently introduce a novel uniformity metric that satisfies all of these desiderata and exhibits sensitivity to dimensional collapse. When applied as an auxiliary loss in various established self-supervised methods, our proposed uniformity metric consistently enhances their performance in downstream tasks.Our code was released at https://github.com/sunset-clouds/WassersteinUniformityMetric.

The generation of 3D scenes from user-specified conditions offers a promising avenue for alleviating the production burden in 3D applications. Previous studies required significant effort to realize the desired scene, owing to limited control conditions. We propose a method for controlling and generating 3D scenes under multimodal conditions using partial images, layout information represented in the top view, and text prompts. Combining these conditions to generate a 3D scene involves the following significant difficulties: (1) the creation of large datasets, (2) reflection on the interaction of multimodal conditions, and (3) domain dependence of the layout conditions. We decompose the process of 3D scene generation into 2D image generation from the given conditions and 3D scene generation from 2D images. 2D image generation is achieved by fine-tuning a pretrained text-to-image model with a small artificial dataset of partial images and layouts, and 3D scene generation is achieved by layout-conditioned depth estimation and neural radiance fields (NeRF), thereby avoiding the creation of large datasets. The use of a common representation of spatial information using 360-degree images allows for the consideration of multimodal condition interactions and reduces the domain dependence of the layout control. The experimental results qualitatively and quantitatively demonstrated that the proposed method can generate 3D scenes in diverse domains, from indoor to outdoor, according to multimodal conditions.

Understanding which information is encoded in deep models of spoken and written language has been the focus of much research in recent years, as it is crucial for debugging and improving these architectures. Most previous work has focused on probing for speaker characteristics, acoustic and phonological information in models of spoken language, and for syntactic information in models of written language. Here we focus on the encoding of syntax in several self-supervised and visually grounded models of spoken language. We employ two complementary probing methods, combined with baselines and reference representations to quantify the degree to which syntactic structure is encoded in the activations of the target models. We show that syntax is captured most prominently in the middle layers of the networks, and more explicitly within models with more parameters.

In recent years, we have witnessed the great advancement of Deep neural networks (DNNs) in image restoration. However, a critical limitation is that they cannot generalize well to real-world degradations with different degrees or types. In this paper, we are the first to propose a novel training strategy for image restoration from the causality perspective, to improve the generalization ability of DNNs for unknown degradations. Our method, termed Distortion Invariant representation Learning (DIL), treats each distortion type and degree as one specific confounder, and learns the distortion-invariant representation by eliminating the harmful confounding effect of each degradation. We derive our DIL with the back-door criterion in causality by modeling the interventions of different distortions from the optimization perspective. Particularly, we introduce counterfactual distortion augmentation to simulate the virtual distortion types and degrees as the confounders. Then, we instantiate the intervention of each distortion with a virtual model updating based on corresponding distorted images, and eliminate them from the meta-learning perspective. Extensive experiments demonstrate the effectiveness of our DIL on the generalization capability for unseen distortion types and degrees. Our code will be available at https://github.com/lixinustc/Causal-IR-DIL.

In community-based question answering (CQA) platforms, automatic answer ranking for a given question is critical for finding potentially popular answers in early times. The mainstream approaches learn to generate answer ranking scores based on the matching degree between question and answer representations as well as the influence of respondents. However, they encounter two main limitations: (1) Correlations between answers in the same question are often overlooked. (2) Question and respondent representations are built independently of specific answers before affecting answer representations. To address the limitations, we devise a novel graph-based tri-attention network, namely GTAN, which has two innovations. First, GTAN proposes to construct a graph for each question and learn answer correlations from each graph through graph neural networks (GNNs). Second, based on the representations learned from GNNs, an alternating tri-attention method is developed to alternatively build target-aware respondent representations, answer-specific question representations, and context-aware answer representations by attention computation. GTAN finally integrates the above representations to generate answer ranking scores. Experiments on three real-world CQA datasets demonstrate GTAN significantly outperforms state-of-the-art answer ranking methods, validating the rationality of the network architecture.

This paper analyses the degree to which dialect classifiers based on syntactic representations remain stable over space and time. While previous work has shown that the combination of grammar induction and geospatial text classification produces robust dialect models, we do not know what influence both changing grammars and changing populations have on dialect models. This paper constructs a test set for 12 dialects of English that spans three years at monthly intervals with a fixed spatial distribution across 1,120 cities. Syntactic representations are formulated within the usage-based Construction Grammar paradigm (CxG). The decay rate of classification performance for each dialect over time allows us to identify regions undergoing syntactic change. And the distribution of classification accuracy within dialect regions allows us to identify the degree to which the grammar of a dialect is internally heterogeneous. The main contribution of this paper is to show that a rigorous evaluation of dialect classification models can be used to find both variation over space and change over time.

The field of natural language processing (NLP) has witnessed significant progress in recent years, with a notable focus on improving large language models' (LLM) performance through innovative prompting techniques. Among these, prompt engineering coupled with structures has emerged as a promising paradigm, with designs such as Chain-of-Thought, Tree of Thoughts, or Graph of Thoughts, in which the overall LLM reasoning is guided by a structure such as a graph. As illustrated with numerous examples, this paradigm significantly enhances the LLM's capability to solve numerous tasks, ranging from logical or mathematical reasoning to planning or creative writing. To facilitate the understanding of this growing field and pave the way for future developments, we devise a general blueprint for effective and efficient LLM reasoning schemes. For this, we conduct an in-depth analysis of the prompt execution pipeline, clarifying and clearly defining different concepts. We then build the first taxonomy of structure-enhanced LLM reasoning schemes. We focus on identifying fundamental classes of harnessed structures, and we analyze the representations of these structures, algorithms executed with these structures, and many others. We refer to these structures as reasoning topologies, because their representation becomes to a degree spatial, as they are contained within the LLM context. Our study compares existing prompting schemes using the proposed taxonomy, discussing how certain design choices lead to different patterns in performance and cost. We also outline theoretical underpinnings, relationships between prompting and others parts of the LLM ecosystem such as knowledge bases, and the associated research challenges. Our work will help to advance future prompt engineering techniques.

Recent advances in deep learning methods such as LLMs and Diffusion models have created a need for improved quantization methods that can meet the computational demands of these modern architectures while maintaining accuracy. Towards this goal, we study the advantages of FP8 data formats for post-training quantization across 75 unique network architectures covering a wide range of tasks, including machine translation, language modeling, text generation, image classification, generation, and segmentation. We examine three different FP8 representations (E5M2, E4M3, and E3M4) to study the effects of varying degrees of trade-off between dynamic range and precision on model accuracy. Based on our extensive study, we developed a quantization workflow that generalizes across different network architectures. Our empirical results show that FP8 formats outperform INT8 in multiple aspects, including workload coverage (92.64% vs. 65.87%), model accuracy and suitability for a broader range of operations. Furthermore, our findings suggest that E4M3 is better suited for NLP models, whereas E3M4 performs marginally better than E4M3 on computer vision tasks. The code is publicly available on Intel Neural Compressor: https://github.com/intel/neural-compressor.

Image representations, from SIFT and Bag of Visual Words to Convolutional Neural Networks (CNNs), are a crucial component of almost any image understanding system. Nevertheless, our understanding of them remains limited. In this paper we conduct a direct analysis of the visual information contained in representations by asking the following question: given an encoding of an image, to which extent is it possible to reconstruct the image itself? To answer this question we contribute a general framework to invert representations. We show that this method can invert representations such as HOG and SIFT more accurately than recent alternatives while being applicable to CNNs too. We then use this technique to study the inverse of recent state-of-the-art CNN image representations for the first time. Among our findings, we show that several layers in CNNs retain photographically accurate information about the image, with different degrees of geometric and photometric invariance.

Recent progress in 3D scene understanding enables scalable learning of representations across large datasets of diverse scenes. As a consequence, generalization to unseen scenes and objects, rendering novel views from just a single or a handful of input images, and controllable scene generation that supports editing, is now possible. However, training jointly on a large number of scenes typically compromises rendering quality when compared to single-scene optimized models such as NeRFs. In this paper, we leverage recent progress in diffusion models to equip 3D scene representation learning models with the ability to render high-fidelity novel views, while retaining benefits such as object-level scene editing to a large degree. In particular, we propose DORSal, which adapts a video diffusion architecture for 3D scene generation conditioned on object-centric slot-based representations of scenes. On both complex synthetic multi-object scenes and on the real-world large-scale Street View dataset, we show that DORSal enables scalable neural rendering of 3D scenes with object-level editing and improves upon existing approaches.

This paper considers the learning of logical (Boolean) functions with focus on the generalization on the unseen (GOTU) setting, a strong case of out-of-distribution generalization. This is motivated by the fact that the rich combinatorial nature of data in certain reasoning tasks (e.g., arithmetic/logic) makes representative data sampling challenging, and learning successfully under GOTU gives a first vignette of an 'extrapolating' or 'reasoning' learner. We then study how different network architectures trained by (S)GD perform under GOTU and provide both theoretical and experimental evidence that for a class of network models including instances of Transformers, random features models, and diagonal linear networks, a min-degree-interpolator (MDI) is learned on the unseen. We also provide evidence that other instances with larger learning rates or mean-field networks reach leaky MDIs. These findings lead to two implications: (1) we provide an explanation to the length generalization problem (e.g., Anil et al. 2022); (2) we introduce a curriculum learning algorithm called Degree-Curriculum that learns monomials more efficiently by incrementing supports.

We provide a construction for categorical representation learning and introduce the foundations of "categorifier". The central theme in representation learning is the idea of everything to vector. Every object in a dataset S can be represented as a vector in R^n by an encoding map E: Obj(S)toR^n. More importantly, every morphism can be represented as a matrix E: Hom(S)toR^{n}_{n}. The encoding map E is generally modeled by a deep neural network. The goal of representation learning is to design appropriate tasks on the dataset to train the encoding map (assuming that an encoding is optimal if it universally optimizes the performance on various tasks). However, the latter is still a set-theoretic approach. The goal of the current article is to promote the representation learning to a new level via a category-theoretic approach. As a proof of concept, we provide an example of a text translator equipped with our technology, showing that our categorical learning model outperforms the current deep learning models by 17 times. The content of the current article is part of the recent US patent proposal (patent application number: 63110906).

Since the introduction of deep learning, a wide scope of representation properties, such as decorrelation, whitening, disentanglement, rank, isotropy, and mutual information, have been studied to improve the quality of representation. However, manipulating such properties can be challenging in terms of implementational effectiveness and general applicability. To address these limitations, we propose to regularize von Neumann entropy~(VNE) of representation. First, we demonstrate that the mathematical formulation of VNE is superior in effectively manipulating the eigenvalues of the representation autocorrelation matrix. Then, we demonstrate that it is widely applicable in improving state-of-the-art algorithms or popular benchmark algorithms by investigating domain-generalization, meta-learning, self-supervised learning, and generative models. In addition, we formally establish theoretical connections with rank, disentanglement, and isotropy of representation. Finally, we provide discussions on the dimension control of VNE and the relationship with Shannon entropy. Code is available at: https://github.com/jaeill/CVPR23-VNE.

Set representation has become ubiquitous in deep learning for modeling the inductive bias of neural networks that are insensitive to the input order. DeepSets is the most widely used neural network architecture for set representation. It involves embedding each set element into a latent space with dimension L, followed by a sum pooling to obtain a whole-set embedding, and finally mapping the whole-set embedding to the output. In this work, we investigate the impact of the dimension L on the expressive power of DeepSets. Previous analyses either oversimplified high-dimensional features to be one-dimensional features or were limited to analytic activations, thereby diverging from practical use or resulting in L that grows exponentially with the set size N and feature dimension D. To investigate the minimal value of L that achieves sufficient expressive power, we present two set-element embedding layers: (a) linear + power activation (LP) and (b) linear + exponential activations (LE). We demonstrate that L being poly(N, D) is sufficient for set representation using both embedding layers. We also provide a lower bound of L for the LP embedding layer. Furthermore, we extend our results to permutation-equivariant set functions and the complex field.

Implicit neural representations (INR) have gained significant popularity for signal and image representation for many end-tasks, such as superresolution, 3D modeling, and more. Most INR architectures rely on sinusoidal positional encoding, which accounts for high-frequency information in data. However, the finite encoding size restricts the model's representational power. Higher representational power is needed to go from representing a single given image to representing large and diverse datasets. Our approach addresses this gap by representing an image with a polynomial function and eliminates the need for positional encodings. Therefore, to achieve a progressively higher degree of polynomial representation, we use element-wise multiplications between features and affine-transformed coordinate locations after every ReLU layer. The proposed method is evaluated qualitatively and quantitatively on large datasets like ImageNet. The proposed Poly-INR model performs comparably to state-of-the-art generative models without any convolution, normalization, or self-attention layers, and with far fewer trainable parameters. With much fewer training parameters and higher representative power, our approach paves the way for broader adoption of INR models for generative modeling tasks in complex domains. The code is available at https://github.com/Rajhans0/Poly_INR

One method for obtaining generalizable solutions to machine learning tasks when presented with diverse training environments is to find invariant representations of the data. These are representations of the covariates such that the best model on top of the representation is invariant across training environments. In the context of linear Structural Equation Models (SEMs), invariant representations might allow us to learn models with out-of-distribution guarantees, i.e., models that are robust to interventions in the SEM. To address the invariant representation problem in a {\em finite sample} setting, we consider the notion of epsilon-approximate invariance. We study the following question: If a representation is approximately invariant with respect to a given number of training interventions, will it continue to be approximately invariant on a larger collection of unseen SEMs? This larger collection of SEMs is generated through a parameterized family of interventions. Inspired by PAC learning, we obtain finite-sample out-of-distribution generalization guarantees for approximate invariance that holds probabilistically over a family of linear SEMs without faithfulness assumptions. Our results show bounds that do not scale in ambient dimension when intervention sites are restricted to lie in a constant size subset of in-degree bounded nodes. We also show how to extend our results to a linear indirect observation model that incorporates latent variables.

We consider concept generalization at a large scale in the diverse and natural visual spectrum. Established computational modes (i.e., rule-based or similarity-based) are primarily studied isolated and focus on confined and abstract problem spaces. In this work, we study these two modes when the problem space scales up, and the complexity of concepts becomes diverse. Specifically, at the representational level, we seek to answer how the complexity varies when a visual concept is mapped to the representation space. Prior psychology literature has shown that two types of complexities (i.e., subjective complexity and visual complexity) (Griffiths and Tenenbaum, 2003) build an inverted-U relation (Donderi, 2006; Sun and Firestone, 2021). Leveraging Representativeness of Attribute (RoA), we computationally confirm the following observation: Models use attributes with high RoA to describe visual concepts, and the description length falls in an inverted-U relation with the increment in visual complexity. At the computational level, we aim to answer how the complexity of representation affects the shift between the rule- and similarity-based generalization. We hypothesize that category-conditioned visual modeling estimates the co-occurrence frequency between visual and categorical attributes, thus potentially serving as the prior for the natural visual world. Experimental results show that representations with relatively high subjective complexity outperform those with relatively low subjective complexity in the rule-based generalization, while the trend is the opposite in the similarity-based generalization.

This paper reformulates a classical result in probability theory from the 1930s in modern categorical terms: de Finetti's representation theorem is redescribed as limit statement for a chain of finite spaces in the Kleisli category of the Giry monad. This new limit is used to identify among exchangeable coalgebras the final one.

We prove a new upper bound on the generalization gap of classifiers that are obtained by first using self-supervision to learn a representation r of the training data, and then fitting a simple (e.g., linear) classifier g to the labels. Specifically, we show that (under the assumptions described below) the generalization gap of such classifiers tends to zero if C(g) ll n, where C(g) is an appropriately-defined measure of the simple classifier g's complexity, and n is the number of training samples. We stress that our bound is independent of the complexity of the representation r. We do not make any structural or conditional-independence assumptions on the representation-learning task, which can use the same training dataset that is later used for classification. Rather, we assume that the training procedure satisfies certain natural noise-robustness (adding small amount of label noise causes small degradation in performance) and rationality (getting the wrong label is not better than getting no label at all) conditions that widely hold across many standard architectures. We show that our bound is non-vacuous for many popular representation-learning based classifiers on CIFAR-10 and ImageNet, including SimCLR, AMDIM and MoCo.

We study the matrix models pi:C(S_N^+)to M_N(C(X)) which are flat, in the sense that the standard generators of C(S_N^+) are mapped to rank 1 projections. Our first result is a generalization of the Pauli matrix construction at N=4, using finite groups and 2-cocycles. Our second result is the construction of a universal representation of C(S_N^+), inspired from the Sinkhorn algorithm, that we conjecture to be inner faithful.

We develop techniques to quantify the degree to which a given (training or testing) example is an outlier in the underlying distribution. We evaluate five methods to score examples in a dataset by how well-represented the examples are, for different plausible definitions of "well-represented", and apply these to four common datasets: MNIST, Fashion-MNIST, CIFAR-10, and ImageNet. Despite being independent approaches, we find all five are highly correlated, suggesting that the notion of being well-represented can be quantified. Among other uses, we find these methods can be combined to identify (a) prototypical examples (that match human expectations); (b) memorized training examples; and, (c) uncommon submodes of the dataset. Further, we show how we can utilize our metrics to determine an improved ordering for curriculum learning, and impact adversarial robustness. We release all metric values on training and test sets we studied.

Recently, many works studied the expressive power of graph neural networks (GNNs) by linking it to the 1-dimensional Weisfeiler--Leman algorithm (1-WL). Here, the 1-WL is a well-studied heuristic for the graph isomorphism problem, which iteratively colors or partitions a graph's vertex set. While this connection has led to significant advances in understanding and enhancing GNNs' expressive power, it does not provide insights into their generalization performance, i.e., their ability to make meaningful predictions beyond the training set. In this paper, we study GNNs' generalization ability through the lens of Vapnik--Chervonenkis (VC) dimension theory in two settings, focusing on graph-level predictions. First, when no upper bound on the graphs' order is known, we show that the bitlength of GNNs' weights tightly bounds their VC dimension. Further, we derive an upper bound for GNNs' VC dimension using the number of colors produced by the 1-WL. Secondly, when an upper bound on the graphs' order is known, we show a tight connection between the number of graphs distinguishable by the 1-WL and GNNs' VC dimension. Our empirical study confirms the validity of our theoretical findings.

We show that deep neural networks (DNNs) can efficiently learn any composition of functions with bounded F_{1}-norm, which allows DNNs to break the curse of dimensionality in ways that shallow networks cannot. More specifically, we derive a generalization bound that combines a covering number argument for compositionality, and the F_{1}-norm (or the related Barron norm) for large width adaptivity. We show that the global minimizer of the regularized loss of DNNs can fit for example the composition of two functions f^{*}=hcirc g from a small number of observations, assuming g is smooth/regular and reduces the dimensionality (e.g. g could be the modulo map of the symmetries of f^{*}), so that h can be learned in spite of its low regularity. The measures of regularity we consider is the Sobolev norm with different levels of differentiability, which is well adapted to the F_{1} norm. We compute scaling laws empirically and observe phase transitions depending on whether g or h is harder to learn, as predicted by our theory.

This paper investigates discrepancies in how neural networks learn from different imaging domains, which are commonly overlooked when adopting computer vision techniques from the domain of natural images to other specialized domains such as medical images. Recent works have found that the generalization error of a trained network typically increases with the intrinsic dimension (d_{data}) of its training set. Yet, the steepness of this relationship varies significantly between medical (radiological) and natural imaging domains, with no existing theoretical explanation. We address this gap in knowledge by establishing and empirically validating a generalization scaling law with respect to d_{data}, and propose that the substantial scaling discrepancy between the two considered domains may be at least partially attributed to the higher intrinsic ``label sharpness'' (K_F) of medical imaging datasets, a metric which we propose. Next, we demonstrate an additional benefit of measuring the label sharpness of a training set: it is negatively correlated with the trained model's adversarial robustness, which notably leads to models for medical images having a substantially higher vulnerability to adversarial attack. Finally, we extend our d_{data} formalism to the related metric of learned representation intrinsic dimension (d_{repr}), derive a generalization scaling law with respect to d_{repr}, and show that d_{data} serves as an upper bound for d_{repr}. Our theoretical results are supported by thorough experiments with six models and eleven natural and medical imaging datasets over a range of training set sizes. Our findings offer insights into the influence of intrinsic dataset properties on generalization, representation learning, and robustness in deep neural networks. Code link: https://github.com/mazurowski-lab/intrinsic-properties

The Linear Representation Hypothesis (LRH) states that neural networks learn to encode concepts as directions in activation space, and a strong version of the LRH states that models learn only such encodings. In this paper, we present a counterexample to this strong LRH: when trained to repeat an input token sequence, gated recurrent neural networks (RNNs) learn to represent the token at each position with a particular order of magnitude, rather than a direction. These representations have layered features that are impossible to locate in distinct linear subspaces. To show this, we train interventions to predict and manipulate tokens by learning the scaling factor corresponding to each sequence position. These interventions indicate that the smallest RNNs find only this magnitude-based solution, while larger RNNs have linear representations. These findings strongly indicate that interpretability research should not be confined by the LRH.

We study the realizability of simplicial complexes with a given pair of integer sequences, representing the node degree distribution and the facet size distribution, respectively. While the s-uniform variant of the problem is NP-complete when s geq 3, we identify two populations of input sequences, most of which can be solved in polynomial time using a recursive algorithm that we contribute. Combining with a sampler for the simplicial configuration model [J.-G. Young et al., Phys. Rev. E 96, 032312 (2017)], we facilitate the efficient sampling of simplicial ensembles from arbitrary degree and size distributions. We find that, contrary to expectations based on dyadic networks, increasing the nodes' degrees reduces the number of loops in simplicial complexes. Our work unveils a fundamental constraint on the degree-size sequences and sheds light on further analysis of higher-order phenomena based on local structures.

State-of-the-art methods for conditional average treatment effect (CATE) estimation make widespread use of representation learning. Here, the idea is to reduce the variance of the low-sample CATE estimation by a (potentially constrained) low-dimensional representation. However, low-dimensional representations can lose information about the observed confounders and thus lead to bias, because of which the validity of representation learning for CATE estimation is typically violated. In this paper, we propose a new, representation-agnostic framework for estimating bounds on the representation-induced confounding bias that comes from dimensionality reduction (or other constraints on the representations) in CATE estimation. First, we establish theoretically under which conditions CATEs are non-identifiable given low-dimensional (constrained) representations. Second, as our remedy, we propose to perform partial identification of CATEs or, equivalently, aim at estimating of lower and upper bounds of the representation-induced confounding bias. We demonstrate the effectiveness of our bounds in a series of experiments. In sum, our framework is of direct relevance in practice where the validity of CATE estimation is of importance.

A central question for neuroscience is how to characterize brain representations of perceptual and cognitive content. An ideal characterization should distinguish different functional regions with robustness to noise and idiosyncrasies of individual brains that do not correspond to computational differences. Previous studies have characterized brain representations by their representational geometry, which is defined by the representational dissimilarity matrix (RDM), a summary statistic that abstracts from the roles of individual neurons (or responses channels) and characterizes the discriminability of stimuli. Here we explore a further step of abstraction: from the geometry to the topology of brain representations. We propose topological representational similarity analysis (tRSA), an extension of representational similarity analysis (RSA) that uses a family of geo-topological summary statistics that generalizes the RDM to characterize the topology while de-emphasizing the geometry. We evaluate this new family of statistics in terms of the sensitivity and specificity for model selection using both simulations and functional MRI (fMRI) data. In the simulations, the ground truth is a data-generating layer representation in a neural network model and the models are the same and other layers in different model instances (trained from different random seeds). In fMRI, the ground truth is a visual area and the models are the same and other areas measured in different subjects. Results show that topology-sensitive characterizations of population codes are robust to noise and interindividual variability and maintain excellent sensitivity to the unique representational signatures of different neural network layers and brain regions.

We introduce a novel method for representation learning that uses an artificial supervision signal based on counting visual primitives. This supervision signal is obtained from an equivariance relation, which does not require any manual annotation. We relate transformations of images to transformations of the representations. More specifically, we look for the representation that satisfies such relation rather than the transformations that match a given representation. In this paper, we use two image transformations in the context of counting: scaling and tiling. The first transformation exploits the fact that the number of visual primitives should be invariant to scale. The second transformation allows us to equate the total number of visual primitives in each tile to that in the whole image. These two transformations are combined in one constraint and used to train a neural network with a contrastive loss. The proposed task produces representations that perform on par or exceed the state of the art in transfer learning benchmarks.

We argue that representations in AI models, particularly deep networks, are converging. First, we survey many examples of convergence in the literature: over time and across multiple domains, the ways by which different neural networks represent data are becoming more aligned. Next, we demonstrate convergence across data modalities: as vision models and language models get larger, they measure distance between datapoints in a more and more alike way. We hypothesize that this convergence is driving toward a shared statistical model of reality, akin to Plato's concept of an ideal reality. We term such a representation the platonic representation and discuss several possible selective pressures toward it. Finally, we discuss the implications of these trends, their limitations, and counterexamples to our analysis.

Humans view the world through many sensory channels, e.g., the long-wavelength light channel, viewed by the left eye, or the high-frequency vibrations channel, heard by the right ear. Each view is noisy and incomplete, but important factors, such as physics, geometry, and semantics, tend to be shared between all views (e.g., a "dog" can be seen, heard, and felt). We investigate the classic hypothesis that a powerful representation is one that models view-invariant factors. We study this hypothesis under the framework of multiview contrastive learning, where we learn a representation that aims to maximize mutual information between different views of the same scene but is otherwise compact. Our approach scales to any number of views, and is view-agnostic. We analyze key properties of the approach that make it work, finding that the contrastive loss outperforms a popular alternative based on cross-view prediction, and that the more views we learn from, the better the resulting representation captures underlying scene semantics. Our approach achieves state-of-the-art results on image and video unsupervised learning benchmarks. Code is released at: http://github.com/HobbitLong/CMC/.

Understanding the internal representations learned by neural networks is a cornerstone challenge in the science of machine learning. While there have been significant recent strides in some cases towards understanding how neural networks implement specific target functions, this paper explores a complementary question -- why do networks arrive at particular computational strategies? Our inquiry focuses on the algebraic learning tasks of modular addition, sparse parities, and finite group operations. Our primary theoretical findings analytically characterize the features learned by stylized neural networks for these algebraic tasks. Notably, our main technique demonstrates how the principle of margin maximization alone can be used to fully specify the features learned by the network. Specifically, we prove that the trained networks utilize Fourier features to perform modular addition and employ features corresponding to irreducible group-theoretic representations to perform compositions in general groups, aligning closely with the empirical observations of Nanda et al. and Chughtai et al. More generally, we hope our techniques can help to foster a deeper understanding of why neural networks adopt specific computational strategies.

Representational analysis explores how input data of a neural system are encoded in high dimensional spaces of its distributed neural activations, and how we can compare different systems, for instance, artificial neural networks and brains, on those grounds. While existing methods offer important insights, they typically do not account for local intrinsic geometrical properties within the high-dimensional representation spaces. To go beyond these limitations, we explore Ollivier-Ricci curvature and Ricci flow as tools to study the alignment of representations between humans and artificial neural systems on a geometric level. As a proof-of-principle study, we compared the representations of face stimuli between VGG-Face, a human-aligned version of VGG-Face, and corresponding human similarity judgments from a large online study. Using this discrete geometric framework, we were able to identify local structural similarities and differences by examining the distributions of node and edge curvature and higher-level properties by detecting and comparing community structure in the representational graphs.

Representation learning, and interpreting learned representations, are key areas of focus in machine learning and neuroscience. Both fields generally use representations as a means to understand or improve a system's computations. In this work, however, we explore surprising dissociations between representation and computation that may pose challenges for such efforts. We create datasets in which we attempt to match the computational role that different features play, while manipulating other properties of the features or the data. We train various deep learning architectures to compute these multiple abstract features about their inputs. We find that their learned feature representations are systematically biased towards representing some features more strongly than others, depending upon extraneous properties such as feature complexity, the order in which features are learned, and the distribution of features over the inputs. For example, features that are simpler to compute or learned first tend to be represented more strongly and densely than features that are more complex or learned later, even if all features are learned equally well. We also explore how these biases are affected by architectures, optimizers, and training regimes (e.g., in transformers, features decoded earlier in the output sequence also tend to be represented more strongly). Our results help to characterize the inductive biases of gradient-based representation learning. These results also highlight a key challenge for interpretability - or for comparing the representations of models and brains - disentangling extraneous biases from the computationally important aspects of a system's internal representations.

We study how multilingual sentence representations capture European countries and occupations and how this differs across European languages. We prompt the models with templated sentences that we machine-translate into 12 European languages and analyze the most prominent dimensions in the embeddings.Our analysis reveals that the most prominent feature in the embedding is the geopolitical distinction between Eastern and Western Europe and the country's economic strength in terms of GDP. When prompted specifically for job prestige, the embedding space clearly distinguishes high and low-prestige jobs. The occupational dimension is uncorrelated with the most dominant country dimensions in three out of four studied models. The exception is a small distilled model that exhibits a connection between occupational prestige and country of origin, which is a potential source of nationality-based discrimination. Our findings are consistent across languages.

Causal representation learning has showed a variety of settings in which we can disentangle latent variables with identifiability guarantees (up to some reasonable equivalence class). Common to all of these approaches is the assumption that (1) the latent variables are represented as d-dimensional vectors, and (2) that the observations are the output of some injective generative function of these latent variables. While these assumptions appear benign, we show that when the observations are of multiple objects, the generative function is no longer injective and disentanglement fails in practice. We can address this failure by combining recent developments in object-centric learning and causal representation learning. By modifying the Slot Attention architecture arXiv:2006.15055, we develop an object-centric architecture that leverages weak supervision from sparse perturbations to disentangle each object's properties. This approach is more data-efficient in the sense that it requires significantly fewer perturbations than a comparable approach that encodes to a Euclidean space and we show that this approach successfully disentangles the properties of a set of objects in a series of simple image-based disentanglement experiments.

Informally, the 'linear representation hypothesis' is the idea that high-level concepts are represented linearly as directions in some representation space. In this paper, we address two closely related questions: What does "linear representation" actually mean? And, how do we make sense of geometric notions (e.g., cosine similarity or projection) in the representation space? To answer these, we use the language of counterfactuals to give two formalizations of "linear representation", one in the output (word) representation space, and one in the input (sentence) space. We then prove these connect to linear probing and model steering, respectively. To make sense of geometric notions, we use the formalization to identify a particular (non-Euclidean) inner product that respects language structure in a sense we make precise. Using this causal inner product, we show how to unify all notions of linear representation. In particular, this allows the construction of probes and steering vectors using counterfactual pairs. Experiments with LLaMA-2 demonstrate the existence of linear representations of concepts, the connection to interpretation and control, and the fundamental role of the choice of inner product.

Information Retrieval using dense low-dimensional representations recently became popular and showed out-performance to traditional sparse-representations like BM25. However, no previous work investigated how dense representations perform with large index sizes. We show theoretically and empirically that the performance for dense representations decreases quicker than sparse representations for increasing index sizes. In extreme cases, this can even lead to a tipping point where at a certain index size sparse representations outperform dense representations. We show that this behavior is tightly connected to the number of dimensions of the representations: The lower the dimension, the higher the chance for false positives, i.e. returning irrelevant documents.

Why do brains have inhibitory connections? Why do deep networks have negative weights? We propose an answer from the perspective of representation capacity. We believe representing functions is the primary role of both (i) the brain in natural intelligence, and (ii) deep networks in artificial intelligence. Our answer to why there are inhibitory/negative weights is: to learn more functions. We prove that, in the absence of negative weights, neural networks with non-decreasing activation functions are not universal approximators. While this may be an intuitive result to some, to the best of our knowledge, there is no formal theory, in either machine learning or neuroscience, that demonstrates why negative weights are crucial in the context of representation capacity. Further, we provide insights on the geometric properties of the representation space that non-negative deep networks cannot represent. We expect these insights will yield a deeper understanding of more sophisticated inductive priors imposed on the distribution of weights that lead to more efficient biological and machine learning.

Small sample sizes are common in many disciplines, which necessitates pooling roughly similar datasets across multiple institutions to study weak but relevant associations between images and disease outcomes. Such data often manifest shift/imbalance in covariates (i.e., secondary non-imaging data). Controlling for such nuisance variables is common within standard statistical analysis, but the ideas do not directly apply to overparameterized models. Consequently, recent work has shown how strategies from invariant representation learning provides a meaningful starting point, but the current repertoire of methods is limited to accounting for shifts/imbalances in just a couple of covariates at a time. In this paper, we show how viewing this problem from the perspective of Category theory provides a simple and effective solution that completely avoids elaborate multi-stage training pipelines that would otherwise be needed. We show the effectiveness of this approach via extensive experiments on real datasets. Further, we discuss how this style of formulation offers a unified perspective on at least 5+ distinct problem settings, from self-supervised learning to matching problems in 3D reconstruction.

Isogenies occur throughout the theory of elliptic curves. Recently, the cryptographic protocols based on isogenies are considered as candidates of quantum-resistant cryptographic protocols. Given two elliptic curves E_1, E_2 defined over a finite field k with the same trace, there is a nonconstant isogeny beta from E_2 to E_1 defined over k. This study gives out the index of Hom_{it k}(it E_{rm 1},E_{rm 2})beta as a left ideal in End_{it k}(it E_{rm 2}) and figures out the correspondence between isogenies and kernel ideals. In addition, some results about the non-trivial minimal degree of isogenies between the two elliptic curves are also provided.

Does the dominant approach to learn representations (as a side effect of optimizing an expected cost for a single training distribution) remain a good approach when we are dealing with multiple distributions? Our thesis is that such scenarios are better served by representations that are richer than those obtained with a single optimization episode. We support this thesis with simple theoretical arguments and with experiments utilizing an apparently na\"{\i}ve ensembling technique: concatenating the representations obtained from multiple training episodes using the same data, model, algorithm, and hyper-parameters, but different random seeds. These independently trained networks perform similarly. Yet, in a number of scenarios involving new distributions, the concatenated representation performs substantially better than an equivalently sized network trained with a single training run. This proves that the representations constructed by multiple training episodes are in fact different. Although their concatenation carries little additional information about the training task under the training distribution, it becomes substantially more informative when tasks or distributions change. Meanwhile, a single training episode is unlikely to yield such a redundant representation because the optimization process has no reason to accumulate features that do not incrementally improve the training performance.

Universality is a key hypothesis in mechanistic interpretability -- that different models learn similar features and circuits when trained on similar tasks. In this work, we study the universality hypothesis by examining how small neural networks learn to implement group composition. We present a novel algorithm by which neural networks may implement composition for any finite group via mathematical representation theory. We then show that networks consistently learn this algorithm by reverse engineering model logits and weights, and confirm our understanding using ablations. By studying networks of differing architectures trained on various groups, we find mixed evidence for universality: using our algorithm, we can completely characterize the family of circuits and features that networks learn on this task, but for a given network the precise circuits learned -- as well as the order they develop -- are arbitrary.

The input space of a neural network with ReLU-like activations is partitioned into multiple linear regions, each corresponding to a specific activation pattern of the included ReLU-like activations. We demonstrate that this partition exhibits the following encoding properties across a variety of deep learning models: (1) {\it determinism}: almost every linear region contains at most one training example. We can therefore represent almost every training example by a unique activation pattern, which is parameterized by a {\it neural code}; and (2) {\it categorization}: according to the neural code, simple algorithms, such as K-Means, K-NN, and logistic regression, can achieve fairly good performance on both training and test data. These encoding properties surprisingly suggest that {\it normal neural networks well-trained for classification behave as hash encoders without any extra efforts.} In addition, the encoding properties exhibit variability in different scenarios. {Further experiments demonstrate that {\it model size}, {\it training time}, {\it training sample size}, {\it regularization}, and {\it label noise} contribute in shaping the encoding properties, while the impacts of the first three are dominant.} We then define an {\it activation hash phase chart} to represent the space expanded by {model size}, training time, training sample size, and the encoding properties, which is divided into three canonical regions: {\it under-expressive regime}, {\it critically-expressive regime}, and {\it sufficiently-expressive regime}. The source code package is available at https://github.com/LeavesLei/activation-code.