Daily Papers

Label noise is one of the key factors that lead to the poor generalization of deep learning models. Existing label-noise learning methods usually assume that the ground-truth classes of the training data are balanced. However, the real-world data is often imbalanced, leading to the inconsistency between observed and intrinsic class distribution with label noises. In this case, it is hard to distinguish clean samples from noisy samples on the intrinsic tail classes with the unknown intrinsic class distribution. In this paper, we propose a learning framework for label-noise learning with intrinsically long-tailed data. Specifically, we propose two-stage bi-dimensional sample selection (TABASCO) to better separate clean samples from noisy samples, especially for the tail classes. TABASCO consists of two new separation metrics that complement each other to compensate for the limitation of using a single metric in sample separation. Extensive experiments on benchmarks demonstrate the effectiveness of our method. Our code is available at https://github.com/Wakings/TABASCO.

Large vision language models (LVLMs) often suffer from object hallucination, producing objects not present in the given images. While current benchmarks for object hallucination primarily concentrate on the presence of a single object class rather than individual entities, this work systematically investigates multi-object hallucination, examining how models misperceive (e.g., invent nonexistent objects or become distracted) when tasked with focusing on multiple objects simultaneously. We introduce Recognition-based Object Probing Evaluation (ROPE), an automated evaluation protocol that considers the distribution of object classes within a single image during testing and uses visual referring prompts to eliminate ambiguity. With comprehensive empirical studies and analysis of potential factors leading to multi-object hallucination, we found that (1) LVLMs suffer more hallucinations when focusing on multiple objects compared to a single object. (2) The tested object class distribution affects hallucination behaviors, indicating that LVLMs may follow shortcuts and spurious correlations.(3) Hallucinatory behaviors are influenced by data-specific factors, salience and frequency, and model intrinsic behaviors. We hope to enable LVLMs to recognize and reason about multiple objects that often occur in realistic visual scenes, provide insights, and quantify our progress towards mitigating the issues.

Collections of probability distributions arise in a variety of applications ranging from user activity pattern analysis to brain connectomics. In practice these distributions can be defined over diverse domain types including finite intervals, circles, cylinders, spheres, other manifolds, and graphs. This paper introduces an approach for detecting differences between two collections of distributions over such general domains. To this end, we propose the intrinsic slicing construction that yields a novel class of Wasserstein distances on manifolds and graphs. These distances are Hilbert embeddable, allowing us to reduce the distribution collection comparison problem to a more familiar mean testing problem in a Hilbert space. We provide two testing procedures one based on resampling and another on combining p-values from coordinate-wise tests. Our experiments in various synthetic and real data settings show that the resulting tests are powerful and the p-values are well-calibrated.

We investigate statistical properties of a likelihood approach to nonparametric estimation of a singular distribution using deep generative models. More specifically, a deep generative model is used to model high-dimensional data that are assumed to concentrate around some low-dimensional structure. Estimating the distribution supported on this low-dimensional structure, such as a low-dimensional manifold, is challenging due to its singularity with respect to the Lebesgue measure in the ambient space. In the considered model, a usual likelihood approach can fail to estimate the target distribution consistently due to the singularity. We prove that a novel and effective solution exists by perturbing the data with an instance noise, which leads to consistent estimation of the underlying distribution with desirable convergence rates. We also characterize the class of distributions that can be efficiently estimated via deep generative models. This class is sufficiently general to contain various structured distributions such as product distributions, classically smooth distributions and distributions supported on a low-dimensional manifold. Our analysis provides some insights on how deep generative models can avoid the curse of dimensionality for nonparametric distribution estimation. We conduct a thorough simulation study and real data analysis to empirically demonstrate that the proposed data perturbation technique improves the estimation performance significantly.

Feature attribution methods identify which features of an input most influence a model's output. Most widely-used feature attribution methods (such as SHAP, LIME, and Grad-CAM) are "class-dependent" methods in that they generate a feature attribution vector as a function of class. In this work, we demonstrate that class-dependent methods can "leak" information about the selected class, making that class appear more likely than it is. Thus, an end user runs the risk of drawing false conclusions when interpreting an explanation generated by a class-dependent method. In contrast, we introduce "distribution-aware" methods, which favor explanations that keep the label's distribution close to its distribution given all features of the input. We introduce SHAP-KL and FastSHAP-KL, two baseline distribution-aware methods that compute Shapley values. Finally, we perform a comprehensive evaluation of seven class-dependent and three distribution-aware methods on three clinical datasets of different high-dimensional data types: images, biosignals, and text.

In the real world, the frequency of occurrence of objects is naturally skewed forming long-tail class distributions, which results in poor performance on the statistically rare classes. A promising solution is to mine tail-class examples to balance the training dataset. However, mining tail-class examples is a very challenging task. For instance, most of the otherwise successful uncertainty-based mining approaches struggle due to distortion of class probabilities resulting from skewness in data. In this work, we propose an effective, yet simple, approach to overcome these challenges. Our framework enhances the subdued tail-class activations and, thereafter, uses a one-class data-centric approach to effectively identify tail-class examples. We carry out an exhaustive evaluation of our framework on three datasets spanning over two computer vision tasks. Substantial improvements in the minority-class mining and fine-tuned model's performance strongly corroborate the value of our proposed solution.

Real-world datasets follow an imbalanced distribution, which poses significant challenges in rare-category object detection. Recent studies tackle this problem by developing re-weighting and re-sampling methods, that utilise the class frequencies of the dataset. However, these techniques focus solely on the frequency statistics and ignore the distribution of the classes in image space, missing important information. In contrast to them, we propose FRActal CALibration (FRACAL): a novel post-calibration method for long-tailed object detection. FRACAL devises a logit adjustment method that utilises the fractal dimension to estimate how uniformly classes are distributed in image space. During inference, it uses the fractal dimension to inversely downweight the probabilities of uniformly spaced class predictions achieving balance in two axes: between frequent and rare categories, and between uniformly spaced and sparsely spaced classes. FRACAL is a post-processing method and it does not require any training, also it can be combined with many off-the-shelf models such as one-stage sigmoid detectors and two-stage instance segmentation models. FRACAL boosts the rare class performance by up to 8.6% and surpasses all previous methods on LVIS dataset, while showing good generalisation to other datasets such as COCO, V3Det and OpenImages. We provide the code at https://github.com/kostas1515/FRACAL.

Most applications of machine learning to classification assume a closed set of balanced classes. This is at odds with the real world, where class occurrence statistics often follow a long-tailed power-law distribution and it is unlikely that all classes are seen in a single sample. Nonparametric Bayesian models naturally capture this phenomenon, but have significant practical barriers to widespread adoption, namely implementation complexity and computational inefficiency. To address this, we present a method for extracting the inductive bias from a nonparametric Bayesian model and transferring it to an artificial neural network. By simulating data with a nonparametric Bayesian prior, we can metalearn a sequence model that performs inference over an unlimited set of classes. After training, this "neural circuit" has distilled the corresponding inductive bias and can successfully perform sequential inference over an open set of classes. Our experimental results show that the metalearned neural circuit achieves comparable or better performance than particle filter-based methods for inference in these models while being faster and simpler to use than methods that explicitly incorporate Bayesian nonparametric inference.

We study the problem of predicting numeric labels that are constrained to the integers or to a subrange of the integers. For example, the number of up-votes on social media posts, or the number of bicycles available at a public rental station. While it is possible to model these as continuous values, and to apply traditional regression, this approach changes the underlying distribution on the labels from discrete to continuous. Discrete distributions have certain benefits, which leads us to the question whether such integer labels can be modeled directly by a discrete distribution, whose parameters are predicted from the features of a given instance. Moreover, we focus on the use case of output distributions of neural networks, which adds the requirement that the parameters of the distribution be continuous so that backpropagation and gradient descent may be used to learn the weights of the network. We investigate several options for such distributions, some existing and some novel, and test them on a range of tasks, including tabular learning, sequential prediction and image generation. We find that overall the best performance comes from two distributions: Bitwise, which represents the target integer in bits and places a Bernoulli distribution on each, and a discrete analogue of the Laplace distribution, which uses a distribution with exponentially decaying tails around a continuous mean.

This paper explores the problem of class-agnostic anomaly detection (AD), where the objective is to train one class-agnostic AD model that can generalize to detect anomalies in diverse new classes from different domains without any retraining or fine-tuning on the target data. When applied for new classes, the performance of current single- and multi-class AD methods is still unsatisfactory. One fundamental reason is that representation learning in existing methods is still class-related, namely, feature correlation. To address this issue, we propose residual features and construct a simple but effective framework, termed ResAD. Our core insight is to learn the residual feature distribution rather than the initial feature distribution. Residual features are formed by matching and then subtracting normal reference features. In this way, we can effectively realize feature decorrelation. Even in new classes, the distribution of normal residual features would not remarkably shift from the learned distribution. In addition, we think that residual features still have one issue: scale correlation. To this end, we propose a feature hypersphere constraining approach, which learns to constrain initial normal residual features into a spatial hypersphere for enabling the feature scales of different classes as consistent as possible. Furthermore, we propose a novel logbarrier bidirectional contraction OCC loss and vector quantization based feature distribution matching module to enhance ResAD, leading to the improved version of ResAD (ResAD++). Comprehensive experiments on eight real-world AD datasets demonstrate that our ResAD++ can achieve remarkable AD results when directly used in new classes, outperforming state-of-the-art competing methods and also surpassing ResAD. The code is available at https://github.com/xcyao00/ResAD.

Class imbalance (CI) is a longstanding problem in machine learning, slowing down training and reducing performances. Although empirical remedies exist, it is often unclear which ones work best and when, due to the lack of an overarching theory. We address a common case of imbalance, that of anomaly (or outlier) detection. We provide a theoretical framework to analyze, interpret and address CI. It is based on an exact solution of the teacher-student perceptron model, through replica theory. Within this framework, one can distinguish several sources of CI: either intrinsic, train or test imbalance. Our analysis reveals that the optimal train imbalance is generally different from 50%, with a non trivial dependence on the intrinsic imbalance, the abundance of data and on the noise in the learning. Moreover, there is a crossover between a small noise training regime where results are independent of the noise level to a high noise regime where performances quickly degrade with noise. Our results challenge some of the conventional wisdom on CI and offer practical guidelines to address it.

Although deep neural networks achieve tremendous success on various classification tasks, the generalization ability drops sheer when training datasets exhibit long-tailed distributions. One of the reasons is that the learned representations (i.e. features) from the imbalanced datasets are less effective than those from balanced datasets. Specifically, the learned representation under class-balanced distribution will present the Neural Collapse (NC) phenomena. NC indicates the features from the same category are close to each other and from different categories are maximally distant, showing an optimal linear separable state of classification. However, the pattern differs on imbalanced datasets and is partially responsible for the reduced performance of the model. In this work, we propose two explicit feature regularization terms to learn high-quality representation for class-imbalanced data. With the proposed regularization, NC phenomena will appear under the class-imbalanced distribution, and the generalization ability can be significantly improved. Our method is easily implemented, highly effective, and can be plugged into most existing methods. The extensive experimental results on widely-used benchmarks show the effectiveness of our method

We apply the Weibull distribution -- a two-parameter family from extreme-value theory -- as a diagnostic framework for element-wise weight magnitude distributions in transformers. At initialization, i.i.d. Gaussian weights give |w| ~ HalfNormal, yielding k ~ 1.20 via middle-80% probability-plot fit (the protocol used throughout this work). This anchor makes k a principled, architecture-independent measuring stick for training dynamics; fitting each weight matrix independently at every layer at every checkpoint enables per-component, per-layer, and per-step diagnostics that aggregate statistics cannot resolve. Applying this framework to 12 model entries spanning 7 architectural families (Pythia, OLMo-1/2, LLaMA-3, Mistral, Qwen2.5/3) reveals three findings. First, FFN modules and the attention output projection W_o -- the Transmission Class -- fall in a narrow k band: median terminal k in [1.186, 1.204] across 12 entries (cross-family CV = 0.51%), shared across SwiGLU/GeLU activations, Pre-LN/QK-Norm placements, and 70M-14B sizes. Second, the attention input projections W_q, W_k -- the Selection Class -- depart from the Weibull family, with severity shaped by storage: separately-stored Q/K (OLMo-1, OLMo-2) yields k in [0.76, 0.99] (deep); GQA models yield k in [1.10, 1.16] (mild); Pythia's merged W_qkv occupies a transitional zone tracking training budget T/tau monotonically. Third, lambda grows substantially during training and scales with sqrt(eta/lambda_wd) within the Pythia family (Pearson r = 0.94, three Transmission kinds), directionally consistent with Fan et al. (2025). The two parameters carry independent information: k labels the functional class, lambda labels training progress. We release npm-weibull-py v0.4 (Python library) and DATABASE_v9_1 at https://github.com/tiexinding/NPM-Weibull-public .

Class-wise characteristics of training examples affect the performance of deep classifiers. A well-studied example is when the number of training examples of classes follows a long-tailed distribution, a situation that is likely to yield sub-optimal performance for under-represented classes. This class imbalance problem is conventionally addressed by approaches relying on the class-wise cardinality of training examples, such as data resampling. In this paper, we demonstrate that considering solely the cardinality of classes does not cover all issues causing class imbalance. To measure class imbalance, we propose "Class Uncertainty" as the average predictive uncertainty of the training examples, and we show that this novel measure captures the differences across classes better than cardinality. We also curate SVCI-20 as a novel dataset in which the classes have equal number of training examples but they differ in terms of their hardness; thereby causing a type of class imbalance which cannot be addressed by the approaches relying on cardinality. We incorporate our "Class Uncertainty" measure into a diverse set of ten class imbalance mitigation methods to demonstrate its effectiveness on long-tailed datasets as well as on our SVCI-20. Code and datasets will be made available.

Medical image segmentation is critical for computer-aided diagnosis. However, dense pixel-level annotation is time-consuming and expensive, and medical datasets often exhibit severe class imbalance. Such imbalance causes minority structures to be overwhelmed by dominant classes in feature representations, hindering the learning of discriminative features and making reliable segmentation particularly challenging. To address this, we propose the Semantic Class Distribution Learning (SCDL) framework, a plug-and-play module that mitigates supervision and representation biases by learning structured class-conditional feature distributions. SCDL integrates Class Distribution Bidirectional Alignment (CDBA) to align embeddings with learnable class proxies and leverages Semantic Anchor Constraints (SAC) to guide proxies using labeled data. Experiments on the Synapse and AMOS datasets demonstrate that SCDL significantly improves segmentation performance across both overall and class-level metrics, with particularly strong gains on minority classes, achieving state-of-the-art results. Our code is released at https://github.com/Zyh55555/SCDL.

When the performance of a machine learning model varies over groups defined by sensitive attributes (e.g., gender or ethnicity), the performance disparity can be expressed in terms of the probability distributions of the input and output variables over each group. In this paper, we exploit this fact to reduce the disparate impact of a fixed classification model over a population of interest. Given a black-box classifier, we aim to eliminate the performance gap by perturbing the distribution of input variables for the disadvantaged group. We refer to the perturbed distribution as a counterfactual distribution, and characterize its properties for common fairness criteria. We introduce a descent algorithm to learn a counterfactual distribution from data. We then discuss how the estimated distribution can be used to build a data preprocessor that can reduce disparate impact without training a new model. We validate our approach through experiments on real-world datasets, showing that it can repair different forms of disparity without a significant drop in accuracy.

We address the neural network robustness problem by adding Similarity (i.e., correctly predicted depth-matches into training)-awareness and Distance-to-training-distribution-awareness to the existing output Magnitude (i.e., decision-boundary)-awareness of the softmax function. The resulting SDM activation function provides strong signals of the relative epistemic (reducible) predictive uncertainty. We use this novel behavior to further address the complementary HCI problem of mapping the output to human-interpretable summary statistics over relevant partitions of a held-out calibration set. Estimates of prediction-conditional uncertainty are obtained via a parsimonious learned transform over the class-conditional empirical CDFs of the output of a final-layer SDM activation function. For decision-making and as an intrinsic model check, estimates of class-conditional accuracy are obtained by further partitioning the high-probability regions of this calibrated output into class-conditional, region-specific CDFs. The uncertainty estimates from SDM calibration are remarkably robust to test-time distribution shifts and out-of-distribution inputs; incorporate awareness of the effective sample size; provide estimates of uncertainty from the learning and data splitting processes; and are well-suited for selective classification and conditional branching for additional test-time compute based on the predictive uncertainty, as for selective LLM generation, routing, and composition over multiple models and retrieval. Finally, we construct SDM networks, LLMs with uncertainty-aware verification and interpretability-by-exemplar as intrinsic properties. We provide open-source software implementing these results.

We introduce a boosting algorithm to pre-process data for fairness. Starting from an initial fair but inaccurate distribution, our approach shifts towards better data fitting while still ensuring a minimal fairness guarantee. To do so, it learns the sufficient statistics of an exponential family with boosting-compliant convergence. Importantly, we are able to theoretically prove that the learned distribution will have a representation rate and statistical rate data fairness guarantee. Unlike recent optimization based pre-processing methods, our approach can be easily adapted for continuous domain features. Furthermore, when the weak learners are specified to be decision trees, the sufficient statistics of the learned distribution can be examined to provide clues on sources of (un)fairness. Empirical results are present to display the quality of result on real-world data.

Recent empirical studies have demonstrated that diffusion models can effectively learn the image distribution and generate new samples. Remarkably, these models can achieve this even with a small number of training samples despite a large image dimension, circumventing the curse of dimensionality. In this work, we provide theoretical insights into this phenomenon by leveraging key empirical observations: (i) the low intrinsic dimensionality of image data, (ii) a union of manifold structure of image data, and (iii) the low-rank property of the denoising autoencoder in trained diffusion models. These observations motivate us to assume the underlying data distribution of image data as a mixture of low-rank Gaussians and to parameterize the denoising autoencoder as a low-rank model according to the score function of the assumed distribution. With these setups, we rigorously show that optimizing the training loss of diffusion models is equivalent to solving the canonical subspace clustering problem over the training samples. Based on this equivalence, we further show that the minimal number of samples required to learn the underlying distribution scales linearly with the intrinsic dimensions under the above data and model assumptions. This insight sheds light on why diffusion models can break the curse of dimensionality and exhibit the phase transition in learning distributions. Moreover, we empirically establish a correspondence between the subspaces and the semantic representations of image data, facilitating image editing. We validate these results with corroborated experimental results on both simulated distributions and image datasets.

We propose to reinterpret a standard discriminative classifier of p(y|x) as an energy based model for the joint distribution p(x,y). In this setting, the standard class probabilities can be easily computed as well as unnormalized values of p(x) and p(x|y). Within this framework, standard discriminative architectures may beused and the model can also be trained on unlabeled data. We demonstrate that energy based training of the joint distribution improves calibration, robustness, andout-of-distribution detection while also enabling our models to generate samplesrivaling the quality of recent GAN approaches. We improve upon recently proposed techniques for scaling up the training of energy based models and presentan approach which adds little overhead compared to standard classification training. Our approach is the first to achieve performance rivaling the state-of-the-artin both generative and discriminative learning within one hybrid model.

The modeling of probability distributions, specifically generative modeling and density estimation, has become an immensely popular subject in recent years by virtue of its outstanding performance on sophisticated data such as images and texts. Nevertheless, a theoretical understanding of its success is still incomplete. One mystery is the paradox between memorization and generalization: In theory, the model is trained to be exactly the same as the empirical distribution of the finite samples, whereas in practice, the trained model can generate new samples or estimate the likelihood of unseen samples. Likewise, the overwhelming diversity of distribution learning models calls for a unified perspective on this subject. This paper provides a mathematical framework such that all the well-known models can be derived based on simple principles. To demonstrate its efficacy, we present a survey of our results on the approximation error, training error and generalization error of these models, which can all be established based on this framework. In particular, the aforementioned paradox is resolved by proving that these models enjoy implicit regularization during training, so that the generalization error at early-stopping avoids the curse of dimensionality. Furthermore, we provide some new results on landscape analysis and the mode collapse phenomenon.

Long-tailed recognition with imbalanced class distribution naturally emerges in practical machine learning applications. Existing methods such as data reweighing, resampling, and supervised contrastive learning enforce the class balance with a price of introducing imbalance between instances of head class and tail class, which may ignore the underlying rich semantic substructures of the former and exaggerate the biases in the latter. We overcome these drawbacks by a novel ``subclass-balancing contrastive learning (SBCL)'' approach that clusters each head class into multiple subclasses of similar sizes as the tail classes and enforce representations to capture the two-layer class hierarchy between the original classes and their subclasses. Since the clustering is conducted in the representation space and updated during the course of training, the subclass labels preserve the semantic substructures of head classes. Meanwhile, it does not overemphasize tail class samples, so each individual instance contribute to the representation learning equally. Hence, our method achieves both the instance- and subclass-balance, while the original class labels are also learned through contrastive learning among subclasses from different classes. We evaluate SBCL over a list of long-tailed benchmark datasets and it achieves the state-of-the-art performance. In addition, we present extensive analyses and ablation studies of SBCL to verify its advantages.

When facing data with imbalanced classes or groups, practitioners follow an intriguing strategy to achieve best results. They throw away examples until the classes or groups are balanced in size, and then perform empirical risk minimization on the reduced training set. This opposes common wisdom in learning theory, where the expected error is supposed to decrease as the dataset grows in size. In this work, we leverage extreme value theory to address this apparent contradiction. Our results show that the tails of the data distribution play an important role in determining the worst-group-accuracy of linear classifiers. When learning on data with heavy tails, throwing away data restores the geometric symmetry of the resulting classifier, and therefore improves its worst-group generalization.

Discovering novel concepts in unlabelled datasets and in a continuous manner is an important desideratum of lifelong learners. In the literature such problems have been partially addressed under very restricted settings, where novel classes are learned by jointly accessing a related labelled set (e.g., NCD) or by leveraging only a supervisedly pre-trained model (e.g., class-iNCD). In this work we challenge the status quo in class-iNCD and propose a learning paradigm where class discovery occurs continuously and truly unsupervisedly, without needing any related labelled set. In detail, we propose to exploit the richer priors from strong self-supervised pre-trained models (PTM). To this end, we propose simple baselines, composed of a frozen PTM backbone and a learnable linear classifier, that are not only simple to implement but also resilient under longer learning scenarios. We conduct extensive empirical evaluation on a multitude of benchmarks and show the effectiveness of our proposed baselines when compared with sophisticated state-of-the-art methods. The code is open source.

Real-world data tends to follow a long-tailed distribution, where the class imbalance results in dominance of the head classes during training. In this paper, we propose a frustratingly simple but effective step-wise learning framework to gradually enhance the capability of the model in detecting all categories of long-tailed datasets. Specifically, we build smooth-tail data where the long-tailed distribution of categories decays smoothly to correct the bias towards head classes. We pre-train a model on the whole long-tailed data to preserve discriminability between all categories. We then fine-tune the class-agnostic modules of the pre-trained model on the head class dominant replay data to get a head class expert model with improved decision boundaries from all categories. Finally, we train a unified model on the tail class dominant replay data while transferring knowledge from the head class expert model to ensure accurate detection of all categories. Extensive experiments on long-tailed datasets LVIS v0.5 and LVIS v1.0 demonstrate the superior performance of our method, where we can improve the AP with ResNet-50 backbone from 27.0% to 30.3% AP, and especially for the rare categories from 15.5% to 24.9% AP. Our best model using ResNet-101 backbone can achieve 30.7% AP, which suppresses all existing detectors using the same backbone.

We show via a combination of mathematical arguments and empirical evidence that data distributions sampled from diffusion models satisfy a Concentration of Measure Property saying that any Lipschitz 1-dimensional projection of a random vector is not too far from its mean with high probability. This implies that such models are quite restrictive and gives an explanation for a fact previously observed in the literature that conventional diffusion models cannot capture "heavy-tailed" data (i.e. data x for which the norm |x|_2 does not possess a sub-Gaussian tail) well. We then proceed to train a generalized linear model using stochastic gradient descent (SGD) on the diffusion-generated data for a multiclass classification task and observe empirically that a Gaussian universality result holds for the test error. In other words, the test error depends only on the first and second order statistics of the diffusion-generated data in the linear setting. Results of such forms are desirable because they allow one to assume the data itself is Gaussian for analyzing performance of the trained classifier. Finally, we note that current approaches to proving universality do not apply to this case as the covariance matrices of the data tend to have vanishing minimum singular values for the diffusion-generated data, while the current proofs assume that this is not the case (see Subsection 3.4 for more details). This leaves extending previous mathematical universality results as an intriguing open question.

Existing long-tailed recognition methods, aiming to train class-balanced models from long-tailed data, generally assume the models would be evaluated on the uniform test class distribution. However, practical test class distributions often violate this assumption (e.g., being either long-tailed or even inversely long-tailed), which may lead existing methods to fail in real applications. In this paper, we study a more practical yet challenging task, called test-agnostic long-tailed recognition, where the training class distribution is long-tailed while the test class distribution is agnostic and not necessarily uniform. In addition to the issue of class imbalance, this task poses another challenge: the class distribution shift between the training and test data is unknown. To tackle this task, we propose a novel approach, called Self-supervised Aggregation of Diverse Experts, which consists of two strategies: (i) a new skill-diverse expert learning strategy that trains multiple experts from a single and stationary long-tailed dataset to separately handle different class distributions; (ii) a novel test-time expert aggregation strategy that leverages self-supervision to aggregate the learned multiple experts for handling unknown test class distributions. We theoretically show that our self-supervised strategy has a provable ability to simulate test-agnostic class distributions. Promising empirical results demonstrate the effectiveness of our method on both vanilla and test-agnostic long-tailed recognition. Code is available at https://github.com/Vanint/SADE-AgnosticLT.

Real-world data often exhibit imbalanced distributions, where certain target values have significantly fewer observations. Existing techniques for dealing with imbalanced data focus on targets with categorical indices, i.e., different classes. However, many tasks involve continuous targets, where hard boundaries between classes do not exist. We define Deep Imbalanced Regression (DIR) as learning from such imbalanced data with continuous targets, dealing with potential missing data for certain target values, and generalizing to the entire target range. Motivated by the intrinsic difference between categorical and continuous label space, we propose distribution smoothing for both labels and features, which explicitly acknowledges the effects of nearby targets, and calibrates both label and learned feature distributions. We curate and benchmark large-scale DIR datasets from common real-world tasks in computer vision, natural language processing, and healthcare domains. Extensive experiments verify the superior performance of our strategies. Our work fills the gap in benchmarks and techniques for practical imbalanced regression problems. Code and data are available at https://github.com/YyzHarry/imbalanced-regression.

Intrinsic dimension (ID) is an important tool in modern LLM analysis, informing studies of training dynamics, scaling behavior, and dataset structure, yet its textual determinants remain underexplored. We provide the first comprehensive study grounding ID in interpretable text properties through cross-encoder analysis, linguistic features, and sparse autoencoders (SAEs). In this work, we establish three key findings. First, ID is complementary to entropy-based metrics: after controlling for length, the two are uncorrelated, with ID capturing geometric complexity orthogonal to prediction quality. Second, ID exhibits robust genre stratification: scientific prose shows low ID (~8), encyclopedic content medium ID (~9), and creative/opinion writing high ID (~10.5) across all models tested. This reveals that contemporary LLMs find scientific text "representationally simple" while fiction requires additional degrees of freedom. Third, using SAEs, we identify causal features: scientific signals (formal tone, report templates, statistics) reduce ID; humanized signals (personalization, emotion, narrative) increase it. Steering experiments confirm these effects are causal. Thus, for contemporary models, scientific writing appears comparatively "easy", whereas fiction, opinion, and affect add representational degrees of freedom. Our multi-faceted analysis provides practical guidance for the proper use of ID and the sound interpretation of ID-based results.

With the rapid increase of large-scale, real-world datasets, it becomes critical to address the problem of long-tailed data distribution (i.e., a few classes account for most of the data, while most classes are under-represented). Existing solutions typically adopt class re-balancing strategies such as re-sampling and re-weighting based on the number of observations for each class. In this work, we argue that as the number of samples increases, the additional benefit of a newly added data point will diminish. We introduce a novel theoretical framework to measure data overlap by associating with each sample a small neighboring region rather than a single point. The effective number of samples is defined as the volume of samples and can be calculated by a simple formula (1-β^{n})/(1-β), where n is the number of samples and βin [0,1) is a hyperparameter. We design a re-weighting scheme that uses the effective number of samples for each class to re-balance the loss, thereby yielding a class-balanced loss. Comprehensive experiments are conducted on artificially induced long-tailed CIFAR datasets and large-scale datasets including ImageNet and iNaturalist. Our results show that when trained with the proposed class-balanced loss, the network is able to achieve significant performance gains on long-tailed datasets.

Detecting test samples drawn sufficiently far away from the training distribution statistically or adversarially is a fundamental requirement for deploying a good classifier in many real-world machine learning applications. However, deep neural networks with the softmax classifier are known to produce highly overconfident posterior distributions even for such abnormal samples. In this paper, we propose a simple yet effective method for detecting any abnormal samples, which is applicable to any pre-trained softmax neural classifier. We obtain the class conditional Gaussian distributions with respect to (low- and upper-level) features of the deep models under Gaussian discriminant analysis, which result in a confidence score based on the Mahalanobis distance. While most prior methods have been evaluated for detecting either out-of-distribution or adversarial samples, but not both, the proposed method achieves the state-of-the-art performances for both cases in our experiments. Moreover, we found that our proposed method is more robust in harsh cases, e.g., when the training dataset has noisy labels or small number of samples. Finally, we show that the proposed method enjoys broader usage by applying it to class-incremental learning: whenever out-of-distribution samples are detected, our classification rule can incorporate new classes well without further training deep models.

Major advancements have been made in the field of object detection and segmentation recently. However, when it comes to rare categories, the state-of-the-art methods fail to detect them, resulting in a significant performance gap between rare and frequent categories. In this paper, we identify that Sigmoid or Softmax functions used in deep detectors are a major reason for low performance and are sub-optimal for long-tailed detection and segmentation. To address this, we develop a Gumbel Optimized Loss (GOL), for long-tailed detection and segmentation. It aligns with the Gumbel distribution of rare classes in imbalanced datasets, considering the fact that most classes in long-tailed detection have low expected probability. The proposed GOL significantly outperforms the best state-of-the-art method by 1.1% on AP , and boosts the overall segmentation by 9.0% and detection by 8.0%, particularly improving detection of rare classes by 20.3%, compared to Mask-RCNN, on LVIS dataset. Code available at: https://github.com/kostas1515/GOL

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

Many recently trained neural networks employ large numbers of parameters to achieve good performance. One may intuitively use the number of parameters required as a rough gauge of the difficulty of a problem. But how accurate are such notions? How many parameters are really needed? In this paper we attempt to answer this question by training networks not in their native parameter space, but instead in a smaller, randomly oriented subspace. We slowly increase the dimension of this subspace, note at which dimension solutions first appear, and define this to be the intrinsic dimension of the objective landscape. The approach is simple to implement, computationally tractable, and produces several suggestive conclusions. Many problems have smaller intrinsic dimensions than one might suspect, and the intrinsic dimension for a given dataset varies little across a family of models with vastly different sizes. This latter result has the profound implication that once a parameter space is large enough to solve a problem, extra parameters serve directly to increase the dimensionality of the solution manifold. Intrinsic dimension allows some quantitative comparison of problem difficulty across supervised, reinforcement, and other types of learning where we conclude, for example, that solving the inverted pendulum problem is 100 times easier than classifying digits from MNIST, and playing Atari Pong from pixels is about as hard as classifying CIFAR-10. In addition to providing new cartography of the objective landscapes wandered by parameterized models, the method is a simple technique for constructively obtaining an upper bound on the minimum description length of a solution. A byproduct of this construction is a simple approach for compressing networks, in some cases by more than 100 times.

Identifying Out-of-distribution (OOD) data is becoming increasingly critical as the real-world applications of deep learning methods expand. Post-hoc methods modify softmax scores fine-tuned on outlier data or leverage intermediate feature layers to identify distinctive patterns between In-Distribution (ID) and OOD samples. Other methods focus on employing diverse OOD samples to learn discrepancies between ID and OOD. These techniques, however, are typically dependent on the quality of the outlier samples assumed. Density-based methods explicitly model class-conditioned distributions but this requires long training time or retraining the classifier. To tackle these issues, we introduce FlowCon, a new density-based OOD detection technique. Our main innovation lies in efficiently combining the properties of normalizing flow with supervised contrastive learning, ensuring robust representation learning with tractable density estimation. Empirical evaluation shows the enhanced performance of our method across common vision datasets such as CIFAR-10 and CIFAR-100 pretrained on ResNet18 and WideResNet classifiers. We also perform quantitative analysis using likelihood plots and qualitative visualization using UMAP embeddings and demonstrate the robustness of the proposed method under various OOD contexts. Code will be open-sourced post decision.

Customer Lifetime Value (CLTV) prediction is a critical task in business applications. Accurately predicting CLTV is challenging in real-world business scenarios, as the distribution of CLTV is complex and mutable. Firstly, there is a large number of users without any consumption consisting of a long-tailed part that is too complex to fit. Secondly, the small set of high-value users spent orders of magnitude more than a typical user leading to a wide range of the CLTV distribution which is hard to capture in a single distribution. Existing approaches for CLTV estimation either assume a prior probability distribution and fit a single group of distribution-related parameters for all samples, or directly learn from the posterior distribution with manually predefined buckets in a heuristic manner. However, all these methods fail to handle complex and mutable distributions. In this paper, we propose a novel optimal distribution selection model OptDist for CLTV prediction, which utilizes an adaptive optimal sub-distribution selection mechanism to improve the accuracy of complex distribution modeling. Specifically, OptDist trains several candidate sub-distribution networks in the distribution learning module (DLM) for modeling the probability distribution of CLTV. Then, a distribution selection module (DSM) is proposed to select the sub-distribution for each sample, thus making the selection automatically and adaptively. Besides, we design an alignment mechanism that connects both modules, which effectively guides the optimization. We conduct extensive experiments on both two public and one private dataset to verify that OptDist outperforms state-of-the-art baselines. Furthermore, OptDist has been deployed on a large-scale financial platform for customer acquisition marketing campaigns and the online experiments also demonstrate the effectiveness of OptDist.

Real-world large-scale datasets are both noisily labeled and class-imbalanced. The issues seriously hurt the generalization of trained models. It is hence significant to address the simultaneous incorrect labeling and class-imbalance, i.e., the problem of learning with noisy labels on long-tailed data. Previous works develop several methods for the problem. However, they always rely on strong assumptions that are invalid or hard to be checked in practice. In this paper, to handle the problem and address the limitations of prior works, we propose a representation calibration method RCAL. Specifically, RCAL works with the representations extracted by unsupervised contrastive learning. We assume that without incorrect labeling and class imbalance, the representations of instances in each class conform to a multivariate Gaussian distribution, which is much milder and easier to be checked. Based on the assumption, we recover underlying representation distributions from polluted ones resulting from mislabeled and class-imbalanced data. Additional data points are then sampled from the recovered distributions to help generalization. Moreover, during classifier training, representation learning takes advantage of representation robustness brought by contrastive learning, which further improves the classifier performance. We derive theoretical results to discuss the effectiveness of our representation calibration. Experiments on multiple benchmarks justify our claims and confirm the superiority of the proposed method.

Neural multivariate regression underpins a wide range of domains such as control, robotics, and finance, yet the geometry of its learned representations remains poorly characterized. While neural collapse has been shown to benefit generalization in classification, we find that analogous collapse in regression consistently degrades performance. To explain this contrast, we analyze models through the lens of intrinsic dimension. Across control tasks and synthetic datasets, we estimate the intrinsic dimension of last-layer features (ID_H) and compare it with that of the regression targets (ID_Y). Collapsed models exhibit ID_H < ID_Y, leading to over-compression and poor generalization, whereas non-collapsed models typically maintain ID_H > ID_Y. For the non-collapsed models, performance with respect to ID_H depends on the data quantity and noise levels. From these observations, we identify two regimes (over-compressed and under-compressed) that determine when expanding or reducing feature dimensionality improves performance. Our results provide new geometric insights into neural regression and suggest practical strategies for enhancing generalization.

Machine learning models -- including prominent zero-shot models -- are often trained on datasets whose labels are only a small proportion of a larger label space. Such spaces are commonly equipped with a metric that relates the labels via distances between them. We propose a simple approach to exploit this information to adapt the trained model to reliably predict new classes -- or, in the case of zero-shot prediction, to improve its performance -- without any additional training. Our technique is a drop-in replacement of the standard prediction rule, swapping argmax with the Fr\'echet mean. We provide a comprehensive theoretical analysis for this approach, studying (i) learning-theoretic results trading off label space diameter, sample complexity, and model dimension, (ii) characterizations of the full range of scenarios in which it is possible to predict any unobserved class, and (iii) an optimal active learning-like next class selection procedure to obtain optimal training classes for when it is not possible to predict the entire range of unobserved classes. Empirically, using easily-available external metrics, our proposed approach, Loki, gains up to 29.7% relative improvement over SimCLR on ImageNet and scales to hundreds of thousands of classes. When no such metric is available, Loki can use self-derived metrics from class embeddings and obtains a 10.5% improvement on pretrained zero-shot models such as CLIP.

Within-dataset class-split evaluation is widely used as a proxy for fully unconditional out-of-distribution anomaly detection. We show that this protocol can become ill-posed when the held-out anomaly class overlaps the normal mixture in representation space. In this regime, anomaly scores may collapse toward chance or even invert, and the preferred score direction can depend on the unknown anomaly class. We introduce a simple training-free diagnostic, neighborhood class leakage, and show that it predicts score-direction instability across Fashion-MNIST, CIFAR-10, and Imagenette, in both pixel and VAE latent spaces. Our results suggest that class-split AD benchmarks should be treated as geometry-dependent stress tests rather than unconditional evidence of anomaly-detection ability.

The Distributional Principal Autoencoder (DPA) combines distributionally correct reconstruction with principal-component-like interpretability of the encodings. In this work, we provide exact theoretical guarantees on both fronts. First, we derive a closed-form relation linking each optimal level-set geometry to the data-distribution score. This result explains DPA's empirical ability to disentangle factors of variation of the data, as well as allows the score to be recovered directly from samples. When the data follows the Boltzmann distribution, we demonstrate that this relation yields an approximation of the minimum free-energy path for the Mueller-Brown potential in a single fit. Second, we prove that if the data lies on a manifold that can be approximated by the encoder, latent components beyond the manifold dimension are conditionally independent of the data distribution - carrying no additional information - and thus reveal the intrinsic dimension. Together, these results show that a single model can learn the data distribution and its intrinsic dimension with exact guarantees simultaneously, unifying two longstanding goals of unsupervised learning.

Data collected from the real world typically exhibit long-tailed distributions, where frequent classes contain abundant data while rare ones have only a limited number of samples. While existing supervised learning approaches have been proposed to tackle such data imbalance, the requirement of label supervision would limit their applicability to real-world scenarios in which label annotation might not be available. Without the access to class labels nor the associated class frequencies, we propose Frequency-Aware Self-Supervised Learning (FASSL) in this paper. Targeting at learning from unlabeled data with inherent long-tailed distributions, the goal of FASSL is to produce discriminative feature representations for downstream classification tasks. In FASSL, we first learn frequency-aware prototypes, reflecting the associated long-tailed distribution. Particularly focusing on rare-class samples, the relationships between image data and the derived prototypes are further exploited with the introduced self-supervised learning scheme. Experiments on long-tailed image datasets quantitatively and qualitatively verify the effectiveness of our learning scheme.

The infinitely wide neural network has been proven a useful and manageable mathematical model that enables the understanding of many phenomena appearing in deep learning. One example is the convergence of random deep networks to Gaussian processes that allows a rigorous analysis of the way the choice of activation function and network weights impacts the training dynamics. In this paper, we extend the seminal proof of Matthews et al. (2018) to a larger class of initial weight distributions (which we call PSEUDO-IID), including the established cases of IID and orthogonal weights, as well as the emerging low-rank and structured sparse settings celebrated for their computational speed-up benefits. We show that fully-connected and convolutional networks initialized with PSEUDO-IID distributions are all effectively equivalent up to their variance. Using our results, one can identify the Edge-of-Chaos for a broader class of neural networks and tune them at criticality in order to enhance their training. Moreover, they enable the posterior distribution of Bayesian Neural Networks to be tractable across these various initialization schemes.

Deep neural networks often generalize well despite heavy over-parameterization, challenging classical parameter-based analyses. We study generalization from a representation-centric perspective and analyze how the geometry of learned embeddings controls predictive performance for a fixed trained model. We show that population risk can be bounded by two factors: (i) the intrinsic dimension of the embedding distribution, which determines the convergence rate of empirical embedding distribution to the population distribution in Wasserstein distance, and (ii) the sensitivity of the downstream mapping from embeddings to predictions, characterized by Lipschitz constants. Together, these yield an embedding-dependent error bound that does not rely on parameter counts or hypothesis class complexity. At the final embedding layer, architectural sensitivity vanishes and the bound is dominated by embedding dimension, explaining its strong empirical correlation with generalization performance. Experiments across architectures and datasets validate the theory and demonstrate the utility of embedding-based diagnostics.

Compositional generalization requires models to produce novel configurations from familiar parts. In diffusion models, prior compositional generation methods typically assume that the relevant concepts or conditioning signals are already available. We instead ask whether a pretrained diffusion model can discover query-specific concepts from the time-indexed scores it learns for the noisy marginals p_t(x_t) and compose them at test time. Given a single out-of-distribution query, our method performs gradient ascent on s_θ(x_t,t) approx nabla_{x_t}log p_t(x_t) at multiple noising timesteps to recover local density modes, maps these modes into clean-space Gaussians, greedily selects relevant prototypes with a submodular likelihood objective, and combines them into a product-of-experts (PoE) teacher model with an analytic score. This teacher model can be sampled directly through classifier-free guidance or used to generate a sample pool for training a new class embedding and low-rank adapter. On held-out composition benchmarks built from ColorMNIST and CelebA, both the analytic PoE sampler and the low-rank adapted model outperform query-only and nearest trained-class baselines. These results suggest that the time-indexed score geometry of the diffusion model contains reusable density-mode concepts that support test-time compositional generation without a predefined concept library.

Industrial defect detection systems face critical limitations when confined to one-class anomaly detection paradigms, which assume uniform outlier distributions and struggle with data scarcity in realworld manufacturing environments. We present ExDD (Explicit Dual Distribution), a novel framework that transcends these limitations by explicitly modeling dual feature distributions. Our approach leverages parallel memory banks that capture the distinct statistical properties of both normality and anomalous patterns, addressing the fundamental flaw of uniform outlier assumptions. To overcome data scarcity, we employ latent diffusion models with domain-specific textual conditioning, generating in-distribution synthetic defects that preserve industrial context. Our neighborhood-aware ratio scoring mechanism elegantly fuses complementary distance metrics, amplifying signals in regions exhibiting both deviation from normality and similarity to known defect patterns. Experimental validation on KSDD2 demonstrates superior performance (94.2% I-AUROC, 97.7% P-AUROC), with optimal augmentation at 100 synthetic samples.

This paper studies Kernel density estimation for a high-dimensional distribution rho(x). Traditional approaches have focused on the limit of large number of data points n and fixed dimension d. We analyze instead the regime where both the number n of data points y_i and their dimensionality d grow with a fixed ratio alpha=(log n)/d. Our study reveals three distinct statistical regimes for the kernel-based estimate of the density hat rho_h^{D}(x)=1{n h^d}sum_{i=1}^n Kleft(x-y_i{h}right), depending on the bandwidth h: a classical regime for large bandwidth where the Central Limit Theorem (CLT) holds, which is akin to the one found in traditional approaches. Below a certain value of the bandwidth, h_{CLT}(alpha), we find that the CLT breaks down. The statistics of hat rho_h^{D}(x) for a fixed x drawn from rho(x) is given by a heavy-tailed distribution (an alpha-stable distribution). In particular below a value h_G(alpha), we find that hat rho_h^{D}(x) is governed by extreme value statistics: only a few points in the database matter and give the dominant contribution to the density estimator. We provide a detailed analysis for high-dimensional multivariate Gaussian data. We show that the optimal bandwidth threshold based on Kullback-Leibler divergence lies in the new statistical regime identified in this paper. Our findings reveal limitations of classical approaches, show the relevance of these new statistical regimes, and offer new insights for Kernel density estimation in high-dimensional settings.

Imbalanced data are frequently encountered in real-world classification tasks. Previous works on imbalanced learning mostly focused on learning with a minority class of few samples. However, the notion of imbalance also applies to cases where the minority class contains abundant samples, which is usually the case for industrial applications like fraud detection in the area of financial risk management. In this paper, we take a population-level approach to imbalanced learning by proposing a new formulation called ultra-imbalanced classification (UIC). Under UIC, loss functions behave differently even if infinite amount of training samples are available. To understand the intrinsic difficulty of UIC problems, we borrow ideas from information theory and establish a framework to compare different loss functions through the lens of statistical information. A novel learning objective termed Tunable Boosting Loss is developed which is provably resistant against data imbalance under UIC, as well as being empirically efficient verified by extensive experimental studies on both public and industrial datasets.

Detecting out-of-distribution inputs for visual recognition models has become critical in safe deep learning. This paper proposes a novel hierarchical visual category modeling scheme to separate out-of-distribution data from in-distribution data through joint representation learning and statistical modeling. We learn a mixture of Gaussian models for each in-distribution category. There are many Gaussian mixture models to model different visual categories. With these Gaussian models, we design an in-distribution score function by aggregating multiple Mahalanobis-based metrics. We don't use any auxiliary outlier data as training samples, which may hurt the generalization ability of out-of-distribution detection algorithms. We split the ImageNet-1k dataset into ten folds randomly. We use one fold as the in-distribution dataset and the others as out-of-distribution datasets to evaluate the proposed method. We also conduct experiments on seven popular benchmarks, including CIFAR, iNaturalist, SUN, Places, Textures, ImageNet-O, and OpenImage-O. Extensive experiments indicate that the proposed method outperforms state-of-the-art algorithms clearly. Meanwhile, we find that our visual representation has a competitive performance when compared with features learned by classical methods. These results demonstrate that the proposed method hasn't weakened the discriminative ability of visual recognition models and keeps high efficiency in detecting out-of-distribution samples.

Out-of-distribution (OOD) detection aims to detect testing samples far away from the in-distribution (ID) training data, which is crucial for the safe deployment of machine learning models in the real world. Distance-based OOD detection methods have emerged with enhanced deep representation learning. They identify unseen OOD samples by measuring their distances from ID class centroids or prototypes. However, existing approaches learn the representation relying on oversimplified data assumptions, e.g, modeling ID data of each class with one centroid class prototype or using loss functions not designed for OOD detection, which overlook the natural diversities within the data. Naively enforcing data samples of each class to be compact around only one prototype leads to inadequate modeling of realistic data and limited performance. To tackle these issues, we propose PrototypicAl Learning with a Mixture of prototypes (PALM) which models each class with multiple prototypes to capture the sample diversities, and learns more faithful and compact samples embeddings to enhance OOD detection. Our method automatically identifies and dynamically updates prototypes, assigning each sample to a subset of prototypes via reciprocal neighbor soft assignment weights. PALM optimizes a maximum likelihood estimation (MLE) loss to encourage the sample embeddings to be compact around the associated prototypes, as well as a contrastive loss on all prototypes to enhance intra-class compactness and inter-class discrimination at the prototype level. Moreover, the automatic estimation of prototypes enables our approach to be extended to the challenging OOD detection task with unlabelled ID data. Extensive experiments demonstrate the superiority of PALM, achieving state-of-the-art average AUROC performance of 93.82 on the challenging CIFAR-100 benchmark. Code is available at https://github.com/jeff024/PALM.

Certain tasks in high-dimensional statistics become easier when the underlying distribution satisfies a local-to-global property called approximate tensorization of entropy (ATE). For example, the Glauber dynamics Markov chain of an ATE distribution mixes fast and can produce approximate samples in a small amount of time, since such a distribution satisfies a modified log-Sobolev inequality. Moreover, identity-testing for an ATE distribution requires few samples if the tester is given coordinate conditional access to the unknown distribution, as shown by Blanca, Chen, Štefankovič, and Vigoda (COLT 2023). A natural class of distributions that do not satisfy ATE consists of mixtures of (few) distributions that do satisfy ATE. We study the complexity of identity-testing and sampling for these distributions. Our main results are the following: 1. We show fast mixing of Glauber dynamics from a data-based initialization, with optimal sample complexity, for mixtures of distributions satisfying modified log-Sobolev inequalities. This extends work of Huang, Koehler, Lee, Mohanty, Rajaraman, Vuong, and Wu (STOC 2025, COLT 2025) for mixtures of distributions satisfying Poincaré inequalities. 2. Answering an open question posed by Blanca et al., we give efficient identity-testers for mixtures of ATE distributions in the coordinate-conditional sampling access model. We also give some simplifications and improvements to the original algorithm of Blanca et al.

Strictly proper scoring rules identify the true conditional class distribution at population level, but their curvature can alter optimization and finite-sample behavior. We study three multiclass objectives: a class-aware quadratic Bregman score (CAPM), a strongly convex generator with constrained log-cosh ridges (HPG), and an HPG objective with an annealed probability-margin penalty (APMS). CAPM is treated as a structured instance of established quadratic scoring-rule theory. We derive conditional-regret, curvature, range, and logit-gradient bounds for CAPM and HPG, and prove exact penalty-range and conditional-target displacement bounds for APMS. Controlled five-seed experiments use Digits, Wisconsin breast cancer, and synthetic confusion and long-tail problems under clean labels, symmetric and pair-flip corruption, class imbalance, calibration evaluation, input corruption, and first-order adversarial perturbations. The candidates are close to cross-entropy on clean data and show descriptive gains in some noisy-label cells, but the five-seed comparisons are interpreted descriptively rather than as significance evidence. The selected noisy-label baselines perform better on Digits with 40% symmetric label noise, and explicit prior-adjustment methods perform better in the 30:1 synthetic long-tail experiment. Ablations do not show a consistent benefit from the candidate-specific graph, ridge, or margin components. The mathematical analysis establishes the stated properties, and the experiments delimit the empirical evidence; together they do not support a claim of general superiority.

The problem of Novel Class Discovery (NCD) consists in extracting knowledge from a labeled set of known classes to accurately partition an unlabeled set of novel classes. While NCD has recently received a lot of attention from the community, it is often solved on computer vision problems and under unrealistic conditions. In particular, the number of novel classes is usually assumed to be known in advance, and their labels are sometimes used to tune hyperparameters. Methods that rely on these assumptions are not applicable in real-world scenarios. In this work, we focus on solving NCD in tabular data when no prior knowledge of the novel classes is available. To this end, we propose to tune the hyperparameters of NCD methods by adapting the k-fold cross-validation process and hiding some of the known classes in each fold. Since we have found that methods with too many hyperparameters are likely to overfit these hidden classes, we define a simple deep NCD model. This method is composed of only the essential elements necessary for the NCD problem and performs impressively well under realistic conditions. Furthermore, we find that the latent space of this method can be used to reliably estimate the number of novel classes. Additionally, we adapt two unsupervised clustering algorithms (k-means and Spectral Clustering) to leverage the knowledge of the known classes. Extensive experiments are conducted on 7 tabular datasets and demonstrate the effectiveness of the proposed method and hyperparameter tuning process, and show that the NCD problem can be solved without relying on knowledge from the novel classes.

Estimating truncated density models is difficult, as these models have intractable normalising constants and hard to satisfy boundary conditions. Score matching can be adapted to solve the truncated density estimation problem, but requires a continuous weighting function which takes zero at the boundary and is positive elsewhere. Evaluation of such a weighting function (and its gradient) often requires a closed-form expression of the truncation boundary and finding a solution to a complicated optimisation problem. In this paper, we propose approximate Stein classes, which in turn leads to a relaxed Stein identity for truncated density estimation. We develop a novel discrepancy measure, truncated kernelised Stein discrepancy (TKSD), which does not require fixing a weighting function in advance, and can be evaluated using only samples on the boundary. We estimate a truncated density model by minimising the Lagrangian dual of TKSD. Finally, experiments show the accuracy of our method to be an improvement over previous works even without the explicit functional form of the boundary.

Inference-Time Scaling (ITS) has largely succeeded in verifiable domains like math and coding, where cheap verification enables scalable output selection. However, extending ITS to tasks prone to systematic failure - driven by faulty initial assumptions or unmet multidimensional constraints - typically relies on costly external solvers or brittle, model-based verifiers. Our key insight is that the intrinsic statistics of parallel sample sets, specifically length-adjusted tail entropy, provide a robust discriminative signal for solution quality without access to ground truth. Crucially, these statistics serve as a difficulty gate for adaptive compute allocation, dynamically routing problems across scaling regimes. First, Intrinsic Selection (iS) ranks candidates post-hoc, matching consensus-based algorithms across three domains and improving engineering design selection by 20% over pass@1 baselines. Second, Intrinsic Particle Filtering (iPF) generalizes this to step-level resampling, guiding generation toward high-confidence reasoning trajectories to improve pass@1 by 6.1 points on average on hard math problems. Finally, Particle Distillation (dPF) injects privileged guidance via early logit blending and KL-guided resampling, steering generation past systematic reasoning errors to satisfy expert rubrics, yielding up to 26.5% gains on complex clinical responses. Our pipeline applies seamlessly across broad-purpose, domain-specialized, and multimodal architectures, successfully extending ITS to open-ended domains without requiring trained reward models or exact ground-truth verification.

We tackle the problem of novel class discovery, which aims to learn novel classes without supervision based on labeled data from known classes. A key challenge lies in transferring the knowledge in the known-class data to the learning of novel classes. Previous methods mainly focus on building a shared representation space for knowledge transfer and often ignore modeling class relations. To address this, we introduce a class relation representation for the novel classes based on the predicted class distribution of a model trained on known classes. Empirically, we find that such class relation becomes less informative during typical discovery training. To prevent such information loss, we propose a novel knowledge distillation framework, which utilizes our class-relation representation to regularize the learning of novel classes. In addition, to enable a flexible knowledge distillation scheme for each data point in novel classes, we develop a learnable weighting function for the regularization, which adaptively promotes knowledge transfer based on the semantic similarity between the novel and known classes. To validate the effectiveness and generalization of our method, we conduct extensive experiments on multiple benchmarks, including CIFAR100, Stanford Cars, CUB, and FGVC-Aircraft datasets. Our results demonstrate that the proposed method outperforms the previous state-of-the-art methods by a significant margin on almost all benchmarks. Code is available at https://github.com/kleinzcy/Cr-KD-NCD{here}.

Data imbalance and open-ended distribution are two intrinsic characteristics of the real visual world. Though encouraging progress has been made in tackling each challenge separately, few works dedicated to combining them towards real-world scenarios. While several previous works have focused on classifying close-set samples and detecting open-set samples during testing, it's still essential to be able to classify unknown subjects as human beings. In this paper, we formally define a more realistic task as distribution-agnostic generalized category discovery (DA-GCD): generating fine-grained predictions for both close- and open-set classes in a long-tailed open-world setting. To tackle the challenging problem, we propose a Self-Balanced Co-Advice contrastive framework (BaCon), which consists of a contrastive-learning branch and a pseudo-labeling branch, working collaboratively to provide interactive supervision to resolve the DA-GCD task. In particular, the contrastive-learning branch provides reliable distribution estimation to regularize the predictions of the pseudo-labeling branch, which in turn guides contrastive learning through self-balanced knowledge transfer and a proposed novel contrastive loss. We compare BaCon with state-of-the-art methods from two closely related fields: imbalanced semi-supervised learning and generalized category discovery. The effectiveness of BaCon is demonstrated with superior performance over all baselines and comprehensive analysis across various datasets. Our code is publicly available.

Exemplar-free class-incremental learning (EFCIL) presents a significant challenge as the old class samples are absent for new task learning. Due to the severe imbalance between old and new class samples, the learned classifiers can be easily biased toward the new ones. Moreover, continually updating the feature extractor under EFCIL can compromise the discriminative power of old class features, e.g., leading to less compact and more overlapping distributions across classes. Existing methods mainly focus on handling biased classifier learning. In this work, both cases are considered using the proposed method. Specifically, we first introduce a Distribution-Based Global Classifier (DBGC) to avoid bias factors in existing methods, such as data imbalance and sampling. More importantly, the compromised distributions of old classes are simulated via a simple operation, variance enlarging (VE). Incorporating VE based on DBGC results in a novel classification loss for EFCIL. This loss is proven equivalent to an Adaptive Margin Softmax Cross Entropy (AMarX). The proposed method is thus called Adaptive Margin Global Classifier (AMGC). AMGC is simple yet effective. Extensive experiments show that AMGC achieves superior image classification results on its own under a challenging EFCIL setting. Detailed analysis is also provided for further demonstration.

Class-incremental learning (CIL) is a particularly challenging variant of continual learning, where the goal is to learn to discriminate between all classes presented in an incremental fashion. Existing approaches often suffer from excessive forgetting and imbalance of the scores assigned to classes that have not been seen together during training. In this study, we introduce a novel approach, Prediction Error-based Classification (PEC), which differs from traditional discriminative and generative classification paradigms. PEC computes a class score by measuring the prediction error of a model trained to replicate the outputs of a frozen random neural network on data from that class. The method can be interpreted as approximating a classification rule based on Gaussian Process posterior variance. PEC offers several practical advantages, including sample efficiency, ease of tuning, and effectiveness even when data are presented one class at a time. Our empirical results show that PEC performs strongly in single-pass-through-data CIL, outperforming other rehearsal-free baselines in all cases and rehearsal-based methods with moderate replay buffer size in most cases across multiple benchmarks.

Prompt-based classifiers are an attractive approach for zero-shot classification. However, the precise choice of the prompt template and label words can largely influence performance, with semantically equivalent settings often showing notable performance difference. This discrepancy can be partly attributed to word biases, where the classifier may be biased towards classes. To address this problem, it is possible to optimise classification thresholds on a labelled data set, however, this mitigates some of the advantages of prompt-based classifiers. This paper instead approaches this problem by examining the expected marginal probabilities of the classes. Here, probabilities are reweighted to have a uniform prior over classes, in an unsupervised fashion. Further, we draw a theoretical connection between the class priors and the language models' word prior, and offer the ability to set a threshold in a zero-resource fashion. We show that matching class priors correlates strongly with the oracle upper bound performance and demonstrate large consistent performance gains for prompt settings over a range of NLP tasks.

We present prompt distribution learning for effectively adapting a pre-trained vision-language model to address downstream recognition tasks. Our method not only learns low-bias prompts from a few samples but also captures the distribution of diverse prompts to handle the varying visual representations. In this way, we provide high-quality task-related content for facilitating recognition. This prompt distribution learning is realized by an efficient approach that learns the output embeddings of prompts instead of the input embeddings. Thus, we can employ a Gaussian distribution to model them effectively and derive a surrogate loss for efficient training. Extensive experiments on 12 datasets demonstrate that our method consistently and significantly outperforms existing methods. For example, with 1 sample per category, it relatively improves the average result by 9.1% compared to human-crafted prompts.

Arbitrary style transfer (AST) and domain generalization (DG) are important yet challenging visual learning tasks, which can be cast as a feature distribution matching problem. With the assumption of Gaussian feature distribution, conventional feature distribution matching methods usually match the mean and standard deviation of features. However, the feature distributions of real-world data are usually much more complicated than Gaussian, which cannot be accurately matched by using only the first-order and second-order statistics, while it is computationally prohibitive to use high-order statistics for distribution matching. In this work, we, for the first time to our best knowledge, propose to perform Exact Feature Distribution Matching (EFDM) by exactly matching the empirical Cumulative Distribution Functions (eCDFs) of image features, which could be implemented by applying the Exact Histogram Matching (EHM) in the image feature space. Particularly, a fast EHM algorithm, named Sort-Matching, is employed to perform EFDM in a plug-and-play manner with minimal cost. The effectiveness of our proposed EFDM method is verified on a variety of AST and DG tasks, demonstrating new state-of-the-art results. Codes are available at https://github.com/YBZh/EFDM.

Data collection is often difficult in critical fields such as medicine, physics, and chemistry. As a result, classification methods usually perform poorly with these small datasets, leading to weak predictive performance. Increasing the training set with additional synthetic data, similar to data augmentation in images, is commonly believed to improve downstream classification performance. However, current tabular generative methods that learn either the joint distribution p(x, y) or the class-conditional distribution p(x mid y) often overfit on small datasets, resulting in poor-quality synthetic data, usually worsening classification performance compared to using real data alone. To solve these challenges, we introduce TabEBM, a novel class-conditional generative method using Energy-Based Models (EBMs). Unlike existing methods that use a shared model to approximate all class-conditional densities, our key innovation is to create distinct EBM generative models for each class, each modelling its class-specific data distribution individually. This approach creates robust energy landscapes, even in ambiguous class distributions. Our experiments show that TabEBM generates synthetic data with higher quality and better statistical fidelity than existing methods. When used for data augmentation, our synthetic data consistently improves the classification performance across diverse datasets of various sizes, especially small ones. Code is available at https://github.com/andreimargeloiu/TabEBM.

Out-of-distribution (OOD) detection is crucial for model reliability, as it identifies samples from unknown classes and reduces errors due to unexpected inputs. Vision-Language Models (VLMs) such as CLIP are emerging as powerful tools for OOD detection by integrating multi-modal information. However, the practical application of such systems is challenged by manual prompt engineering, which demands domain expertise and is sensitive to linguistic nuances. In this paper, we introduce Label-driven Automated Prompt Tuning (LAPT), a novel approach to OOD detection that reduces the need for manual prompt engineering. We develop distribution-aware prompts with in-distribution (ID) class names and negative labels mined automatically. Training samples linked to these class labels are collected autonomously via image synthesis and retrieval methods, allowing for prompt learning without manual effort. We utilize a simple cross-entropy loss for prompt optimization, with cross-modal and cross-distribution mixing strategies to reduce image noise and explore the intermediate space between distributions, respectively. The LAPT framework operates autonomously, requiring only ID class names as input and eliminating the need for manual intervention. With extensive experiments, LAPT consistently outperforms manually crafted prompts, setting a new standard for OOD detection. Moreover, LAPT not only enhances the distinction between ID and OOD samples, but also improves the ID classification accuracy and strengthens the generalization robustness to covariate shifts, resulting in outstanding performance in challenging full-spectrum OOD detection tasks. Codes are available at https://github.com/YBZh/LAPT.

Open-set test-time adaptation (TTA) updates models on new data in the presence of input shifts and unknown output classes. While recent methods have made progress on improving in-distribution (InD) accuracy for known classes, their ability to accurately detect out-of-distribution (OOD) unknown classes remains underexplored. We benchmark robust and open-set TTA methods (SAR, OSTTA, UniEnt, and SoTTA) on the standard corruption benchmarks of CIFAR-10-C at the small scale and ImageNet-C at the large scale. For CIFAR-10-C, we use OOD data from SVHN and CIFAR-100 in their respective corrupted forms of SVHN-C and CIFAR-100-C. For ImageNet-C, we use OOD data from ImageNet-O and Textures in their respective corrupted forms of ImageNet-O-C and Textures-C. ImageNet-O is nearer to ImageNet, as unknown but related object classes (like ''garlic bread'' vs. ''hot dog'' for food, or ''highway'' vs. ''dam'' for infrastructure), while Textures is farther from ImageNet, as non-object patterns (like ''cracked'' mud, ''porous'' sponge, ''veined'' leaves). We evaluate the accuracy and confidence of TTA methods for InD vs. OOD recognition on CIFAR-10-C and ImageNet-C. We verify the accuracy of each method's own OOD detection technique on CIFAR-10-C. We also evaluate on ImageNet-C and report both accuracy and standard OOD detection metrics. We further examine more realistic settings, in which the proportions and rates of OOD data can vary. To explore the trade-off between InD recognition and OOD rejection, we propose a new baseline that replaces softmax/multi-class output with sigmoid/multi-label output. Our analysis shows for the first time that current open-set TTA methods struggle to balance InD and OOD accuracy and that they only imperfectly filter OOD data for their own adaptation updates.

Ensuring the reliability and safety of automated decision-making is crucial. It is well-known that data distribution shifts in machine learning can produce unreliable outcomes. This paper proposes a new approach for measuring the reliability of predictions under distribution shifts. We analyze how the outputs of a trained neural network change using clustering to measure distances between outputs and class centroids. We propose this distance as a metric to evaluate the confidence of predictions under distribution shifts. We assign each prediction to a cluster with centroid representing the mean softmax output for all correct predictions of a given class. We then define a safety threshold for a class as the smallest distance from an incorrect prediction to the given class centroid. We evaluate the approach on the MNIST and CIFAR-10 datasets using a Convolutional Neural Network and a Vision Transformer, respectively. The results show that our approach is consistent across these data sets and network models, and indicate that the proposed metric can offer an efficient way of determining when automated predictions are acceptable and when they should be deferred to human operators given a distribution shift.

Real-world data often exhibits long tail distributions with heavy class imbalance, where the majority classes can dominate the training process and alter the decision boundaries of the minority classes. Recently, researchers have investigated the potential of supervised contrastive learning for long-tailed recognition, and demonstrated that it provides a strong performance gain. In this paper, we show that while supervised contrastive learning can help improve performance, past baselines suffer from poor uniformity brought in by imbalanced data distribution. This poor uniformity manifests in samples from the minority class having poor separability in the feature space. To address this problem, we propose targeted supervised contrastive learning (TSC), which improves the uniformity of the feature distribution on the hypersphere. TSC first generates a set of targets uniformly distributed on a hypersphere. It then makes the features of different classes converge to these distinct and uniformly distributed targets during training. This forces all classes, including minority classes, to maintain a uniform distribution in the feature space, improves class boundaries, and provides better generalization even in the presence of long-tail data. Experiments on multiple datasets show that TSC achieves state-of-the-art performance on long-tailed recognition tasks.

Existing long-tailed classification (LT) methods only focus on tackling the class-wise imbalance that head classes have more samples than tail classes, but overlook the attribute-wise imbalance. In fact, even if the class is balanced, samples within each class may still be long-tailed due to the varying attributes. Note that the latter is fundamentally more ubiquitous and challenging than the former because attributes are not just implicit for most datasets, but also combinatorially complex, thus prohibitively expensive to be balanced. Therefore, we introduce a novel research problem: Generalized Long-Tailed classification (GLT), to jointly consider both kinds of imbalances. By "generalized", we mean that a GLT method should naturally solve the traditional LT, but not vice versa. Not surprisingly, we find that most class-wise LT methods degenerate in our proposed two benchmarks: ImageNet-GLT and MSCOCO-GLT. We argue that it is because they over-emphasize the adjustment of class distribution while neglecting to learn attribute-invariant features. To this end, we propose an Invariant Feature Learning (IFL) method as the first strong baseline for GLT. IFL first discovers environments with divergent intra-class distributions from the imperfect predictions and then learns invariant features across them. Promisingly, as an improved feature backbone, IFL boosts all the LT line-up: one/two-stage re-balance, augmentation, and ensemble. Codes and benchmarks are available on Github: https://github.com/KaihuaTang/Generalized-Long-Tailed-Benchmarks.pytorch

While neural networks are used for classification tasks across domains, a long-standing open problem in machine learning is determining whether neural networks trained using standard procedures are optimal for classification, i.e., whether such models minimize the probability of misclassification for arbitrary data distributions. In this work, we identify and construct an explicit set of neural network classifiers that achieve optimality. Since effective neural networks in practice are typically both wide and deep, we analyze infinitely wide networks that are also infinitely deep. In particular, using the recent connection between infinitely wide neural networks and Neural Tangent Kernels, we provide explicit activation functions that can be used to construct networks that achieve optimality. Interestingly, these activation functions are simple and easy to implement, yet differ from commonly used activations such as ReLU or sigmoid. More generally, we create a taxonomy of infinitely wide and deep networks and show that these models implement one of three well-known classifiers depending on the activation function used: (1) 1-nearest neighbor (model predictions are given by the label of the nearest training example); (2) majority vote (model predictions are given by the label of the class with greatest representation in the training set); or (3) singular kernel classifiers (a set of classifiers containing those that achieve optimality). Our results highlight the benefit of using deep networks for classification tasks, in contrast to regression tasks, where excessive depth is harmful.

Region-of-Interest (ROI)-based image compression allocates bits unevenly according to the semantic importance of different regions. Such differentiated coding typically induces a sharp-peaked and heavy-tailed distribution. This distribution characteristic mathematically necessitates a probability model with adaptable shape parameters for accurate description. However, existing methods commonly use a Gaussian model to fit this distribution, resulting in a loss of coding performance. To systematically analyze the impact of this distribution on ROI coding, we develop a unified rate-distortion optimization theoretical paradigm. Building on this paradigm, we propose a novel Generalized Gaussian Model (GGM) to achieve flexible modeling of the latent variables distribution. To support stable optimization of GGM, we introduce effective differentiable functions and further propose a dynamic lower bound to alleviate train-test mismatch. Moreover, finite differences are introduced to solve the gradient computation after GGM fits the distribution. Experiments on COCO2017 demonstrate that our method achieves state-of-the-art in both ROI reconstruction and downstream tasks (e.g., Segmentation, Object Detection). Furthermore, compared to classical probability models, our GGM provides a more precise fit to feature distributions and achieves superior coding performance. The project page is at https://github.com/hukai-tju/ROIGGM.

Unsupervised reinforcement learning with verifiable rewards (URLVR) offers a pathway to scale LLM training beyond the supervision bottleneck by deriving rewards without ground truth labels. Recent works leverage model intrinsic signals, showing promising early gains, yet their potential and limitations remain unclear. In this work, we revisit URLVR and provide a comprehensive analysis spanning taxonomy, theory and extensive experiments. We first classify URLVR methods into intrinsic versus external based on reward sources, then establish a unified theoretical framework revealing that all intrinsic methods converge toward sharpening the model's initial distribution This sharpening mechanism succeeds when initial confidence aligns with correctness but fails catastrophically when misaligned. Through systematic experiments, we show intrinsic rewards consistently follow a rise-then-fall pattern across methods, with collapse timing determined by model prior rather than engineering choices. Despite these scaling limits, we find intrinsic rewards remain valuable in test-time training on small datasets, and propose Model Collapse Step to measure model prior, serving as a practical indicator for RL trainability. Finally, we explore external reward methods that ground verification in computational asymmetries, showing preliminary evidence they may escape the confidence-correctness ceiling. Our findings chart boundaries for intrinsic URLVR while motivating paths toward scalable alternatives.

Straight line equation y=mx with slope m, when singularly perturbed as ay^3+y=mx with a positive parameter a, results in S-shaped curves or S-curves on a real plane. As arightarrow 0, we get back y=mx which is a cumulative distribution function of a continuous uniform distribution that describes the occurrence of every event in an interval to be equally probable. As arightarrowinfty, the derivative of y has finite support only at y=0 resembling a degenerate distribution. Based on these arguments, in this work, we propose that these S-curves can represent maximum entropy uniform distribution to a zero entropy single value. We also argue that these S-curves are superposable as they are only parametrically nonlinear but fundamentally linear. So far, the superposed forms have been used to capture the patterns of natural systems such as nonlinear dynamics of biological growth and kinetics of enzyme reactions. Here, we attempt to use the S-curve and its superposed form as statistical models. We fit the models on a classical dataset containing flower measurements of iris plants and analyze their usefulness in pattern recognition. Based on these models, we claim that any non-uniform pattern can be represented as a singular perturbation to uniform distribution. However, our parametric estimation procedure have some limitations such as sensitivity to initial conditions depending on the data at hand.

Neural scaling laws (NSL) refer to the phenomenon where model performance improves with scale. Sharma & Kaplan analyzed NSL using approximation theory and predict that MSE losses decay as N^{-alpha}, alpha=4/d, where N is the number of model parameters, and d is the intrinsic input dimension. Although their theory works well for some cases (e.g., ReLU networks), we surprisingly find that a simple 1D problem y=x^2 manifests a different scaling law (alpha=1) from their predictions (alpha=4). We opened the neural networks and found that the new scaling law originates from lottery ticket ensembling: a wider network on average has more "lottery tickets", which are ensembled to reduce the variance of outputs. We support the ensembling mechanism by mechanistically interpreting single neural networks, as well as studying them statistically. We attribute the N^{-1} scaling law to the "central limit theorem" of lottery tickets. Finally, we discuss its potential implications for large language models and statistical physics-type theories of learning.

On-the-fly category discovery (OCD) aims to recognize known categories while simultaneously discovering novel ones from an unlabeled online stream, using a model trained only on labeled data. Existing approaches freeze the feature extractor trained offline and employ a hash-based framework that quantizes features into binary codes as class prototypes. However, discovering novel categories with a fixed knowledge base is counterintuitive, as the learning potential of incoming data is entirely neglected. In addition, feature quantization introduces information loss, diminishes representational expressiveness, and amplifies intra-class variance. It often results in category explosion, where a single class is fragmented into multiple pseudo-classes. To overcome these limitations, we propose a test-time adaptation framework that enables learning through discovery. It incorporates two complementary strategies: a semantic-aware prototype update and a stable test-time encoder update. The former dynamically refines class prototypes to enhance classification, whereas the latter integrates new information directly into the parameter space. Together, these components allow the model to continuously expand its knowledge base with newly encountered samples. Furthermore, we introduce a margin-aware logit calibration in the offline stage to enlarge inter-class margins and improve intra-class compactness, thereby reserving embedding space for future class discovery. Experiments on standard OCD benchmarks demonstrate that our method substantially outperforms existing hash-based state-of-the-art approaches, yielding notable improvements in novel-class accuracy and effectively mitigating category explosion. The code is publicly available at blue{https://github.com/ynanwu/TALON}.

The traditional kernel density estimator of an unknown density is by construction completely nonparametric, in the sense that it has no preferences and will work reasonably well for all shapes. The present paper develops a class of semiparametric methods that are designed to work better than the kernel estimator in a broad nonparametric neighbourhood of a given parametric class of densities, for example the normal, while not losing much in precision when the true density is far from the parametric class. The idea is to multiply an initial parametric density estimate with a kernel type estimate of the necessary correction factor. This works well in cases where the correction factor function is less rough than the original density itself. Extensive comparisons with the kernel estimator are carried out, including exact analysis for the class of all normal mixtures. The new method, with a normal start, wins quite often, even in many cases where the true density is far from normal. Procedures for choosing the smoothing parameter of the estimator are also discussed. The new estimator should be particularly useful in higher dimensions, where the usual nonparametric methods have problems. The idea is also spelled out for nonparametric regression.

Entropy and its various generalizations are important in many fields, including mathematical statistics, communication theory, physics and computer science, for characterizing the amount of information associated with a probability distribution. In this paper we propose goodness-of-fit statistics for the multivariate Student and multivariate Pearson type II distributions, based on the maximum entropy principle and a class of estimators for Renyi entropy based on nearest neighbour distances. We prove the L2-consistency of these statistics using results on the subadditivity of Euclidean functionals on nearest neighbour graphs, and investigate their rate of convergence and asymptotic distribution using Monte Carlo methods.

It is impossible today to pretend that the practice of machine learning is compatible with the idea that training and testing data follow the same distribution. Several authors have recently used ensemble techniques to show how scenarios involving multiple data distributions are best served by representations that are both richer than those obtained by regularizing for the best in-distribution performance, and richer than those obtained under the influence of the implicit sparsity bias of common stochastic gradient procedures. This contribution investigates the use of very high dropout rates instead of ensembles to obtain such rich representations. Although training a deep network from scratch using such dropout rates is virtually impossible, fine-tuning a large pre-trained model under such conditions is not only possible but also achieves out-of-distribution performances that exceed those of both ensembles and weight averaging methods such as model soups. This result has practical significance because the importance of the fine-tuning scenario has considerably grown in recent years. This result also provides interesting insights on the nature of rich representations and on the intrinsically linear nature of fine-tuning a large network using a comparatively small dataset.

This paper studies the prediction of a target z from a pair of random variables (x,y), where the ground-truth predictor is additive E[z mid x,y] = f_star(x) +g_{star}(y). We study the performance of empirical risk minimization (ERM) over functions f+g, f in F and g in G, fit on a given training distribution, but evaluated on a test distribution which exhibits covariate shift. We show that, when the class F is "simpler" than G (measured, e.g., in terms of its metric entropy), our predictor is more resilient to heterogenous covariate shifts in which the shift in x is much greater than that in y. These results rely on a novel H\"older style inequality for the Dudley integral which may be of independent interest. Moreover, we corroborate our theoretical findings with experiments demonstrating improved resilience to shifts in "simpler" features across numerous domains.

We consider the problem of weakly supervised object detection, where the training samples are annotated using only image-level labels that indicate the presence or absence of an object category. In order to model the uncertainty in the location of the objects, we employ a dissimilarity coefficient based probabilistic learning objective. The learning objective minimizes the difference between an annotation agnostic prediction distribution and an annotation aware conditional distribution. The main computational challenge is the complex nature of the conditional distribution, which consists of terms over hundreds or thousands of variables. The complexity of the conditional distribution rules out the possibility of explicitly modeling it. Instead, we exploit the fact that deep learning frameworks rely on stochastic optimization. This allows us to use a state of the art discrete generative model that can provide annotation consistent samples from the conditional distribution. Extensive experiments on PASCAL VOC 2007 and 2012 data sets demonstrate the efficacy of our proposed approach.

To learn intrinsic low-dimensional structures from high-dimensional data that most discriminate between classes, we propose the principle of Maximal Coding Rate Reduction (MCR^2), an information-theoretic measure that maximizes the coding rate difference between the whole dataset and the sum of each individual class. We clarify its relationships with most existing frameworks such as cross-entropy, information bottleneck, information gain, contractive and contrastive learning, and provide theoretical guarantees for learning diverse and discriminative features. The coding rate can be accurately computed from finite samples of degenerate subspace-like distributions and can learn intrinsic representations in supervised, self-supervised, and unsupervised settings in a unified manner. Empirically, the representations learned using this principle alone are significantly more robust to label corruptions in classification than those using cross-entropy, and can lead to state-of-the-art results in clustering mixed data from self-learned invariant features.

Large transformer-based models are able to perform in-context few-shot learning, without being explicitly trained for it. This observation raises the question: what aspects of the training regime lead to this emergent behavior? Here, we show that this behavior is driven by the distributions of the training data itself. In-context learning emerges when the training data exhibits particular distributional properties such as burstiness (items appear in clusters rather than being uniformly distributed over time) and having large numbers of rarely occurring classes. In-context learning also emerges more strongly when item meanings or interpretations are dynamic rather than fixed. These properties are exemplified by natural language, but are also inherent to naturalistic data in a wide range of other domains. They also depart significantly from the uniform, i.i.d. training distributions typically used for standard supervised learning. In our initial experiments, we found that in-context learning traded off against more conventional weight-based learning, and models were unable to achieve both simultaneously. However, our later experiments uncovered that the two modes of learning could co-exist in a single model when it was trained on data following a skewed Zipfian distribution -- another common property of naturalistic data, including language. In further experiments, we found that naturalistic data distributions were only able to elicit in-context learning in transformers, and not in recurrent models. In sum, our findings indicate how the transformer architecture works together with particular properties of the training data to drive the intriguing emergent in-context learning behaviour of large language models, and how future work might encourage both in-context and in-weights learning in domains beyond language.

Deep learning has achieved tremendous success with independent and identically distributed (i.i.d.) data. However, the performance of neural networks often degenerates drastically when encountering out-of-distribution (OoD) data, i.e., when training and test data are sampled from different distributions. While a plethora of algorithms have been proposed for OoD generalization, our understanding of the data used to train and evaluate these algorithms remains stagnant. In this work, we first identify and measure two distinct kinds of distribution shifts that are ubiquitous in various datasets. Next, through extensive experiments, we compare OoD generalization algorithms across two groups of benchmarks, each dominated by one of the distribution shifts, revealing their strengths on one shift as well as limitations on the other shift. Overall, we position existing datasets and algorithms from different research areas seemingly unconnected into the same coherent picture. It may serve as a foothold that can be resorted to by future OoD generalization research. Our code is available at https://github.com/ynysjtu/ood_bench.

Learning distribution families over R^d is a fundamental problem in unsupervised learning and statistics. A central question in this setting is whether a given family of distributions possesses sufficient structure to be (at least) information-theoretically learnable and, if so, to characterize its sample complexity. In 2018, Ashtiani et al. reframed sample compressibility, originally due to Littlestone and Warmuth (1986), as a structural property of distribution classes, proving that it guarantees PAC-learnability. This discovery subsequently enabled a series of recent advancements in deriving nearly tight sample complexity bounds for various high-dimensional open problems. It has been further conjectured that the converse also holds: every learnable class admits a tight sample compression scheme. In this work, we establish that sample compressible families remain learnable even from perturbed samples, subject to a set of necessary and sufficient conditions. We analyze two models of data perturbation: (i) an additive independent noise model, and (ii) an adversarial corruption model, where an adversary manipulates a limited subset of the samples unknown to the learner. Our results are general and rely on as minimal assumptions as possible. We develop a perturbation-quantization framework that interfaces naturally with the compression scheme and leads to sample complexity bounds that scale gracefully with the noise level and corruption budget. As concrete applications, we establish new sample complexity bounds for learning finite mixtures of high-dimensional uniform distributions under both noise and adversarial perturbations, as well as for learning Gaussian mixture models from adversarially corrupted samples, resolving two open problems in the literature.

Semi-supervised learning based methods are current SOTA solutions to the noisy-label learning problem, which rely on learning an unsupervised label cleaner first to divide the training samples into a labeled set for clean data and an unlabeled set for noise data. Typically, the cleaner is obtained via fitting a mixture model to the distribution of per-sample training losses. However, the modeling procedure is class agnostic and assumes the loss distributions of clean and noise samples are the same across different classes. Unfortunately, in practice, such an assumption does not always hold due to the varying learning difficulty of different classes, thus leading to sub-optimal label noise partition criteria. In this work, we reveal this long-ignored problem and propose a simple yet effective solution, named Class Prototype-based label noise Cleaner (CPC). Unlike previous works treating all the classes equally, CPC fully considers loss distribution heterogeneity and applies class-aware modulation to partition the clean and noise data. CPC takes advantage of loss distribution modeling and intra-class consistency regularization in feature space simultaneously and thus can better distinguish clean and noise labels. We theoretically justify the effectiveness of our method by explaining it from the Expectation-Maximization (EM) framework. Extensive experiments are conducted on the noisy-label benchmarks CIFAR-10, CIFAR-100, Clothing1M and WebVision. The results show that CPC consistently brings about performance improvement across all benchmarks. Codes and pre-trained models will be released at https://github.com/hjjpku/CPC.git.

The presence of distribution shifts poses a significant challenge for deploying modern machine learning models in real-world applications. This work focuses on the target shift problem in a regression setting (Zhang et al., 2013; Nguyen et al., 2016). More specifically, the target variable y (also known as the response variable), which is continuous, has different marginal distributions in the training source and testing domain, while the conditional distribution of features x given y remains the same. While most literature focuses on classification tasks with finite target space, the regression problem has an infinite dimensional target space, which makes many of the existing methods inapplicable. In this work, we show that the continuous target shift problem can be addressed by estimating the importance weight function from an ill-posed integral equation. We propose a nonparametric regularized approach named ReTaSA to solve the ill-posed integral equation and provide theoretical justification for the estimated importance weight function. The effectiveness of the proposed method has been demonstrated with extensive numerical studies on synthetic and real-world datasets.

Instance segmentation is data-hungry, and as model capacity increases, data scale becomes crucial for improving the accuracy. Most instance segmentation datasets today require costly manual annotation, limiting their data scale. Models trained on such data are prone to overfitting on the training set, especially for those rare categories. While recent works have delved into exploiting generative models to create synthetic datasets for data augmentation, these approaches do not efficiently harness the full potential of generative models. To address these issues, we introduce a more efficient strategy to construct generative datasets for data augmentation, termed DiverGen. Firstly, we provide an explanation of the role of generative data from the perspective of distribution discrepancy. We investigate the impact of different data on the distribution learned by the model. We argue that generative data can expand the data distribution that the model can learn, thus mitigating overfitting. Additionally, we find that the diversity of generative data is crucial for improving model performance and enhance it through various strategies, including category diversity, prompt diversity, and generative model diversity. With these strategies, we can scale the data to millions while maintaining the trend of model performance improvement. On the LVIS dataset, DiverGen significantly outperforms the strong model X-Paste, achieving +1.1 box AP and +1.1 mask AP across all categories, and +1.9 box AP and +2.5 mask AP for rare categories.

While score-based generative models (SGMs) have achieved remarkable success in enormous image generation tasks, their mathematical foundations are still limited. In this paper, we analyze the approximation and generalization of SGMs in learning a family of sub-Gaussian probability distributions. We introduce a notion of complexity for probability distributions in terms of their relative density with respect to the standard Gaussian measure. We prove that if the log-relative density can be locally approximated by a neural network whose parameters can be suitably bounded, then the distribution generated by empirical score matching approximates the target distribution in total variation with a dimension-independent rate. We illustrate our theory through examples, which include certain mixtures of Gaussians. An essential ingredient of our proof is to derive a dimension-free deep neural network approximation rate for the true score function associated with the forward process, which is interesting in its own right.

We propose a goodness-of-fit measure for probability densities modeling observations with varying dimensionality, such as text documents of differing lengths or variable-length sequences. The proposed measure is an instance of the kernel Stein discrepancy (KSD), which has been used to construct goodness-of-fit tests for unnormalized densities. The KSD is defined by its Stein operator: current operators used in testing apply to fixed-dimensional spaces. As our main contribution, we extend the KSD to the variable-dimension setting by identifying appropriate Stein operators, and propose a novel KSD goodness-of-fit test. As with the previous variants, the proposed KSD does not require the density to be normalized, allowing the evaluation of a large class of models. Our test is shown to perform well in practice on discrete sequential data benchmarks.

Addressing imbalanced or long-tailed data is a major challenge in visual recognition tasks due to disparities between training and testing distributions and issues with data noise. We propose the Wrapped Cauchy Distributed Angular Softmax (WCDAS), a novel softmax function that incorporates data-wise Gaussian-based kernels into the angular correlation between feature representations and classifier weights, effectively mitigating noise and sparse sampling concerns. The class-wise distribution of angular representation becomes a sum of these kernels. Our theoretical analysis reveals that the wrapped Cauchy distribution excels the Gaussian distribution in approximating mixed distributions. Additionally, WCDAS uses trainable concentration parameters to dynamically adjust the compactness and margin of each class. Empirical results confirm label-aware behavior in these parameters and demonstrate WCDAS's superiority over other state-of-the-art softmax-based methods in handling long-tailed visual recognition across multiple benchmark datasets. The code is public available.

Black-box machine learning models are now routinely used in high-risk settings, like medical diagnostics, which demand uncertainty quantification to avoid consequential model failures. Conformal prediction is a user-friendly paradigm for creating statistically rigorous uncertainty sets/intervals for the predictions of such models. Critically, the sets are valid in a distribution-free sense: they possess explicit, non-asymptotic guarantees even without distributional assumptions or model assumptions. One can use conformal prediction with any pre-trained model, such as a neural network, to produce sets that are guaranteed to contain the ground truth with a user-specified probability, such as 90%. It is easy-to-understand, easy-to-use, and general, applying naturally to problems arising in the fields of computer vision, natural language processing, deep reinforcement learning, and so on. This hands-on introduction is aimed to provide the reader a working understanding of conformal prediction and related distribution-free uncertainty quantification techniques with one self-contained document. We lead the reader through practical theory for and examples of conformal prediction and describe its extensions to complex machine learning tasks involving structured outputs, distribution shift, time-series, outliers, models that abstain, and more. Throughout, there are many explanatory illustrations, examples, and code samples in Python. With each code sample comes a Jupyter notebook implementing the method on a real-data example; the notebooks can be accessed and easily run using our codebase.

We consider the problem of estimating how well a model class is capable of fitting a distribution of labeled data. We show that it is often possible to accurately estimate this "learnability" even when given an amount of data that is too small to reliably learn any accurate model. Our first result applies to the setting where the data is drawn from a d-dimensional distribution with isotropic covariance (or known covariance), and the label of each datapoint is an arbitrary noisy function of the datapoint. In this setting, we show that with O(d) samples, one can accurately estimate the fraction of the variance of the label that can be explained via the best linear function of the data. In contrast to this sublinear sample size, finding an approximation of the best-fit linear function requires on the order of d samples. Our sublinear sample results and approach also extend to the non-isotropic setting, where the data distribution has an (unknown) arbitrary covariance matrix: we show that, if the label y of point x is a linear function with independent noise, y = langle x , βrangle + noise with |β| bounded, the variance of the noise can be estimated to error ε with O(d^{1-1/1/ε}) if the covariance matrix has bounded condition number, or O(d^{1-sqrtε}) if there are no bounds on the condition number. We also establish that these sample complexities are optimal, to constant factors. Finally, we extend these techniques to the setting of binary classification, where we obtain analogous sample complexities for the problem of estimating the prediction error of the best linear classifier, in a natural model of binary labeled data. We demonstrate the practical viability of our approaches on several real and synthetic datasets.

Chain-of-thought (CoT) reasoning and its variants have substantially improved the performance of language models on complex reasoning tasks, yet the precise mechanisms by which different strategies facilitate generalization remain poorly understood. While current explanations often point to increased test-time computation or structural guidance, establishing a consistent, quantifiable link between these factors and generalization remains challenging. In this work, we identify intrinsic dimensionality as a quantitative measure for characterizing the effectiveness of reasoning chains. Intrinsic dimensionality quantifies the minimum number of model dimensions needed to reach a given accuracy threshold on a given task. By keeping the model architecture fixed and varying the task formulation through different reasoning strategies, we demonstrate that effective reasoning strategies consistently reduce the intrinsic dimensionality of the task. Validating this on GSM8K with Gemma-3 1B and 4B, we observe a strong inverse correlation between the intrinsic dimensionality of a reasoning strategy and its generalization performance on both in-distribution and out-of-distribution data. Our findings suggest that effective reasoning chains facilitate learning by better compressing the task using fewer parameters, offering a new quantitative metric for analyzing reasoning processes.

Anomalies are rare and anomaly detection is often therefore framed as One-Class Classification (OCC), i.e. trained solely on normalcy. Leading OCC techniques constrain the latent representations of normal motions to limited volumes and detect as abnormal anything outside, which accounts satisfactorily for the openset'ness of anomalies. But normalcy shares the same openset'ness property since humans can perform the same action in several ways, which the leading techniques neglect. We propose a novel generative model for video anomaly detection (VAD), which assumes that both normality and abnormality are multimodal. We consider skeletal representations and leverage state-of-the-art diffusion probabilistic models to generate multimodal future human poses. We contribute a novel conditioning on the past motion of people and exploit the improved mode coverage capabilities of diffusion processes to generate different-but-plausible future motions. Upon the statistical aggregation of future modes, an anomaly is detected when the generated set of motions is not pertinent to the actual future. We validate our model on 4 established benchmarks: UBnormal, HR-UBnormal, HR-STC, and HR-Avenue, with extensive experiments surpassing state-of-the-art results.

Outlier detection refers to the identification of data points that deviate from a general data distribution. Existing unsupervised approaches often suffer from high computational cost, complex hyperparameter tuning, and limited interpretability, especially when working with large, high-dimensional datasets. To address these issues, we present a simple yet effective algorithm called ECOD (Empirical-Cumulative-distribution-based Outlier Detection), which is inspired by the fact that outliers are often the "rare events" that appear in the tails of a distribution. In a nutshell, ECOD first estimates the underlying distribution of the input data in a nonparametric fashion by computing the empirical cumulative distribution per dimension of the data. ECOD then uses these empirical distributions to estimate tail probabilities per dimension for each data point. Finally, ECOD computes an outlier score of each data point by aggregating estimated tail probabilities across dimensions. Our contributions are as follows: (1) we propose a novel outlier detection method called ECOD, which is both parameter-free and easy to interpret; (2) we perform extensive experiments on 30 benchmark datasets, where we find that ECOD outperforms 11 state-of-the-art baselines in terms of accuracy, efficiency, and scalability; and (3) we release an easy-to-use and scalable (with distributed support) Python implementation for accessibility and reproducibility.

We consider the problem of approximating a function from L^2 by an element of a given m-dimensional space V_m, associated with some feature map varphi, using evaluations of the function at random points x_1,dots,x_n. After recalling some results on optimal weighted least-squares using independent and identically distributed points, we consider weighted least-squares using projection determinantal point processes (DPP) or volume sampling. These distributions introduce dependence between the points that promotes diversity in the selected features varphi(x_i). We first provide a generalized version of volume-rescaled sampling yielding quasi-optimality results in expectation with a number of samples n = O(mlog(m)), that means that the expected L^2 error is bounded by a constant times the best approximation error in L^2. Also, further assuming that the function is in some normed vector space H continuously embedded in L^2, we further prove that the approximation is almost surely bounded by the best approximation error measured in the H-norm. This includes the cases of functions from L^infty or reproducing kernel Hilbert spaces. Finally, we present an alternative strategy consisting in using independent repetitions of projection DPP (or volume sampling), yielding similar error bounds as with i.i.d. or volume sampling, but in practice with a much lower number of samples. Numerical experiments illustrate the performance of the different strategies.

Post-hoc out-of-distribution (OOD) detection has garnered intensive attention in reliable machine learning. Many efforts have been dedicated to deriving score functions based on logits, distances, or rigorous data distribution assumptions to identify low-scoring OOD samples. Nevertheless, these estimate scores may fail to accurately reflect the true data density or impose impractical constraints. To provide a unified perspective on density-based score design, we propose a novel theoretical framework grounded in Bregman divergence, which extends distribution considerations to encompass an exponential family of distributions. Leveraging the conjugation constraint revealed in our theorem, we introduce a ConjNorm method, reframing density function design as a search for the optimal norm coefficient p against the given dataset. In light of the computational challenges of normalization, we devise an unbiased and analytically tractable estimator of the partition function using the Monte Carlo-based importance sampling technique. Extensive experiments across OOD detection benchmarks empirically demonstrate that our proposed ConjNorm has established a new state-of-the-art in a variety of OOD detection setups, outperforming the current best method by up to 13.25% and 28.19% (FPR95) on CIFAR-100 and ImageNet-1K, respectively.

We introduce a new approach to nonlinear sufficient dimension reduction in cases where both the predictor and the response are distributional data, modeled as members of a metric space. Our key step is to build universal kernels (cc-universal) on the metric spaces, which results in reproducing kernel Hilbert spaces for the predictor and response that are rich enough to characterize the conditional independence that determines sufficient dimension reduction. For univariate distributions, we construct the universal kernel using the Wasserstein distance, while for multivariate distributions, we resort to the sliced Wasserstein distance. The sliced Wasserstein distance ensures that the metric space possesses similar topological properties to the Wasserstein space while also offering significant computation benefits. Numerical results based on synthetic data show that our method outperforms possible competing methods. The method is also applied to several data sets, including fertility and mortality data and Calgary temperature data.

Although pretrained language models can be fine-tuned to produce state-of-the-art results for a very wide range of language understanding tasks, the dynamics of this process are not well understood, especially in the low data regime. Why can we use relatively vanilla gradient descent algorithms (e.g., without strong regularization) to tune a model with hundreds of millions of parameters on datasets with only hundreds or thousands of labeled examples? In this paper, we argue that analyzing fine-tuning through the lens of intrinsic dimension provides us with empirical and theoretical intuitions to explain this remarkable phenomenon. We empirically show that common pre-trained models have a very low intrinsic dimension; in other words, there exists a low dimension reparameterization that is as effective for fine-tuning as the full parameter space. For example, by optimizing only 200 trainable parameters randomly projected back into the full space, we can tune a RoBERTa model to achieve 90\% of the full parameter performance levels on MRPC. Furthermore, we empirically show that pre-training implicitly minimizes intrinsic dimension and, perhaps surprisingly, larger models tend to have lower intrinsic dimension after a fixed number of pre-training updates, at least in part explaining their extreme effectiveness. Lastly, we connect intrinsic dimensionality with low dimensional task representations and compression based generalization bounds to provide intrinsic-dimension-based generalization bounds that are independent of the full parameter count.

We introduce a rank-statistic approximation of f-divergences that avoids explicit density-ratio estimation by working directly with the distribution of ranks. For a resolution parameter K, we map the mismatch between two univariate distributions μ and ν to a rank histogram on { 0, ldots, K} and measure its deviation from uniformity via a discrete f-divergence, yielding a rank-statistic divergence estimator. We prove that the resulting estimator of the divergence is monotone in K, is always a lower bound of the true f-divergence, and we establish quantitative convergence rates for Ktoinfty under mild regularity of the quantile-domain density ratio. To handle high-dimensional data, we define the sliced rank-statistic f-divergence by averaging the univariate construction over random projections, and we provide convergence results for the sliced limit as well. We also derive finite-sample deviation bounds along with asymptotic normality results for the estimator. Finally, we empirically validate the approach by benchmarking against neural baselines and illustrating its use as a learning objective in generative modeling experiments.

We develop an econometric framework integrating heavy-tailed Student's t distributions with behavioral probability weighting while preserving infinite divisibility. Using 432{,}752 observations across 86 assets (2004--2024), we demonstrate Student's t specifications outperform Gaussian models in 88.4\% of cases. Bounded probability-weighting transformations preserve mathematical properties required for dynamic pricing. Gaussian models underestimate 99\% Value-at-Risk by 19.7\% versus 3.2\% for our specification. Joint estimation procedures identify tail and behavioral parameters with established asymptotic properties. Results provide robust inference for asset-pricing applications where heavy tails and behavioral distortions coexist.

We present a novel approach for generating minority samples that live on low-density regions of a data manifold. Our framework is built upon diffusion models, leveraging the principle of guided sampling that incorporates an arbitrary energy-based guidance during inference time. The key defining feature of our sampler lies in its self-contained nature, \ie, implementable solely with a pretrained model. This distinguishes our sampler from existing techniques that require expensive additional components (like external classifiers) for minority generation. Specifically, we first estimate the likelihood of features within an intermediate latent sample by evaluating a reconstruction loss w.r.t. its posterior mean. The generation then proceeds with the minimization of the estimated likelihood, thereby encouraging the emergence of minority features in the latent samples of subsequent timesteps. To further improve the performance of our sampler, we provide several time-scheduling techniques that properly manage the influence of guidance over inference steps. Experiments on benchmark real datasets demonstrate that our approach can greatly improve the capability of creating realistic low-likelihood minority instances over the existing techniques without the reliance on costly additional elements. Code is available at https://github.com/soobin-um/sg-minority.

An approach to the construction of classifiers from imbalanced datasets is described. A dataset is imbalanced if the classification categories are not approximately equally represented. Often real-world data sets are predominately composed of "normal" examples with only a small percentage of "abnormal" or "interesting" examples. It is also the case that the cost of misclassifying an abnormal (interesting) example as a normal example is often much higher than the cost of the reverse error. Under-sampling of the majority (normal) class has been proposed as a good means of increasing the sensitivity of a classifier to the minority class. This paper shows that a combination of our method of over-sampling the minority (abnormal) class and under-sampling the majority (normal) class can achieve better classifier performance (in ROC space) than only under-sampling the majority class. This paper also shows that a combination of our method of over-sampling the minority class and under-sampling the majority class can achieve better classifier performance (in ROC space) than varying the loss ratios in Ripper or class priors in Naive Bayes. Our method of over-sampling the minority class involves creating synthetic minority class examples. Experiments are performed using C4.5, Ripper and a Naive Bayes classifier. The method is evaluated using the area under the Receiver Operating Characteristic curve (AUC) and the ROC convex hull strategy.

We introduce a novel generative model, the Discrete Distribution Networks (DDN), that approximates data distribution using hierarchical discrete distributions. We posit that since the features within a network inherently capture distributional information, enabling the network to generate multiple samples simultaneously, rather than a single output, may offer an effective way to represent distributions. Therefore, DDN fits the target distribution, including continuous ones, by generating multiple discrete sample points. To capture finer details of the target data, DDN selects the output that is closest to the Ground Truth (GT) from the coarse results generated in the first layer. This selected output is then fed back into the network as a condition for the second layer, thereby generating new outputs more similar to the GT. As the number of DDN layers increases, the representational space of the outputs expands exponentially, and the generated samples become increasingly similar to the GT. This hierarchical output pattern of discrete distributions endows DDN with unique properties: more general zero-shot conditional generation and 1D latent representation. We demonstrate the efficacy of DDN and its intriguing properties through experiments on CIFAR-10 and FFHQ. The code is available at https://discrete-distribution-networks.github.io/

Categorical variables are a natural choice for representing discrete structure in the world. However, stochastic neural networks rarely use categorical latent variables due to the inability to backpropagate through samples. In this work, we present an efficient gradient estimator that replaces the non-differentiable sample from a categorical distribution with a differentiable sample from a novel Gumbel-Softmax distribution. This distribution has the essential property that it can be smoothly annealed into a categorical distribution. We show that our Gumbel-Softmax estimator outperforms state-of-the-art gradient estimators on structured output prediction and unsupervised generative modeling tasks with categorical latent variables, and enables large speedups on semi-supervised classification.

In contrast to classical reinforcement learning, distributional reinforcement learning algorithms aim to learn the distribution of returns rather than their expected value. Since the nature of the return distribution is generally unknown a priori or arbitrarily complex, a common approach finds approximations within a set of representable, parametric distributions. Typically, this involves a projection of the unconstrained distribution onto the set of simplified distributions. We argue that this projection step entails a strong inductive bias when coupled with neural networks and gradient descent, thereby profoundly impacting the generalization behavior of learned models. In order to facilitate reliable uncertainty estimation through diversity, this work studies the combination of several different projections and representations in a distributional ensemble. We establish theoretical properties of such projection ensembles and derive an algorithm that uses ensemble disagreement, measured by the average 1-Wasserstein distance, as a bonus for deep exploration. We evaluate our algorithm on the behavior suite benchmark and find that diverse projection ensembles lead to significant performance improvements over existing methods on a wide variety of tasks with the most pronounced gains in directed exploration problems.

We propose a deep learning framework for modeling complex high-dimensional densities called Non-linear Independent Component Estimation (NICE). It is based on the idea that a good representation is one in which the data has a distribution that is easy to model. For this purpose, a non-linear deterministic transformation of the data is learned that maps it to a latent space so as to make the transformed data conform to a factorized distribution, i.e., resulting in independent latent variables. We parametrize this transformation so that computing the Jacobian determinant and inverse transform is trivial, yet we maintain the ability to learn complex non-linear transformations, via a composition of simple building blocks, each based on a deep neural network. The training criterion is simply the exact log-likelihood, which is tractable. Unbiased ancestral sampling is also easy. We show that this approach yields good generative models on four image datasets and can be used for inpainting.

Reward models are crucial for aligning large language models (LLMs) with human values and intentions. Existing approaches follow either Generative (GRMs) or Discriminative (DRMs) paradigms, yet both suffer from limitations: GRMs typically demand costly point-wise supervision, while DRMs produce uncalibrated relative scores that lack probabilistic interpretation. To address these challenges, we introduce a novel reward modeling paradigm: Probabilistic Reward Model (PRM). Instead of modeling reward as a deterministic scalar, our approach treats it as a random variable, learning a full probability distribution for the quality of each response. To make this paradigm practical, we present its closed-form, discrete realization: the Ordinal Probabilistic Reward Model (OPRM), which discretizes the quality score into a finite set of ordinal ratings. Building on OPRM, we propose a data-efficient training strategy called Region Flooding Tuning (RgFT). It enables rewards to better reflect absolute text quality by incorporating quality-level annotations, which guide the model to concentrate the probability mass within corresponding rating sub-regions. Experiments on various reward model benchmarks show that our method improves accuracy by 2.9%sim7.4% compared to prior reward models, demonstrating strong performance and data efficiency. Analysis of the score distribution provides evidence that our method captures not only relative rankings but also absolute quality.

Many real-world applications reveal difficulties in learning classifiers from imbalanced data. The rising big data era has been witnessing more classification tasks with large-scale but extremely imbalance and low-quality datasets. Most of existing learning methods suffer from poor performance or low computation efficiency under such a scenario. To tackle this problem, we conduct deep investigations into the nature of class imbalance, which reveals that not only the disproportion between classes, but also other difficulties embedded in the nature of data, especially, noises and class overlapping, prevent us from learning effective classifiers. Taking those factors into consideration, we propose a novel framework for imbalance classification that aims to generate a strong ensemble by self-paced harmonizing data hardness via under-sampling. Extensive experiments have shown that this new framework, while being very computationally efficient, can lead to robust performance even under highly overlapping classes and extremely skewed distribution. Note that, our methods can be easily adapted to most of existing learning methods (e.g., C4.5, SVM, GBDT and Neural Network) to boost their performance on imbalanced data.

We introduce UnpredictaBench, an evaluation that tests the ability of large language models (LLMs) to capture true underlying distributions. As LLMs are increasingly used as substitutes for other entities (e.g., for humans in economic simulations), the tendency of many models to collapse towards a single plausible answer means a failure to capture the unpredictability of real systems. Recent work on improving output diversity is insufficient for this setting: simulation requires samples that are calibrated to a target distribution, not merely varied outputs. UnpredictaBench isolates a simplified but fundamental version of this problem: sampling outcomes from individual target distributions, including canonical statistical distributions, distributions induced by stochastic programs, and natural-language scenarios that describe random processes. We introduce 448 such problems together with KS@N, a general-purpose evaluation metric that quantifies how well a model outputs approximate black-box target distributions via the Kolmogorov-Smirnov statistical test. This is the rate at which we fail to reject model samples of size N against ground-truth samples, with larger N indicating greater difficulty. Tested across open and proprietary models, we find a large spread in distributional capabilities. For instance, when models generate samples of size 100 (KS@100, our standard metric), scores range from near 0 to over 20%. No model is able to achieve over 40% at KS@100, showing significant headroom in distributional sampling as a capability. Although adding reasoning can somewhat increase scores, we find no immediate solution for this issue. UnpredictaBench shows that even simple distributional simulation remains challenging, making it a necessary first step toward using LLMs as stand-ins for complex systems.

We develop a rigorous mathematical analysis of zero-shot learning with attributes. In this setting, the goal is to label novel classes with no training data, only detectors for attributes and a description of how those attributes are correlated with the target classes, called the class-attribute matrix. We develop the first non-trivial lower bound on the worst-case error of the best map from attributes to classes for this setting, even with perfect attribute detectors. The lower bound characterizes the theoretical intrinsic difficulty of the zero-shot problem based on the available information -- the class-attribute matrix -- and the bound is practically computable from it. Our lower bound is tight, as we show that we can always find a randomized map from attributes to classes whose expected error is upper bounded by the value of the lower bound. We show that our analysis can be predictive of how standard zero-shot methods behave in practice, including which classes will likely be confused with others.

Hierarchical classification is a crucial task in many applications, where objects are organized into multiple levels of categories. However, conventional classification approaches often neglect inherent inter-class relationships at different hierarchy levels, thus missing important supervisory signals. Thus, we propose two novel hierarchical contrastive learning (HMLC) methods. The first, leverages a Gaussian Mixture Model (G-HMLC) and the second uses an attention mechanism to capture hierarchy-specific features (A-HMLC), imitating human processing. Our approach explicitly models inter-class relationships and imbalanced class distribution at higher hierarchy levels, enabling fine-grained clustering across all hierarchy levels. On the competitive CIFAR100 and ModelNet40 datasets, our method achieves state-of-the-art performance in linear evaluation, outperforming existing hierarchical contrastive learning methods by 2 percentage points in terms of accuracy. The effectiveness of our approach is backed by both quantitative and qualitative results, highlighting its potential for applications in computer vision and beyond.

A prominent family of methods for learning data distributions relies on density ratio estimation (DRE), where a model is trained to classify between data samples and samples from some reference distribution. DRE-based models can directly output the likelihood for any given input, a highly desired property that is lacking in most generative techniques. Nevertheless, to date, DRE methods have failed in accurately capturing the distributions of complex high-dimensional data, like images, and have thus been drawing reduced research attention in recent years. In this work we present classification diffusion models (CDMs), a DRE-based generative method that adopts the formalism of denoising diffusion models (DDMs) while making use of a classifier that predicts the level of noise added to a clean signal. Our method is based on an analytical connection that we derive between the MSE-optimal denoiser for removing white Gaussian noise and the cross-entropy-optimal classifier for predicting the noise level. Our method is the first DRE-based technique that can successfully generate images beyond the MNIST dataset. Furthermore, it can output the likelihood of any input in a single forward pass, achieving state-of-the-art negative log likelihood (NLL) among methods with this property. Code is available on the project's webpage in https://shaharYadin.github.io/CDM/ .

Multiple instance learning (MIL) is a variation of supervised learning where a single class label is assigned to a bag of instances. In this paper, we state the MIL problem as learning the Bernoulli distribution of the bag label where the bag label probability is fully parameterized by neural networks. Furthermore, we propose a neural network-based permutation-invariant aggregation operator that corresponds to the attention mechanism. Notably, an application of the proposed attention-based operator provides insight into the contribution of each instance to the bag label. We show empirically that our approach achieves comparable performance to the best MIL methods on benchmark MIL datasets and it outperforms other methods on a MNIST-based MIL dataset and two real-life histopathology datasets without sacrificing interpretability.

Continuous latent variables (LVs) are a key ingredient of many generative models, as they allow modelling expressive mixtures with an uncountable number of components. In contrast, probabilistic circuits (PCs) are hierarchical discrete mixtures represented as computational graphs composed of input, sum and product units. Unlike continuous LV models, PCs provide tractable inference but are limited to discrete LVs with categorical (i.e. unordered) states. We bridge these model classes by introducing probabilistic integral circuits (PICs), a new language of computational graphs that extends PCs with integral units representing continuous LVs. In the first place, PICs are symbolic computational graphs and are fully tractable in simple cases where analytical integration is possible. In practice, we parameterise PICs with light-weight neural nets delivering an intractable hierarchical continuous mixture that can be approximated arbitrarily well with large PCs using numerical quadrature. On several distribution estimation benchmarks, we show that such PIC-approximating PCs systematically outperform PCs commonly learned via expectation-maximization or SGD.

Recently, Qiao, Duan, and Cheng~(2019) proposed a distributed nearest-neighbor classification method, in which a massive dataset is split into smaller groups, each processed with a k-nearest-neighbor classifier, and the final class label is predicted by a majority vote among these groupwise class labels. This paper shows that the distributed algorithm with k=1 over a sufficiently large number of groups attains a minimax optimal error rate up to a multiplicative logarithmic factor under some regularity conditions, for both regression and classification problems. Roughly speaking, distributed 1-nearest-neighbor rules with M groups has a performance comparable to standard Theta(M)-nearest-neighbor rules. In the analysis, alternative rules with a refined aggregation method are proposed and shown to attain exact minimax optimal rates.

We develop a method to generate predictive regions that cover a multivariate response variable with a user-specified probability. Our work is composed of two components. First, we use a deep generative model to learn a representation of the response that has a unimodal distribution. Existing multiple-output quantile regression approaches are effective in such cases, so we apply them on the learned representation, and then transform the solution to the original space of the response. This process results in a flexible and informative region that can have an arbitrary shape, a property that existing methods lack. Second, we propose an extension of conformal prediction to the multivariate response setting that modifies any method to return sets with a pre-specified coverage level. The desired coverage is theoretically guaranteed in the finite-sample case for any distribution. Experiments conducted on both real and synthetic data show that our method constructs regions that are significantly smaller compared to existing techniques.

The Beta kernel estimator offers a theoretically superior alternative to the Gaussian kernel for unit interval data, eliminating boundary bias without requiring reflection or transformation. However, its adoption remains limited by the lack of a reliable bandwidth selector; practitioners currently rely on iterative optimization methods that are computationally expensive and prone to instability. We derive the ``Beta Reference Rule,'' a fast, closed-form bandwidth selector based on the unweighted Asymptotic Mean Integrated Squared Error (AMISE) of a beta reference distribution. To address boundary integrability issues, we introduce a principled heuristic for U-shaped and J-shaped distributions. By employing a method-of-moments approximation, we reduce the bandwidth selection complexity from iterative optimization to O(1). Extensive Monte Carlo simulations demonstrate that our rule matches the accuracy of numerical optimization while delivering a speedup of over 35,000 times. Real-world validation on socioeconomic data shows that it avoids the ``vanishing boundary'' and ``shoulder'' artifacts common to Gaussian-based methods. We provide a comprehensive, open-source Python package to facilitate the immediate adoption of the Beta kernel as a drop-in replacement for standard density estimation tools.

Loss functions serve as the foundation of supervised learning and are often chosen prior to model development. To avoid potentially ad hoc choices of losses, statistical decision theory describes a desirable property for losses known as properness, which asserts that Bayes' rule is optimal. Recent works have sought to learn losses and models jointly. Existing methods do this by fitting an inverse canonical link function which monotonically maps R to [0,1] to estimate probabilities for binary problems. In this paper, we extend monotonicity to maps between R^{C-1} and the projected probability simplex Delta^{C-1} by using monotonicity of gradients of convex functions. We present {\sc LegendreTron} as a novel and practical method that jointly learns proper canonical losses and probabilities for multiclass problems. Tested on a benchmark of domains with up to 1,000 classes, our experimental results show that our method consistently outperforms the natural multiclass baseline under a t-test at 99% significance on all datasets with greater than 10 classes.

Large-scale commercial platforms usually involve numerous business domains for diverse business strategies and expect their recommendation systems to provide click-through rate (CTR) predictions for multiple domains simultaneously. Existing promising and widely-used multi-domain models discover domain relationships by explicitly constructing domain-specific networks, but the computation and memory boost significantly with the increase of domains. To reduce computational complexity, manually grouping domains with particular business strategies is common in industrial applications. However, this pre-defined data partitioning way heavily relies on prior knowledge, and it may neglect the underlying data distribution of each domain, hence limiting the model's representation capability. Regarding the above issues, we propose an elegant and flexible multi-distribution modeling paradigm, named Adaptive Distribution Hierarchical Model (AdaptDHM), which is an end-to-end optimization hierarchical structure consisting of a clustering process and classification process. Specifically, we design a distribution adaptation module with a customized dynamic routing mechanism. Instead of introducing prior knowledge for pre-defined data allocation, this routing algorithm adaptively provides a distribution coefficient for each sample to determine which cluster it belongs to. Each cluster corresponds to a particular distribution so that the model can sufficiently capture the commonalities and distinctions between these distinct clusters. Extensive experiments on both public and large-scale Alibaba industrial datasets verify the effectiveness and efficiency of AdaptDHM: Our model achieves impressive prediction accuracy and its time cost during the training stage is more than 50% less than that of other models.

We explore the relations between the zeta distribution and algorithmic information theory via a new model of the transfer learning problem. The program distribution is approximated by a zeta distribution with parameter near 1. We model the training sequence as a stochastic process. We analyze the upper temporal bound for learning a training sequence and its entropy rates, assuming an oracle for the transfer learning problem. We argue from empirical evidence that power-law models are suitable for natural processes. Four sequence models are proposed. Random typing model is like no-free lunch where transfer learning does not work. Zeta process independently samples programs from the zeta distribution. A model of common sub-programs inspired by genetics uses a database of sub-programs. An evolutionary zeta process samples mutations from Zeta distribution. The analysis of stochastic processes inspired by evolution suggest that AI may be feasible in nature, countering no-free lunch sort of arguments.