Distributions

Understanding the distributional structure of high-dimensional datasets has become an important topic, yet direct visual characterization is difficult. In this work, we develop a geometric framework for characterizing the distributional structure of empirical datasets by quantifying their deviation from the Gaussian family under the geometry induced by optimal transport theory. Building on the cone structure of the relative translation invariant quadratic Wasserstein $(RW_2)$ space, we define two geometric quantities---the \emph{relative Wasserstein angle} and the \emph{orthogonal projection distance}---and show that they are well-defined because of the flat geometry of the filling cone between distributional rays. This formulation recasts the problem of measuring deviation from the Gaussian family as an orthogonal projection problem onto the Gaussian cone and reveals that the commonly used moment-matching Gaussian is, in general, not the $W_2$-nearest Gaussian to a non-Gaussian distribution. In one dimension, we derive closed-form expressions for the proposed quantities and extend closed-form expressions to several other location--scale families, including uniform, Laplace, and logistic distributions. In higher dimensions, we develop a numerical approximation method for the proposed quantities based on empirical optimal transport and covariance-shape optimization. Our experimental results show the empirical convergence and stability of the proposed methods and reveal that the $RW_2$ angle provides a robust and consistent measure of distributional non-Gaussianity. Moreover, these results provide empirical support for its potential use as an indicator of distributional heterogeneity.

While Diffusion Transformers (DiTs) have achieved notable progress in video generation, this long-sequence generation task remains constrained by the quadratic complexity inherent to self-attention mechanisms, creating significant barriers to practical deployment. Although sparse attention methods attempt to address this challenge, existing approaches either rely on oversimplified static patterns or require computationally expensive sampling operations to achieve dynamic sparsity, resulting in inaccurate pattern predictions and degraded generation quality. To overcome these limitations, we propose a \underline{\textbf{M}}ixture-\underline{\textbf{O}}f-\underline{\textbf{D}}istribution \textbf{DiT} (\textbf{MOD-DiT}), a novel sampling-free dynamic attention framework that accurately models evolving attention patterns through a two-stage process. First, MOD-DiT leverages prior information from early denoising steps and adopts a {distributed mixing approach} to model an efficient linear approximation model, which is then used to predict mask patterns for a specific denoising interval. Second, an online block masking strategy dynamically applies these predicted masks while maintaining historical sparsity information, eliminating the need for repetitive sampling operations. Extensive evaluations demonstrate consistent acceleration and quality improvements across multiple benchmarks and model architectures, validating MOD-DiT's effectiveness for efficient, high-quality video generation while overcoming the computational limitations of traditional sparse attention approaches.

As Large Language Models (LLMs) increasingly utilize massive context windows as working memory for autonomous tasks, their reliability fluctuates significantly depending on how information is distributed in real-world corpora. We investigate how fact placement, corpus-level distributions, and anti-hallucination ("Don't Make It Up") prompts influence model behavior by introducing a model-agnostic extended needle-in-a-haystack benchmark designed for scalability, which we apply to evaluate Gemini-2.5-flash, ChatGPT-5-mini, Claude-4.5-haiku, and Deepseek-v3.2-chat. Unlike prior work, we separately evaluate literal extraction, logical inference, and hallucination risk. We identify two critical failure modes: Distributional Collapse, where performance degrades significantly when evidence is dispersed; and a Safety Tax, where anti-hallucination prompts cause over-conservative refusal of present facts and evidence, sharply reducing accuracy. Our results suggest that many failures stem from ineffective context utilization, as models struggle to prioritize relevant information even when it is present. These findings highlight the need for model-specific robustness and effective context management to ensure reliable deployment in long-horizon agentic workflows.

Most visual generative models compress images into a latent space before applying diffusion or autoregressive modelling. Yet, existing approaches such as VAEs and foundation model aligned encoders implicitly constrain the latent space without explicitly shaping its distribution, making it unclear which types of distributions are optimal for modeling. We introduce \textbf{Distribution-Matching VAE} (\textbf{DMVAE}), which explicitly aligns the encoder's latent distribution with an arbitrary reference distribution via a distribution matching constraint. This generalizes beyond the Gaussian prior of conventional VAEs, enabling alignment with distributions derived from self-supervised features, diffusion noise, or other prior distributions. With DMVAE, we can systematically investigate which latent distributions are more conducive to modeling, and we find that SSL-derived distributions provide an excellent balance between reconstruction fidelity and modeling efficiency, reaching gFID equals 3.2 on ImageNet with only 64 training epochs. Our results suggest that choosing a suitable latent distribution structure (achieved via distribution-level alignment), rather than relying on fixed priors, is key to bridging the gap between easy-to-model latents and high-fidelity image synthesis. Code is avaliable at https://github.com/sen-ye/dmvae.

Assessing the importance of individual features in Machine Learning is critical to understand the model's decision-making process. While numerous methods exist, the lack of a definitive ground truth for comparison highlights the need for alternative, well-founded measures. This paper introduces a novel post-hoc local feature importance method called Counterfactual Importance Distribution (CID). We generate two sets of positive and negative counterfactuals, model their distributions using Kernel Density Estimation, and rank features based on a distributional dissimilarity measure. This measure, grounded in a rigorous mathematical framework, satisfies key properties required to function as a valid metric. We showcase the effectiveness of our method by comparing with well-established local feature importance explainers. Our method not only offers complementary perspectives to existing approaches, but also improves performance on faithfulness metrics (both for comprehensiveness and sufficiency), resulting in more faithful explanations of the system. These results highlight its potential as a valuable tool for model analysis. Link to repository: https://github.com/EddieConti/CID

This paper introduces a method for Model Predictive Path Integral (MPPI) control that optimizes sample generation towards an optimal trajectory through Stein Variational Gradient Descent (SVGD). MPPI relies upon predictive rollout of trajectories sampled from a distribution of possible actions. Traditionally, these action distributions are assumed to be unimodal and represented as Gaussian. This assumption can lead to suboptimal rollout predictions due to sample deprivation and, in the case of differentiable simulation, sensitivity to noise in the cost gradients. By introducing SVGD updates in between MPPI environment steps, we present Stein-Optimized Predictive Path-Integral Inference (SOPPI), an MPPI/SVGD algorithm that can dynamically update noise distributions during rollouts to better capture action sampling distributions without an excessive increase in computational requirements. We demonstrate the efficacy of SOPPI through experiments on a planar cart-pole, 7-DOF robot arm, a planar bipedal walker, and a Unitree G1 29-DOF Robot. Results indicate improved inference of an underlying multi-modal action distribution compared to other Stein-based MPPI algorithms, with improved system performance and similar or improved efficiency across a range of hyper-parameters.

Diffusion large language models (dLLMs) are promising alternatives to autoregressive large language models (AR-LLMs), as they potentially allow higher inference throughput. Reinforcement learning (RL) is crucial to enabling dLLMs to achieve performance comparable to that of AR-LLMs on important tasks, such as reasoning. However, RL algorithms well-suited to dLLMs' unique characteristics have yet to be developed. This paper proposes Distribution Matching Policy Optimization (DMPO), a principled and theoretically grounded RL fine-tuning method specifically designed to enhance the reasoning capabilities of dLLMs by matching the dLLM policy distribution to the optimal, reward-tilted one through cross-entropy optimization. We identify a key implementation challenge with small training batch sizes and propose several effective solutions based on a novel weight baseline subtraction technique. DMPO exhibits superior performance on multiple reasoning benchmarks without supervised fine-tuning, achieving up to a $39.63$ percentage-point improvement in accuracy over prior non-DMPO RL baselines and $67.97$ percentage points over the base model, underscoring the effectiveness of the distribution-matching framework. Our code is available at https://github.com/yuchen-zhu-zyc/DMPO.

As Large Language Models (LLMs) demonstrate remarkable capabilities learned from vast corpora, concerns regarding data privacy and safety are receiving increasing attention. LLM unlearning, which aims to remove the influence of specific data while preserving overall model utility, is becoming an important research area. One of the mainstream unlearning classes is optimization-based methods, which achieve forgetting directly through fine-tuning, exemplified by Negative Preference Optimization (NPO). However, NPO's effectiveness is limited by its inherent lack of explicit positive preference signals. Attempts to introduce such signals by constructing preferred responses often necessitate domain-specific knowledge or well-designed prompts, fundamentally restricting their generalizability. In this paper, we shift the focus to the distribution-level, directly targeting the next-token probability distribution instead of entire responses, and derive a novel unlearning algorithm termed \textbf{Di}stribution \textbf{P}reference \textbf{O}ptimization (DiPO). We show that the requisite preference distribution pairs for DiPO, which are distributions over the model's output tokens, can be constructed by selectively amplifying or suppressing the model's high-confidence output logits, thereby effectively overcoming NPO's limitations. We theoretically prove the consistency of DiPO's loss function with the desired unlearning direction. Extensive experiments demonstrate that DiPO achieves a strong trade-off between model utility and forget quality. Notably, DiPO attains the highest forget quality on the TOFU benchmark, and maintains leading scalability and sustainability in utility preservation on the MUSE benchmark.

The factors driving the performance of in-context learning (ICL) in large language models (LLMs) remain poorly understood despite ICL's surprising effectiveness, enabling models to adapt to new tasks from only a handful of examples. To clarify and improve these capabilities, we characterize how the statistical properties of the pretraining distribution (e.g., tail behavior, coverage) shape ICL. We develop a theoretical framework that encompasses generalization and task selection and show how distributional properties govern sample efficiency, task retrieval, and robustness. To this end, we generalize existing concentration results to heavy-tailed priors and dependent sequences, better reflecting the structure of LLM pretraining data. Our framework reveals a fundamental design trade-off: heavy-tailed pretraining distributions facilitate robust task selection under distribution shifts but are detrimental to generalization, especially in low-data regimes. We then empirically evaluate our predictions by studying how ICL performance varies with the pretraining distribution on challenging tasks such as stochastic differential equations and stochastic processes with memory. Together, these findings suggest that controlling key statistical properties of the pretraining distribution is essential for building ICL-capable and reliable LLMs.

Large language models (LLMs) are increasingly used to simulate human behavior in business, economics, and the social sciences, offering a low-cost complement to laboratory experiments, field studies, and surveys. This paper evaluates how well LLMs replicate human behavior in operations management. Using nine published behavioral-operations experiments, we assess LLM performance along two dimensions: whether LLM-generated data reproduce the original hypothesis-test outcomes, and whether their full response distributions align with human data, measured by Wasserstein distance. We find that LLMs often replicate hypothesis-level effects, suggesting that they can capture salient decision biases and behavioral regularities. However, their response distributions frequently diverge from human data, even for strong proprietary models, with dispersion mismatch playing an important role. We also examine two lightweight mitigation strategies: chain-of-thought prompting and hyperparameter tuning. Both can reduce distributional misalignment, and appropriate tuning can sometimes allow smaller or open-source models to match or outperform larger proprietary systems.

Spatio-temporal forecasting remains challenging under non-stationary environments because both data distributions and spatial relations evolve over time. Temporal normalization and de-normalization are widely used to mitigate distribution shifts, but they may distort inter-node relationships and thereby impair spatial dependency modeling. To address these issues, we propose the Distribution and Relation Adaptive Network (DRAN) for spatio-temporal forecasting. DRAN incorporates a Spatial Factor Learner (SFL) module, which enables effective normalization and de-normalization while preserving spatial dependencies in spatio-temporal systems. To model evolving spatial interactions, DRAN further proposes the Dynamic-Static Fusion Learner (DSFL) module. DSFL decomposes features into static and dynamic components and adaptively fuses them according to input variability. Experiments on six benchmark datasets show that DRAN outperforms state-of-the-art baselines. Additional analyses demonstrate that SFL consistently reduces spatial-relation distortion across multiple normalization schemes, whereas DSFL captures complementary static and dynamic dependencies and adjusts their contributions according to temporal variability.

Physically motivated stochastic dynamics are widely used to sample from high-dimensional distributions. However, such samplers often get trapped in metastable states, approximately sampling from a distribution that differs significantly from the desired stationary state. We rigorously show that for multivariable discrete distributions, the true stationary model can nevertheless be recovered from these metastable samples. This relies on a fundamental observation: for distributions satisfying a strong metastability condition, their single-variable conditional probabilities are on average extremely close to those of the true stationary distribution. This remains true even when the two distributions are far apart under global metrics such as Kullback-Leibler divergence. Consequently, we can effectively learn the true model using a conditional-likelihood estimator even when the samples are drawn from a restricted state space. Extending these general results to Ising models, we prove rigorous parameter and structure learning guarantees. Finally, we demonstrate this phenomenon numerically on higher-alphabet spin glass models.

Despite their empirical success across a wide range of generative tasks, the fundamental principles underlying the ability of diffusion models to learn data distributions are poorly understood. In this work, we develop a new mathematical framework that explains how diffusion models can effectively learn low-dimensional distributions from a finite number of training samples without suffering from the curse of dimensionality. Specifically, motivated by the intrinsic low-dimensional structure of image data, we theoretically analyze a setting in which the data distribution is modeled as a mixture of low-rank Gaussians. Under suitable network parameterization, we show that optimizing the training objective of diffusion models is equivalent to solving the canonical subspace clustering problem over the training samples, where each subspace basis corresponds to the low-rank covariance of a Gaussian component. This equivalence allows us to show that the sample complexity for learning the underlying distribution scales linearly with the intrinsic dimension of the data, rather than exponentially with the ambient dimension. Our theoretical findings are further supported by empirical evidence that demonstrates phase transition phenomena in generalization on both synthetic and real-world image datasets. Moreover, we establish a correspondence between the learned subspace bases and semantic attributes of image data, providing a principled foundation for controllable image generation.

Continuous-Time Dynamic Graphs (CTDGs) enable fine-grained modeling of evolving relational systems. However, most existing CTDG representation learning methods are tailored to in-distribution settings and exhibit limited robustness under out-of-distribution (OOD) shifts. Although recent causal approaches learn invariant representations via interventions, they are primarily designed for static or discrete-time graphs and become computationally prohibitive for CTDGs due to the combinatorial explosion of structural and temporal variations. To address these challenges, we propose CIR, a framework grounded in a novel structural causal model termed the ICCM. To avoid exhaustive interventions, we leverage the Normalized Weighted Geometric Mean (NWGM) to efficiently approximate interventional predictions. We further instantiate ICCM within a practical deep learning architecture that jointly captures invariant structural and temporal patterns through dedicated subgraph extractors, and maintains an environment memory bank to model distributional shifts across evolving contexts. Extensive experiments demonstrate that CIR consistently outperforms existing methods under diverse OOD scenarios.

We present four new mathematical methods, two exact and two approximate, along with open-source software, to compute the cdf, pdf and inverse cdf of the generalized chi-square distribution. Some methods are geared for speed, while others are designed to be accurate far into the tails, using which we can also measure large values of the discriminability index $d'$ between multivariate normal distributions. We compare the accuracy and speed of these and previous methods, characterize their advantages and limitations, and identify the best methods to use in different cases.

Federated learning is a distributed learning framework that takes full advantage of private data samples kept on edge devices. In real-world federated learning systems, these data samples are often decentralized and Non-Independently Identically Distributed (Non-IID), causing divergence and performance degradation in the federated learning process. As a new solution, clustered federated learning groups federated clients with similar data distributions to impair the Non-IID effects and train a better model for every cluster. However, existing CFL algorithms are ineffective because they lack an information-sharing mechanism across clusters resulting in low data efficiency and model performance. Meanwhile, their performance is highly subjected to ideal client clustering results which are practically unavailable. This paper proposes StoCFL, a novel clustered federated learning framework for generic Non-IID issues. In detail, StoCFL implements a flexible CFL framework that supports an arbitrary proportion of client participation and newly joined clients for a varying FL system, while maintaining a great improvement in model performance. The intensive experiments are conducted by using four basic Non-IID settings and a real-world dataset. The results show that StoCFL could obtain promising cluster results even when the number of clusters is unknown. Based on the client clustering results, models trained with StoCFL outperform baseline approaches in a variety of scenarios.

Speaker embeddings, or x-vectors, are widely used to represent speaker identity and speaker-related attributes, but existing embedding extractors are typically descriptive rather than generative: they map an observed speech segment to an x-vector, which is then used for downstream applications. We introduce ProPS, Prompted Profile Synthesis, a framework for generating distributions of speaker embeddings conditioned on natural language prompts such as "a thirties male speaker with an Indian accent". ProPS converts human-written profile descriptions into sentence embeddings and uses a mixture density network trained on a large-scale dataset to predict a Gaussian mixture model in the x-vector space. The model is trained by maximizing the likelihood that real speaker embeddings match the requested profile, and its generated distributions are evaluated by negative log-likelihood on held-out x-vectors and by attribute classification accuracies on sampled synthetic x-vectors. Experiments show that ProPS produces profile-conditioned distributions and generates x-vectors that preserve requested speaker attributes such as age, gender, accent, and prosodic characteristics. This design enables controllable speaker-profile synthesis for speech generation systems like Text-To-Speech (TTS) or Voice Conversion (VC) while anchoring generated distributions in observed speaker-embedding structure.