Density Estimation

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8 papers in the last four weeks, up 100% on the four weeks before. 0.1% of all new papers.

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Latest papers 45

Oct 7, 2026cs.LG

Symbolic Density Estimators for Unnormalized Distributions

Estimating the symbolic or analytical form of probability density functions (PDFs) from observed samples is a fundamental challenge in statistical and computational modelling. This process is critical for deriving interpretable and generalizable relationships characterizing the underlying phenomenon. Traditionally, this estimation depends strongly on domain expertise and prior field-specific knowledge, with experts selecting appropriate functional forms or parametric families based on empirical evidence and theoretical understanding. The coefficients of these forms are then typically determined through parameter estimation. In this paper, we develop a framework for estimating symbolic expressions of unnormalized distributions from observed samples using domain-specific prior knowledge, such as the range of interactions and a predefined set of primitive functions. We integrate deep generative models with symbolic regression (SR), incorporating inductive biases, such as factorizing large distributions, to keep the problem tractable. The deep generative models we examine include likelihood-based models, viz., flow models, and score-based models. Experiments show the effectiveness of the proposed framework for estimating density functions for multivariate toy distributions as well as lattices from computational physics, namely, XY model and

φ^4

theory. When applied to the renormalization problem in

φ^4

theory, the proposed framework estimates compact symbolic approximations of the hamiltonian function at different scales directly from samples, yielding expressions that may be challenging to derive using traditional perturbative or analytic approaches in nonperturbative settings.

Vikas Kanaujia, Riyansha Singh, Shashank Sharma +1

Department of Electrical Engineering IIT Kanpur · Department of Computer Science and Engineering IIT Kanpur · Department of Physics IIT Kanpur +1

Density Estimation Symbolic Regression

Oct 7, 2026cs.CV

Gaussian Density Splatting Network

This paper proposes a novel crowd counting approach, the Gaussian Density Splatting Network (GDSNet). Unlike methods that rely on conventional, grid-based density maps and are sensitive to spatial resolution, GDSNet represents a crowd as a superposition of continuous 2D Gaussian primitives. Our approach is built upon two key contributions. First, a control-point-based fitting mechanism is introduced to structure the prediction of Gaussian parameters. A set of control points is adaptively allocated to define local regions, from which features are pooled to regress each primitive's parameters. Second, we adapt a differentiable Gaussian Splatting framework to the counting task by parameterizing each primitive with geometric parameters and a scalar density mass. This formulation allows the network to be trained end-to-end via spatial matching of differentiably rendered density maps, naturally providing both local density supervision and global count optimization. Extensive evaluations on four standard benchmarks show GDSNet consistently outperforms the state of the art. The code is available at https://github.com/infinite0522/GDSNet-Gaussian-Density-Splatting-Network.

Miao Shang, Yabin Wang, Xiaopeng Hong

Faculty of Computing Harbin Institute of Technology

Crowd Counting Density Estimation Gaussian Splatting

Oct 7, 2026math.ST

The Silhouette Operator: Identifiability of Low-Rank Measures from One-Dimensional Projections

Structured recovery phenomena, such as restricted isometry properties in compressed sensing, have shown that high-dimensional objects can often be reconstructed from remarkably low-dimensional linear measurements. This work develops an analogous recovery framework for low-rank signed measures on

\mathbb{R}^2

, defined here as measures that can be expressed as finite sums of product measures with one-dimensional factors. The framework is based on linear operators, termed "silhouette operators," that map a measure to a fixed finite collection of one-dimensional linear pushforwards. The main results show that a suitably chosen collection of

2k

projected marginals suffices to identify every compactly supported rank-

\le k

signed measure, that this number is optimal, and that the projection directions cannot be chosen arbitrarily. The framework is also extended to higher-dimensional sums of product measures by establishing sufficient conditions under which collections of pairwise marginals identify the full model. Building on this framework, a computationally efficient estimator, termed "silhouette mixture estimation" (SME), is introduced for constructing a low-rank empirical measure from data by matching its one-dimensional projected marginals to the corresponding empirical marginals in Wasserstein distance. When combined with one-dimensional density estimators, SME yields an efficient nonparametric density estimator that performs strongly relative to a range of parametric, nonparametric, and deep-learning baselines in settings of moderate dimension and sample size.

Robert A. Vandermeulen

Density Estimation

Oct 7, 2026stat.ML

Scalable Logistic Gaussian Process Density Regression with Kinetic Langevin Sampling

Conditional density estimation targets the full distribution of a response given covariates, as required, for example, for per-galaxy photometric redshifts. We develop a scalable Bayesian estimator based on the logistic Gaussian process. The log conditional density has a separable covariance: a Matérn kernel along the response, represented in a truncated Fourier basis on a circle, and a covariate kernel represented by Nyström features, which accommodate non-stationary kernels with input-dependent amplitudes and length scales. Instead of a Laplace or variational approximation, we sample the latent field of this finite-feature model. Given the hyperparameters, its posterior is strongly log-concave with a uniformly bounded Hessian, and we draw from it by simulating kinetic Langevin dynamics with symmetric minibatch splitting in Kronecker-whitened coordinates. Marginal-likelihood gradients follow from Fisher's identity as posterior expectations. Under the conditions of our analysis their bias is controlled by the sampler's step size and run length, and the predictive averages over the non-Gaussian latent posterior instead of a Gaussian around its mode. On photometric-redshift benchmarks with up to 3.9 million training observations, trained on a single GPU, the estimator is competitive with state-of-the-art tabular foundation models on density and calibration metrics.

Daniel Paulin, Ádám Jung, András A. Benczúr

Nanyang Technological University · Eötvös Lor´and University (ELTE) HUN-REN Institute for Computer Science and Control (SZTAKI)

Conditional Distribution Estimation Density Estimation Gaussian Processes

Oct 6, 2026stat.ML

Uniform Discrete Diffusion Models are Minimax Optimal for Estimating Distributions with Small Effective Support Size

Discrete diffusion models have emerged as a practically successful framework for generative modeling on discrete product spaces, yet their statistical generalization properties remain poorly understood. Discrete real-world data such as text or biological sequences often concentrate on a small fraction of the astronomically large ambient space because of semantic or physical constraints, but existing bounds fail to capture this distributional structure and instead scale with the size of the ambient space, giving rise to almost vacuous error bounds. We address this gap for uniform discrete diffusion, one of the two dominant discrete diffusion paradigms alongside masking diffusion, by deriving statistical guarantees governed by the effective support size

s_n(P_0)

, a sample-size-dependent measure of distributional complexity. Given

n

independent and identically distributed (i.i.d.) samples from an unknown data distribution

P_0

on

[K]^d

, we show that, with appropriate choices of network size and hyperparameters, the expected total variation (TV) loss scales as

O(\sqrt{s_n(P_0)/n})

, while the expected Kullback--Leibler (KL) divergence is bounded by

O(\frac{1}{n}s_n(P_0)\log(eK^d/s_n(P_0))\log n)

. Furthermore, we show that the TV rate is minimax optimal and that the KL rate is minimax optimal up to a factor of

\log n

. Together, these upper and lower bounds show that uniform discrete diffusion successfully avoids the curse of dimensionality for distributions with small effective support size: the TV error rate depends on the ambient state-space size only through

s_n(P_0)

, while the corresponding KL rate incurs only an additional logarithmic dependence on the ambient state-space size.

Dongsun Yoon, Saptarshi Chakraborty

Department of Statistics, University of Michigan

Minimax Estimation Statistical Learning Theory Density Estimation Diffusion Models

Oct 1, 2026stat.ME

Polylogarithmic Sparsity of Randomly Reweighted NPMLEs for Gaussian Mixtures

The nonparametric maximum likelihood estimator (NPMLE) of a Gaussian location mixture maximizes the likelihood over the infinite-dimensional space of mixing distributions. The maximizing mixing distribution can be nonunique, and the classical bound on its number of atoms grows linearly with the sample size

n

. We show that a vanishingly small random perturbation of the likelihood yields exact polylogarithmic sparsity. The resulting randomly reweighted NPMLE maximizes a weighted likelihood whose independent weights, taken to be Gamma in our analysis, concentrate around one as

n

grows. With high probability, it is unique, has

O\{(\log n/\log\log n)^d+\log n\}

atoms in dimension

d

, nearly maximizes the ordinary likelihood, and estimates the mixture density at a Hellinger rate that is parametric up to logarithmic factors. This sparsity holds for the estimator itself, not for an approximation of it, and requires no support penalty. The proof rests on an effective-dimension principle for positive kernel mixtures: low-dimensional variation of the fitted values controls the support of every extreme point of the set of maximizers. Numerical illustrations verify that the reweighted NPMLE has Hellinger risk and support size comparable to those of the ordinary NPMLE.

Hansheng Jiang

University of Toronto

Gaussian Mixture Models Density Estimation

Sep 30, 2026stat.ML

BayesNDE: Bayesian Generative Modeling for Neural Density Estimation

Density estimation is a fundamental problem in statistics and machine learning. In this work, we introduce BayesNDE, a neural density estimator based on Bayesian generative modeling. BayesNDE learns a Bayesian generative model and evaluates its density without requiring invertible networks or Jacobian-determinant computation. For each observation, it infers a sample-specific latent posterior to construct an adaptive proposal that focuses computation on regions contributing most to its density. Bridge sampling then combines samples from this proposal with separate posterior samples to estimate the density. Experiments on nonlinear and multimodal synthetic datasets show improved estimation of density values and better recovery of the density structure compared to the state-of-the-art neural density estimators. Applications to real-world datasets further demonstrate improved anomaly detection. Together, these results highlight BayesNDE as a flexible and effective neural density estimator, demonstrating how posterior inference can turn generative models into tools for density estimation. The code and tutorials are available at https://github.com/liuq-lab/BayesNDE.

Chenglin Li, Qiao Liu

Department of Biostatistics, Yale University New Haven, Connecticut, USA

Generative Modeling Density Estimation

Sep 29, 2026cs.CV

VesselBench-800K: A Large-scale Perception Benchmark for Multimodal Vessel Detection, Counting, and Density Estimation

Vessel perception from space is crucial for a wide range of maritime applications, from traffic monitoring to environmental protection. However, most existing datasets predominantly focus on general object detection tasks in optical remote sensing (RS) images. Relying solely on single-modality optical RS images proves inadequate for effectively perceiving vessel objects in complex maritime scenarios, where ever-changing weather conditions (e.g., clouds and rain), the need for day-and-night coverage, and the inherent limitations of a single imaging modality pose significant challenges. To fill this gap, we introduce VesselBench-800K, the largest-to-date benchmark dataset on a global scale for vessel perception in multimodal RS images. As its name suggests, VesselBench-800K comprises 800,000 images, each at a resolution of 512x512 pixels, specifically curated for vessel perception tasks such as detection, counting, and density estimation. These multimodal image pairs (i.e., optical, SAR) are collected from diverse platforms, sensors, scenes, shooting heights, and synthetic sources, spanning spatial resolutions from 4.5m to 0.1m. Furthermore, we evaluate numerous state-of-the-art detection, counting, and density estimation models on VesselBench-800K through both qualitative and quantitative comparisons. By revealing previously unrecognized cues, this dataset holds immense potential to significantly advance our understanding of marine traffic. Our VesselBench dataset will be publicly available at https://github.com/danfenghong/IEEE_TGRS_VesselBench to support and contribute to community development.

Danfeng Hong, Chenyu Li, Jocelyn Chanussot

School of Automation, Southeast University, Nanjing 211189, China · Univ. Grenoble Alpes, INRAE, LESSEM, 38000 Grenoble, France · Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK, Grenoble 38000, France

Object Counting Remote Sensing Object Detection Remote Sensing Benchmarks Density Estimation

Sep 28, 2026cs.LG

From Data to Program: Fast & Direct Generative Program Inference from Empirical Data

Estimating probability densities from a finite set of samples typically requires dataset-specific model fitting. We introduce PRODiGI, a pretrained data-to-program model that infers an explicit, executable generative program in a single forward pass. Pretrained on synthetic datasets paired with their ground-truth programs, PRODiGI accommodates diverse generative families and data dimensionalities through template prediction and non-autoregressive program parameter decoding. Its inferred programs support direct sampling, density and score evaluation, and inspection independently of the pretrained model. We further introduce program-space fine-tuning, which refines differentiable program parameters by matching generated and empirical samples while keeping model parameters intact. Experiments show that PRODiGI achieves lower average density and score MAE than existing pretrained models, while offering multi-fold speedups over its closest competitors. Program-space fine-tuning further reduces generation MMD by 84%. By turning empirical data into explicit, reusable programs, PRODiGI introduces a new direction for fast, interpretable tabular generative modeling.

Simon Klüttermann, Xueying Ding, Leman Akoglu

Carnegie Mellon University

Program Synthesis Synthetic Tabular Data Generation Density Estimation

Sep 23, 2026cs.LG

Discrete Diffusion Models via Evolving Variational Autoregressive Networks

Conventional score-based diffusion models learn scores without representing normalized densities, whereas tractable normalized models support both sampling and direct likelihood evaluation. A recent tensor-network approach provides such a representation but is largely restricted to low-dimensional lattices. Here we introduce a discrete diffusion model that parameterizes normalized probability distributions using variational autoregressive networks. Explicit Markov jump operators govern the forward noising and reverse denoising dynamics, extending discrete diffusion models with normalized distributions to spin systems on higher-dimensional lattices. We apply this framework to the two- and three-dimensional Ising models across ordered, critical, and disordered regimes, accurately computing thermodynamic quantities including free energy, energy, and magnetization. We further integrate the framework with Monte Carlo sampling, using adaptive diffusion steps to maintain high acceptance rates even at low temperatures while enhancing sample diversity. These results establish a neural-network framework for the discrete diffusion model with normalized probability distributions.

Kewen Pan, Ying Tang

Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu 611731, China · School of Physics, University of Electronic Science and Technology of China, Chengdu 611731, China · Key Laboratory of Quantum Physics and Photonic Quantum Information, Ministry of Education, University of Electronic Science and Technology of China, Chengdu 611731, China +1

Diffusion Sampling Density Estimation Diffusion Models

Sep 21, 2026cs.CV

SAFe: Segment-guided Aggregation of Feature Densities for Anomaly-aware Segmentation

Visual segmentation systems encounter objects outside their training distribution during real-world deployment, hindering reliable autonomous systems that depend on scene parsing in the perception stage. Many recent methods address this by using self-supervised foundation models to train density estimators that yield low likelihood in anomalous image regions. Although promising, these methods suffer from poor feature semantics or they lack spatial consistency, both of which undermine critical downstream decisions. We address this problem with~\method, a generative method based on class-conditional density estimation over self-supervised representations. SAFe trains lightweight normalizing flows that produce class-conditional normalized likelihood estimates over frozen DINOv3 features. We combine density estimates from transformer features with density scores over multi-scale convolutional features to capture both global semantics and local detail. We introduce a method-agnostic post-processing step based on SAM3 that connects per-location likelihoods into spatially coherent segments while suppressing false positives, and enables instance-level anomaly detection without retraining. The post processing further distinguishes novel categories among anomalous objects by a similarity-based agglomerative clustering scheme. SAFe sets a new state of the art on the PANIC, OoDIS, SMIYC ObstacleTrack with strong performance on the ISSU benchmark.

Anja Delić, Jurica Runtas, Marin Oršić +2

University of Zagreb Faculty of Electrical Engineering and Computing Laboratory for Autonomous Systems and Mobile Robotics Zagreb, Croatia

Anomaly Localization Density Estimation Anomaly Detection

Sep 14, 2026stat.ML

A Splitting Method for SDE Terminal-Law Estimation

In many settings involving stochastic differential equations, including in diffusion based generative AI, our aim is to accurately generate samples from a terminal distribution. Typically, this is done by generating i.i.d. samples of diffusion paths. Given a fixed simulation budget, a reasonable way to gain efficiency may be to instead generate a tree of paths through appropriately split partial paths. This suggests improved performance, but one worries about the injected dependence. In this paper, we study this issue comprehensively. With Kolmogorov-Smirnov distance as a measure of accuracy, we identify the limiting errors of the associated empirical distributions as the simulation budget increases to infinity. We characterize a splitting strategy motivated by a corresponding asymptotic optimization problem. The theoretical results bring out the elegant underlying structure in the problem. Practical implementation involves two phases, an initial estimation phase and a final inference phase. Overall, we observe a 10-25% improvement in mean error over i.i.d. samples in many settings. In an exploratory CIFAR-10 study, our method reduces the maximum mean discrepancy by 8-13%.

Rushil Gupta, Sandeep Juneja

Safexpress Centre for Data, Learning and Decision Sciences Ashoka University

Stochastic Differential Equations Diffusion Model Sampling Density Estimation

Sep 8, 2026astro-ph.GA

Closed-Form of the Local Galactic Potential and Stellar Distribution Function from Gaia DR3

The local dark matter density determines the strength of the signal expected in direct-detection experiments, yet published estimates from stellar motions disagree by more than their errors, and the most recent machine-learning analysis of Gaia data finds a local density consistent with zero. According to Jeans' theorem, a distribution function built from integrals of motion satisfies the collisionless Boltzmann equation (CBE) trivially for any choice of potential, so a search that simultaneously fits the distribution function and the potential to the CBE identifies neither. Our pipeline instead estimates the distribution function in isolation, linearizing the equation in terms of accelerations and allowing for direct measurement of the local force field, and then fits closed forms to that field via symbolic regression. Throughout, we find that the usable information lies not in the CBE residual but in the stellar number counts, the observable most distorted by survey selection. Along the vertical profile, our recovered potential agrees with the classical self-gravitating isothermal disc.

Indranil Das, Adam Kamoski, Dora Demiri +3

Department of Physics University of Massachusetts Boston NSF Institute for AI and Fundamental Interactions · Lebedev Physical Institute Russian Academy of Sciences Moscow 119991, Russia

Density Estimation

Sep 2, 2026cs.LG

Source Distribution Estimation by Posterior Averaging

Simulation-based science often requires a distribution over simulator parameters whose push-forward reproduces a set of real observations: this is the source distribution estimation (SDE) problem. Existing methods fit the source against a likelihood surrogate trained once from a fixed proposal prior. Their objective is therefore stated only in terms of the surrogate instead of the true simulator, which may fail for inaccurate areas in parameter space where the surrogate was never trained. We instead solve SDE by expectation maximization: an E-step trains an amortized posterior on fresh simulations from the current source estimate, and an M-step refits the source to the average of that posterior over the observed data. We give two parameterizations, (1) separate source and posterior flows and (2) a single shared conditional flow. We evaluate our method on three benchmark tasks under both broad and misspecified initial priors. Both improve on existing fixed surrogate approaches and on iterated variants of each, most clearly on Lotka--Volterra, where no baseline falls below 0.96 data-space C2ST while our methods reach 0.64-0.68 in three of four initial-prior settings.

Trung-Dung Hoang, Lisa M. Koch

Department of Digital Medicine, University of Bern, Switzerland · Department of Diabetes, Endocrinology, Nutritional Medicine and Metabolism UDEM, Inselspital, Bern University Hospital, University of Bern, Switzerland · Graduate School for Cellular and Biomedical Sciences (GCB), University of Bern, Switzerland +1

Simulation-Based Inference Density Estimation

Aug 31, 2026cs.IT

Minimax bounds for watermarked and masked recursive discrete distribution estimation

Watermarking has been proposed as a way to identify synthetic samples in estimation settings where no metadata is available to distinguish them from real samples, but its precise effects remain unexplored. In the absence of a distinguishing mechanism, it has been shown that adding synthetic samples significantly reduces the marginal efficacy of new real samples. In this work, we study the minimax loss of such recursive discrete distribution estimation in the presence of watermarks in contrast to the unassisted and oracle-assisted losses. When the fraction of real samples vanishes asymptotically, we provide a lower bound that shows that it is impossible to improve performance by adding watermarks unless the false negative rate of detection also vanishes. Additionally, we show that in most regimes, the worst-case losses of a sequence of simple deterministic estimators match the corresponding lower bounds up to constants. Finally, we propose masking, a randomization procedure that narrows the gap in the remaining regimes to a Jensen gap. We conjecture that a tighter lower bound argument can close this gap.

Millen Kanabar, Michael Gastpar

School of Computer and Communication Sciences, EPFL, Switzerland

Dataset Watermarking Minimax Estimation Density Estimation Information-Theoretic Lower Bounds

Aug 13, 2026stat.ML

Wasserstein Filtering: A Sample Selection Method for Robust Distribution Learning

Given a dataset where a portion of the samples are contaminated, our goal is to recover the underlying clean population distribution. To this end, we propose Wasserstein Filtering (WF), a novel sample selection framework that discards a fraction of suspicious samples and estimates the target distribution using the empirical measure of the remaining data. The core insight is to select a subset of samples whose empirical distribution maximizes its Wasserstein distance to the fully contaminated empirical distribution, thereby preferentially isolating and removing geometrically influential outliers. To render this optimization computationally tractable, we introduce three algorithms: a marginal screening scheme, SinkMarg, and two joint optimization algorithms, SinkWF and SlicedWF, leveraging entropic optimal transport and sliced Wasserstein approximations, respectively. On the theoretical front, we introduce the Far Exclusion and Local Projection (FELP) contamination model, which characterizes corruptions consisting of well-separated outliers and locally indistinguishable perturbations. Under this model, we prove that the WF estimator achieves minimax optimality over distribution families with bounded covariance. Extensive numerical experiments on synthetic datasets, benchmark anomaly detection suites, and robust generative learning with diffusion models demonstrate that WF serves as a highly practical, model-agnostic preprocessing tool. It delivers competitive outlier detection performance and provides substantial downstream benefits for generative modeling under heavy contamination.

Yikai Xu, Zhao Chen, Jian Huang

Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China. · School of Data Science, Fudan University, Shanghai, China. · Department of Data Science and Artificial Intelligence, and Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China.

Unsupervised Anomaly Detection Density Estimation

Aug 10, 2026cs.LG

In-Context Density Estimation for Tabular Data

Density estimation underlies many unsupervised tasks on tabular data such as anomaly detection, out-of-distribution detection, and data augmentation. Although all these problems reduce to questions about where probability mass lies, they are typically solved individually by fitting a separate model to each dataset, with its own hyperparameters and tuning budget. We introduce ICED, an in-context, energy-based density estimator that removes this per-dataset cost. ICED is a transformer-based model pretrained once on a synthetic prior built specifically for density estimation under an objective that fits log-density where it is informative and preserves its ordering elsewhere. In the inference, it reads a dataset as context and returns an unnormalized log-density for any query point in a single forward pass, with no fitting, sampling, or hyperparameter selection. A single frozen ICED model then drives four tasks usually handled by four specialized pipelines: density estimation, out-of-distribution detection, unsupervised anomaly detection, and generative augmentation. Across all four, it is competitive with the strongest task-specific method, while being the only approach that needs no retraining, no tuning, and no labels to move between them. The code is available at https://github.com/gmum/iced.

Patryk Marszałek, Jacek Tabor, Marek Śmieja

Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland

Energy-Based Models Unsupervised Anomaly Detection OOD Detection Tabular ICL Density Estimation

Jul 29, 2026cs.LG

Inverse Learning of Latent Risk-Neutral Densities from Irregular Option Quotes

Accurate option prices do not imply accurate recovery of the latent risk-neutral density. We study this distinction with two complementary benchmarks. A controlled benchmark exposes simulator-truth densities for latent evaluation, while a chronological NIFTY benchmark tests only held-out market prices. A two-component lognormal mixture has the lowest aggregate price,

L^1

, Wasserstein, and fixed-tail errors on the synthetic benchmark. Learned operators retain narrower strengths: DeepONet reduces 1% quantile and variance error by 39.0% and 34.6% relative to the mixture, and a quote transformer reduces

L^1

by 16.4% on the structurally misspecified Merton family. A numerical conditioning analysis explains why these rankings can differ: after enforcing mass and forward constraints, 95 of 126 pricing directions are numerically null, and two densities separated by

L^1 = 0.061

produce identical prices on the covered strikes. On 524 held-out NIFTY calls, validation-selected test-time adaptation reduces DeepONet RMSE by 28.3%, but per-expiry mixture and SVI fits remain much more accurate. The evidence supports target-dependent inductive bias, not a universal winner.

Lennon J. Shikhman, Michael Galarnyk, Aadi Dash +1

School of Computer Science Georgia Institute of Technology USA

Quantitative Finance Deep Operator Networks Density Estimation Neural Operators Inverse Problems Finance

Jul 29, 2026stat.ML

Breaking the Curse with BAND: Nonparametric Distribution Estimation in High Dimensions

Minimax-optimal rates for multivariate distribution estimation are known to suffer from the curse of dimensionality. We propose a sparse Bayesian network approach in which each conditional probability is estimated using sparsity-aware conditional mean methods. The resulting estimator, \textit{BAyesian Network Distribution regression} (BAND), handles mixed data types in high-dimensional time series and achieves polynomial total variation convergence rates while allowing the feature dimension to grow polynomially with the sample size. These rates are substantially faster than the classical optimal rates for multivariate histogram density estimators that lack sparsity. Empirical evaluations show that BAND performs competitively for data sampling and confidence region forecasting against a range of state-of-the-art benchmarks.

Shuo-Chieh Huang, Chien-Ming Chi, Jau-er Chen

Department of Statistics, Rutgers University, Piscataway, NJ, U.S.A. · Institute of Statistical Science, Academia Sinica, Taipei, Taiwan · School of Political Science and Economics, National Taiwan University, Taipei, Taiwan

Bayesian Networks Density Estimation

Jul 27, 2026cs.AI

Localized Anomaly Detection via Differentiable D-vine Copulas

Vine copulas provide a flexible framework for modeling complex multivariate distributions through a hierarchical decomposition into bivariate pair-copulas. Fitting a D-vine requires selecting a copula family and parameter configuration for each pair-copula from a set of candidates encoding different dependence patterns. As the number of variables and candidate families increases, the number of possible configurations grows combinatorially. Existing fitting procedures address this challenge through sequential greedy decisions, committing to a single locally optimal family at each step and potentially discarding configurations that would yield a better global fit. To overcome this limitation, we propose a novel estimation framework that combines gradient-based maximum likelihood estimation, enabled by our fully differentiable implementation, with a beam-search strategy that maintains multiple competing D-vine configurations throughout the fitting process. This allows a broader exploration of the configuration space while remaining computationally tractable. Building on the fitted D-vine, we introduce a localized anomaly detection framework that exploits the hierarchical decomposition to produce both global anomaly scores and edge-level explanations. Statistical guarantees are provided through Mondrian conformal prediction, while the pair-copula structure enables the localization of anomalies to specific variable relationships. We evaluate the proposed framework on both benchmark and real-world datasets, demonstrating its effectiveness for interpretable anomaly detection with uncertainty quantification.

Nicholas Andrea Pearson, Francesca Zanello, Davide Russo +2

University of Trieste, Trieste, Italy · Idrostudi srl, Trieste, Italy

Anomaly Localization Density Estimation Anomaly Detection

Jul 15, 2026cs.LG

The Hyperspherical Geometry of CLIP Latent Space: A Semantic Mixture Model

Contrastive Language-Image Pretraining (CLIP) representations form a semantic embedding space governed by cosine similarity, reflecting an intrinsic hyperspherical geometry. However, existing probabilistic interpretations typically rely on Gaussian assumptions, which fail to capture this directional and multimodal structure. We propose a principled density model for the CLIP latent space based on Mixtures of von Mises-Fisher (MovMF) distributions defined on the unit hypersphere. Using the Expectation-Maximization (EM) algorithm, we efficiently learn a probabilistic model in which each mixture component corresponds to a coherent semantic concept. This formulation yields a closed-form likelihood naturally aligned with hyperspherical geometry, enabling accurate and interpretable density estimation. Empirically, our model significantly improves long-tailed and out-of-distribution detection and provides a natural semantic decomposition, representing each embedding as a sparse probabilistic combination of interpretable concepts. These results suggest that CLIP latent space is more faithfully characterized as a hyperspherical semantic mixture rather than an isotropic Gaussian, establishing a simple and geometrically consistent probabilistic framework for modeling and understanding multimodal representations. Project page is available at https://xiaoyuzhizi.github.io/movmf-clip/.

Zijie Yu, Gaowen Liu, Ramana Rao Kompella +2

Tsinghua University · Cisco Research · University of Illinois Chicago

Latent Variable Models Hyperspherical Representation Learning Multimodal Embedding Multimodal Model Interpretability Density Estimation

Jul 10, 2026eess.IV

CHM-Net: Center Heatmap-driven Macro-Micro Modeling Network for MRI-based Microbial Density Stratification

Microbial density is clinically important for tumor assessment and treatment decision-making, and recent advances in deep learning suggest that it can be non-invasively inferred from multimodal MRI. In this work, MRI-based Microbial Density Stratification (MRI-MDS) is first investigated as a patient-level representation learning task, and Center Heatmap-driven Macro-micro modeling Network (CHM-Net) is introduced for this task. CHM-Net first establishes the link between imaging phenotypes and microbial states through center heatmap-guided small-lesion response localization. Building upon this, it constructs patient-level macro-micro evidence from localized heatmap responses for microbial density prediction. Experiments on the novel GBNPC 2026 dataset constructed for MRI-MDS demonstrate the effectiveness of CHM-Net, achieving superior performance over representative baselines with a 12.06% absolute ACC gain over the strongest competing result. Additionally, auxiliary validation on two 3D medical image datasets further verifies its robustness across volumetric medical image classification scenarios. The project is available at https://anonymous.4open.science/r/CHM-Net-942E/.

Jiaming Liang, Haolin Chen, Tingting Li +7

School of Computer Science and Engineering South China University of Technology Guangzhou, China · School of Future Technology South China University of Technology Guangzhou, China · Department of Medical Imaging Affiliated Cancer Hospital, Guangzhou Medical University Guangzhou, China

Medical Image Classification Medical Image Representation Learning Density Estimation

Jul 8, 2026stat.ML

Distributionally Faithful Imputation via Positive Semi-Definite Kernel Density Estimation

Missing values undermine statistical inference and machine learning pipelines, yet most imputation methods rely on heuristics or restrictive parametric assumptions that ignore the joint data distribution. We recast imputation under missing completely at random (MCAR) as density estimation from masked observations: estimate a distribution whose observed marginals exactly match those in the data. Leveraging positive semi definite (PSD) kernel densities we obtain a convex empirical risk problem with closed form marginals, solvable by a Newton interior point method. The resulting PSD Impute model yields both single and multiple imputations from the same fitted density, enjoys statistical consistency with fast adaptive excess risk beating the curse of dimensionality for very regular probabilities. Preliminary experiments on one synthetic and eleven real world datasets already indicate competitive distributional accuracy compared with popular imputation baselines, suggesting strong practical promise.

Andrea Basteri, Carlo Ciliberto, Alessandro Rudi

Inria - Ecole Normale Supériore · Department of Computer Science - UCL · SDA Bocconi School of Management

Incomplete Data Imputation Kernel Density Estimation Density Estimation

Jun 24, 2026cs.CV

Dual Distribution Estimation for Zero-shot Noisy Test-Time Adaptation with VLMs

While test-time adaptation (TTA) empowers vision-language models to adapt without costly retraining, it remains highly vulnerable to out-of-distribution (OOD) outliers prevalent in real-world applications. This discrepancy motivates Noisy TTA (NTTA), an online task to filter noisy OOD samples on the fly while maximizing in-distribution (ID) classification accuracy. Existing zero-shot NTTA approaches typically rely on test-time discriminative training, leading to overconfident misclassifications and significantly degraded inference efficiency. To address these limitations, we propose a novel framework named Dual Distribution Estimation (DDE), shifting the zero-shot NTTA paradigm from instance-level learning to training-free Gaussian distribution modeling. DDE incorporates two novel modules: Positive Feature Distribution Estimation (PFDE) and Negative Label Distribution Estimation (NLDE). PFDE explicitly models class-wise inclusion and exclusion Gaussian distributions to formulate a calibrated contrastive score, robustly enhancing ID accuracy. In parallel, NLDE improves OOD identification by explicitly modeling the negative label distribution to mine highly discriminative labels, effectively mitigating spurious correlations. Extensive experiments show that on the large-scale ImageNet benchmark, DDE achieves an improvement of 3.70% in harmonic mean accuracy and reduces the FPR95 for OOD detection by 6.20%, while ensuring highly scalable and efficient online inference. Furthermore, DDE is zero-shot and training-free, demonstrating remarkable robustness in data-scarce scenarios. Codes are available at https://github.com/ZhuWenjie98/DDE.

Wenjie Zhu, Yabin Zhang, Liang Xu +3

The Hong Kong Polytechnic University · Harbin Institute of Technology (Shenzhen) · Shanghai Jiao Tong University +1

Test-Time Adaptation VLM Adaptation OOD Detection Density Estimation

Jun 17, 2026cs.CV

Multi-Modal Hyper-Graph Fusion for Low-Light Crowd Counting

Crowd counting is a fundamental task in computer vision. However, crowd counting in low-light environments remains largely underexplored, despite its practical importance in the real world. Existing methods mainly focus on well-lit scenes or rely on single-modality Red-Green-Blue (RGB) representations, which often become unreliable under extreme darkness and complex non-uniform illumination. To handle this problem, we construct three new low-light crowd counting benchmarks, which consist of two synthetic datasets, SHA_Dark and SHB_Dark, and a real-world benchmark LC-Crowd (Low-light Crowd Dataset). Inspired by Retinex-based physical modeling, we introduce depth and Canny edge cues as complementary geometric and structural priors to enhance the intrinsic reflectance representation under low-light conditions. We propose a Multi-Modal Hyper-Graph Fusion module, which formulates RGB appearance, depth geometry, and edge structure cues as nodes in a unified hyper-graph and explicitly captures their high-order complementary relationships via dynamic hyperedge construction and message passing. Furthermore, to adaptively allocate computation in dense prediction, we propose a Deformable Rectangular Sparse Attention (DRSA) module, which concentrates computation on informative regions through anchor-aware estimation and adaptive rectangular window modeling. Based on these designs, we develop a unified Low-Light Counting Network (LCNet) for robust low-light crowd counting. Extensive experiments on three benchmarks demonstrate that the proposed method achieves the best overall performance against existing state-of-the-art (SOTA) methods. The code is in the supplementary material. The datasets will be made public upon acceptance.

Hao-Yuan Ma, Li Zhang, Yushi Qiu +3

School of Computer Science and Technology, Soochow University, Suzhou, China

Crowd Counting Deformable Attention Density Estimation Multimodal Sensor Fusion

Jun 12, 2026cs.CV

Pix2Pix-Hybrid: Structure-Guided Conditional Synthesis of Hajj Crowd Images with Multi-Channel Conditioning and Weak Attribute Supervision

Developing accurate crowd-counting models for Hajj pilgrimage scenes remains challenging because domain-specific annotated images are scarce and data collection during large gatherings raises privacy concerns. To address these limitations, this paper proposes Pix2Pix-Hybrid (P2P-H), a hybrid conditional GAN for structure-guided Hajj crowd-image synthesis and data augmentation. P2P-H builds on Pix2Pix and employs a U-Net generator conditioned on eight input channels that jointly encode structural cues (edges and grayscale) and contextual attributes (crowd density and time of day). To capture detailed textures in dense scenes, the framework integrates two multi-scale PatchGAN discriminators operating at different resolutions. The training procedure combines adversarial, perceptual, and feature-matching objectives with adaptive data augmentation and stabilization strategies. The model was trained on 993 real Hajj frames collected from 60 publicly available video sources, with conditioning attributes derived automatically to reduce manual labeling effort. Using this framework, we constructed CrowdH, a synthetic dataset of 10,000 high-resolution Hajj crowd images. Experimental results show that P2P-H improves structure-preserving conditional synthesis quality compared with Pix2Pix and StyleGAN2-ADA baselines and shows favorable transfer to other crowd datasets. To assess downstream utility, we further constructed CrowdH-Mix-469, an annotated mixed real-synthetic dataset comprising 384 real Hajj images and 85 selected synthetic images,and evaluated five crowd-counting models under real-only and real-plus-synthetic training. The selected synthetic data reduced MAE across all five models, with the strongest gain observed for CSRNet.

Amirah F. Alshammari, Bander A. Alzahrani, Nahed A. Alowidi

Information Systems Department, Faculty of Computing and Information Technology, King Abdulaziz University, Jeddah, Saudi Arabia · Computer Science Department, Faculty of Computing and Information Technology, King Abdulaziz University, Jeddah, Saudi Arabia

Synthetic Data Augmentation Conditional Image Generation Crowd Counting Image Generation Density Estimation Generative Adversarial Networks

Jun 11, 2026cs.CV

Overhead Wildlife Locator (OWL): Benchmarking Weakly Supervised Learning for Aerial Wildlife Surveys

Automated aerial wildlife surveys increasingly rely on deep learning, yet standard object detectors require bounding-box annotations, reported to be up to seven times slower and three times more expensive to produce than point-level labels. To address this bottleneck, we introduce the Overhead Wildlife Locator (OWL), a weakly supervised density-estimation framework with three variants: OWL-C, a fully convolutional model for high-throughput screening; OWL-T, a Swin-augmented hybrid for heterogeneous, cluttered scenes; and OWL-D, built on a frozen DINOv3 ViT-H+/16 encoder with a DPT-style fusion decoder. We benchmark all three against POLO, YOLOv11n, and YOLOv11l across five public aerial datasets, from sparse fixed-wing savanna surveys to dense UAV paddock imagery, and against the published HerdNet baseline on its native Delplanque split. OWL-D sets a new state of the art on Delplanque (0.934 AP vs. HerdNet's 0.840) and records the highest AP on four of the five datasets. Performance is regime-dependent: on the extreme-density SheepCounter UAV dataset the hybrid OWL-T leads (0.978 AP) and the convolutional variants attain the lowest counting error, whereas the foundation-based OWL-D degrades, indicating which variant suits which survey type. We further validate operational readiness on the Alaska Department of Fish and Game's 2022 Central Arctic Caribou census: under cross-herd and cross-temporal transfer, OWL-C fine-tuned on the 2017 Porcupine Caribou Herd split attains F1 = 0.965 on a held-out patch test set, with a signed count error of +3.1% aggregated across the released test patches. We release the OWL code, model weights, and the annotated Porcupine Caribou Herd 2017 (PCH) and Central Arctic Herd 2022 (CAH) patches, the first open patch-level datasets for large-scale caribou aerial surveys, at https://github.com/microsoft/MegaDetector-Overhead.

Isai Daniel Chacón, Zhongqi Miao, Bruno Demuro +9

Microsoft AI for Good Lab, Redmond, WA, USA · Center for Research and Formation in Artificial Intelligence (Cinfonia), Universidad de los Andes, Bogotá, Colombia · Conservation X Labs, Washington, D.C., USA +3

Remote Sensing Image Understanding Weakly Supervised Learning Drone-Based Wildlife Monitoring Aerial Object Detection Wildlife Monitoring Density Estimation

Jun 9, 2026cs.DS

Density estimation for Hellinger via minimum-distance estimators: mixtures of Gaussians, log-concave, and more

We study the task of density estimation, where we hope to accurately estimate a probability density from

n

samples. A textbook method for density estimation in total variation distance is the minimum-distance estimator approach, where we conclude both the algorithm and the analysis merely from bounding the VC dimension of a particular concept class (the so-called Yatracos class). While this technique has originally yielded sharp guarantees primarily for total variation distance, in this work we extend the minimum-distance estimator approach for learning within Hellinger distance. Our main observation is that we may produce an analogous recipe for Hellinger (where we only require bounding the VC dimension of a related concept class) by drawing connections to recent results yielding reverse data processing inequalities. This recipe is flexible enough to accommodate fast algorithms originally designed for total variation distance; by modifying the approach of Acharya et al. (2017) we conclude the first near-linear time algorithm for learning classes including univariate mixtures of log-concave densities and mixtures of Gaussians (with arbitrary variances), with near-optimal sample complexity.

Spencer Compton, Jerry Li

Stanford University · University of Washington

Gaussian Mixture Models Density Estimation

May 28, 2026hep-ph

Generative Models and Statistical Validation

Generative machine learning has become an essential tool in theoretical and experimental physics, especially in the context of fast surrogates and density estimators. In this work, we first introduce the underlying framework of modern generative networks and then discuss challenges in quantifying their accuracy, precision, and statistical power.

Sascha Diefenbacher, Sofia Palacios Schweitzer, Gregor Kasieczka

Institut für Theoretische Physik, Universität Heidelberg, Germany · Physics Division, Lawrence Berkeley National Laboratory, Berkeley, USA · NHETC, Department of Physics & Astronomy, Rutgers University, Piscataway, NJ, USA +1

Generative Modeling Density Estimation Generative AI Evaluation

May 25, 2026stat.ME

Geometry Adaptive Counterfactual Distribution Learning with Diffusion-Guided Smoothing

We study counterfactual distribution learning for high-dimensional outcomes whose counterfactual law may concentrate near lower-dimensional structure. Standard isotropic smoothing treats all ambient directions equally, leading to unfavorable scaling and unstable local inference. We propose two diffusion-guided estimators based on semiparametric debiasing: diffusion-informed smoothing for counterfactual densities and diffusion-informed score smoothing for counterfactual scores. The estimators combine causal nuisance adjustment with geometry-adaptive localization driven by diffusion score information, removing first-order nuisance bias while aligning smoothing with local outcome geometry. We establish asymptotic expansions, risk bounds, and inference procedures for smoothed density and score-based targets, with ambient density inference obtained under additional approximation conditions. Under structural geometry conditions, the leading stochastic error is governed by an effective dimension induced by the diffusion-guided kernel, rather than by the ambient dimension. Semi-synthetic experiments based on CelebA show steeper error decay for geometry-adaptive methods, supporting the proposed effective-dimension theory.

Kwangho Kim

Department of Statistics, Korea University

Semiparametric Inference Causal Inference Density Estimation Counterfactual Learning