This paper introduces a new type of probabilistic semiparametric model that takes advantage of data binning to reduce the computational cost of kernel density estimation in nonparametric distributions. Two new conditional probability distributions are developed for the new binned semiparametric Bayesian networks, the sparse binned kernel density estimation and the Fourier kernel density estimation. These two probability distributions address the curse of dimensionality, which typically impacts binned models, by using sparse tensors and restricting the number of parent nodes in conditional probability calculations. To evaluate the proposal, we perform a complexity analysis and conduct several comparative experiments using synthetic data and datasets from the UCI Machine Learning repository. The experiments include different binning rules, parent restrictions, grid sizes, and number of instances to get a holistic view of the model's behavior. As a result, our binned semiparametric Bayesian networks achieve structural learning and log-likelihood estimations with no statistically significant differences compared to the semiparametric Bayesian networks, but at a much higher speed. Thus, the new binned semiparametric Bayesian networks prove to be a reliable and more efficient alternative to their non-binned counterparts.
Density map estimation has become the mainstream paradigm in crowd counting. However, most existing methods overlook the extreme sparsity of ground-truth density maps. In real-world crowd scenes, the vast majority of spatial regions (often over 95%) contain no people, leading to heavily imbalanced count distributions. Ignoring this imbalance can bias models toward overestimating dense regions and underperforming in sparse areas. Furthermore, most loss functions used in density estimation are majorly based on MSE and implicitly assume Gaussian distributions, which are ill-suited for modeling discrete, non-negative count data. In this paper, we propose EBC-ZIP, a crowd counting framework that models the spatial distribution of counts using a Zero-Inflated Poisson (ZIP) regression formulation. Our approach replaces the traditional regression loss with the negative log-likelihood of the ZIP distribution, enabling better handling of zero-heavy distributions while preserving count accuracy. Built upon the recently proposed Enhanced Block Classification (EBC) framework, EBC-ZIP inherits EBC's advantages in preserving the discreteness of targets and ensuring training stability, while further improving performance through a more principled probabilistic loss. We also evaluate EBC-ZIP with backbones of varying computational complexity to assess its scalability. Extensive experiments on four crowd counting benchmarks demonstrate that EBC-ZIP consistently outperforms EBC and achieves state-of-the-art results.
Today, Earth Observation (EO) satellites generate massive volumes of data, with the Copernicus Sentinel-2 constellation alone producing approximately 1.6TB per day. To fully exploit this information, it is essential to pretrain EO Foundation Models (FMs) on large unlabeled datasets, enabling efficient fine-tuning for several different downstream tasks with minimal labeled data. In this work, we present the scaling-up of our recently proposed EO Foundation Model, PhilEO Geo-Aware U-Net, on the unlabeled 23TB dataset MajorTOM, which covers the vast majority of the Earth's surface, as well as on the specialized subset FastTOM 2TB that does not include oceans and ice. We develop and study various PhilEO model variants with different numbers of parameters and architectures. Finally, we fine-tune the models on the PhilEO Bench for road density estimation, building density pixel-wise regression, and land cover semantic segmentation, and we evaluate the performance. Our results demonstrate that for all n-shots for road density regression, the PhilEO 44M MajorTOM 23TB model outperforms PhilEO Globe 0.5TB 44M. We also show that for most n-shots for road density estimation and building density regression, PhilEO 200M FastTOM outperforms all the other models. The effectiveness of both dataset and model scaling is validated using the PhilEO Bench. We also study the impact of architecture scaling, transitioning from U-Net Convolutional Neural Networks (CNN) to Vision Transformers (ViT).
Recent advances in generative modeling -- particularly diffusion models and flow matching -- have achieved remarkable success in synthesizing discrete data such as images and videos. However, adapting these models to physical applications remains challenging, as the quantities of interest are continuous functions governed by complex physical laws. Here, we introduce $\textbf{FunDiff}$, a novel framework for generative modeling in function spaces. FunDiff combines a latent diffusion process with a function autoencoder architecture to handle input functions with varying discretizations, generate continuous functions evaluable at arbitrary locations, and seamlessly incorporate physical priors. These priors are enforced through architectural constraints or physics-informed loss functions, ensuring that generated samples satisfy fundamental physical laws. We theoretically establish minimax optimality guarantees for density estimation in function spaces, showing that diffusion-based estimators achieve optimal convergence rates under suitable regularity conditions. We demonstrate the practical effectiveness of FunDiff across diverse applications in fluid dynamics and solid mechanics. Empirical results show that our method generates physically consistent samples with high fidelity to the target distribution and exhibits robustness to noisy and low-resolution data. Code and datasets are publicly available at https://github.com/sifanexisted/fundiff.
Hollmann et al. (Nature 637 (2025) 319-326) recently introduced TabPFN, a transformer-based deep learning model for regression and classification on tabular data, which they claim "outperforms all previous methods on datasets with up to 10,000 samples by a wide margin, using substantially less training time." Furthermore, they have called TabPFN a "foundation model" for tabular data, as it can support "data generation, density estimation, learning reusable embeddings and fine-tuning". If these statements are well-supported, TabPFN may have the potential to supersede existing modeling approaches on a wide range of statistical tasks, mirroring a similar revolution in other areas of artificial intelligence that began with the advent of large language models. In this paper, we provide a tailored explanation of how TabPFN works for a statistics audience, by emphasizing its interpretation as approximate Bayesian inference. We also provide more evidence of TabPFN's "foundation model" capabilities: We show that an out-of-the-box application of TabPFN vastly outperforms specialized state-of-the-art methods for semi-supervised parameter estimation, prediction under covariate shift, and heterogeneous treatment effect estimation. We further show that TabPFN can outperform LASSO at sparse regression and can break a robustness-efficiency trade-off in classification. All experiments can be reproduced using the code provided at https://github.com/qinglong-tian/tabpfn_study (https://github.com/qinglong-tian/tabpfn_study).
Over-parameterized models have raised concerns about their potential to memorize training data, even when achieving strong generalization. The privacy implications of such memorization are generally unclear, particularly in scenarios where only model outputs are accessible. We study this question in the context of kernel methods, and demonstrate both empirically and theoretically that querying kernel models at various points suffices to reconstruct their training data, even without access to model parameters. Our results hold for a range of kernel methods, including kernel regression, support vector machines, and kernel density estimation. Our hope is that this work can illuminate potential privacy concerns for such models.
Density ratio estimation is fundamental to tasks involving $f$-divergences, yet existing methods often fail under significantly different distributions or inadequately overlap supports, suffering from the density-chasm and the support-chasm problems. Additionally, prior approaches yield divergent time scores near boundaries, leading to instability. We design $\textbf{D}^3\textbf{RE}$, a unified framework for robust, stable and efficient density ratio estimation. We propose the dequantified diffusion bridge interpolant (DDBI), which expands support coverage and stabilizes time scores via diffusion bridges and Gaussian dequantization. Building on DDBI, the proposed dequantified Schr{\"o}dinger bridge interpolant (DSBI) incorporates optimal transport to solve the Schr{\"o}dinger bridge problem, enhancing accuracy and efficiency. Our method offers uniform approximation and bounded time scores in theory, and outperforms baselines empirically in mutual information and density estimation tasks.
In this work, a kernel-based Ensemble Gaussian Mixture Probability Hypothesis Density (EnGM-PHD) filter is presented for multi-target filtering applications. The EnGM-PHD filter combines the Gaussian-mixture-based techniques of the Gaussian Mixture Probability Hypothesis Density (GM-PHD) filter with the particle-based techniques of the Sequential Monte Carlo Probability Hypothesis Density (SMC-PHD) filter. It achieves this by obtaining particles from the posterior intensity function, propagating them through the system dynamics, and then using Kernel Density Estimation (KDE) techniques to approximate the Gaussian mixture of the prior intensity function. This approach guarantees convergence to the true intensity function in the limit of the number of components. Moreover, in the special case of a single target with no births, deaths, clutter, and perfect detection probability, the EnGM-PHD filter reduces to the standard Ensemble Gaussian Mixture Filter (EnGMF). In the presented experiment, the results indicate that the EnGM-PHD filter achieves better multi-target filtering performance than both the GM-PHD and SMC-PHD filters while using the same number of components or particles.
In arbitrary-order language models, it is an open question how to sample tokens in parallel from the correct joint distribution. With discrete diffusion models, the more tokens they generate in parallel, the less their predicted distributions adhere to the originally learned data distribution, as they rely on a conditional independence assumption that only works with infinitesimally small timesteps. We find that a different class of models, any-subset autoregressive models (AS-ARMs), holds the solution. As implied by the name, AS-ARMs can generate tokens in any order, and in parallel. Moreover, AS-ARMs support parallelized joint probability density estimation, allowing them to correct their own parallel-generated token distributions, via our Any-Subset Speculative Decoding (ASSD) algorithm. ASSD provably enables generation of tokens from the correct joint distribution, with the number of neural network calls upper bounded by the number of tokens predicted. We empirically verify that ASSD speeds up language generation, without sacrificing quality. Furthermore, we provide a mathematically justified scheme for training AS-ARMs for generation, and show that AS-ARMs achieve state-of-the-art performance among sub-200M parameter models on infilling benchmark tasks, and nearly match the performance of models 50X larger on code generation. Our theoretical and empirical results indicate that the once-forgotten AS-ARMs are a promising direction of language modeling.
Interactive visualization of embedding projections is a useful technique for understanding data and evaluating machine learning models. Labeling data within these visualizations is critical for interpretation, as labels provide an overview of the projection and guide user navigation. However, most methods for producing labels require clustering the points, which can be computationally expensive as the number of points grows. In this paper, we describe an efficient clustering approach using kernel density estimation in the projected 2D space instead of points. This algorithm can produce high-quality cluster regions from a 2D density map in a few hundred milliseconds, orders of magnitude faster than current approaches. We contribute the design of the algorithm, benchmarks, and applications that demonstrate the utility of the algorithm, including labeling and summarization.
Decision making under uncertainty is a cross-cutting challenge in science and engineering. Most approaches to this challenge employ probabilistic representations of uncertainty. In complicated systems accessible only via data or black-box models, however, these representations are rarely known. We discuss how to characterize and manipulate such representations using triangular transport maps, which approximate any complex probability distribution as a transformation of a simple, well-understood distribution. The particular structure of triangular transport guarantees many desirable mathematical and computational properties that translate well into solving practical problems. Triangular maps are actively used for density estimation, (conditional) generative modelling, Bayesian inference, data assimilation, optimal experimental design, and related tasks. While there is ample literature on the development and theory of triangular transport methods, this manuscript provides a detailed introduction for scientists interested in employing measure transport without assuming a formal mathematical background. We build intuition for the key foundations of triangular transport, discuss many aspects of its practical implementation, and outline the frontiers of this field.
Hierarchical clustering is an effective, interpretable method for analyzing structure in data. It reveals insights at multiple scales without requiring a predefined number of clusters and captures nested patterns and subtle relationships, which are often missed by flat clustering approaches. However, existing hierarchical clustering methods struggle with high-dimensional data, especially when there are no clear density gaps between modes. In this work, we introduce t-NEB, a probabilistically grounded hierarchical clustering method, which yields state-of-the-art clustering performance on naturalistic high-dimensional data. t-NEB consists of three steps: (1) density estimation via overclustering; (2) finding maximum density paths between clusters; (3) creating a hierarchical structure via bottom-up cluster merging. t-NEB uses a probabilistic parametric density model for both overclustering and cluster merging, which yields both high clustering performance and a meaningful hierarchy, making it a valuable tool for exploratory data analysis. Code is available at https://github.com/ecker-lab/tneb clustering.
Reinforcement learning control algorithms face significant challenges due to out-of-distribution and inefficient exploration problems. While model-based reinforcement learning enhances the agent's reasoning and planning capabilities by constructing virtual environments, training such virtual environments can be very complex. In order to build an efficient inference model and enhance the representativeness of learning data, we propose the Counterfactual Experience Augmentation (CEA) algorithm. CEA leverages variational autoencoders to model the dynamic patterns of state transitions and introduces randomness to model non-stationarity. This approach focuses on expanding the learning data in the experience pool through counterfactual inference and performs exceptionally well in environments that follow the bisimulation assumption. Environments with bisimulation properties are usually represented by discrete observation and action spaces, we propose a sampling method based on maximum kernel density estimation entropy to extend CEA to various environments. By providing reward signals for counterfactual state transitions based on real information, CEA constructs a complete counterfactual experience to alleviate the out-of-distribution problem of the learning data, and outperforms general SOTA algorithms in environments with difference properties. Finally, we discuss the similarities, differences and properties of generated counterfactual experiences and real experiences. The code is available at https://github.com/Aegis1863/CEA.
We present a framework for modeling complex, high-dimensional distributions on convex polytopes by leveraging recent advances in discrete and continuous normalizing flows on Riemannian manifolds. We show that any full-dimensional polytope is homeomorphic to a unit ball, and our approach harnesses flows defined on the ball, mapping them back to the original polytope. Furthermore, we introduce a strategy to construct flows when only the vertex representation of a polytope is available, employing maximum entropy barycentric coordinates and Aitchison geometry. Our experiments take inspiration from applications in metabolic flux analysis and demonstrate that our methods achieve competitive density estimation, sampling accuracy, as well as fast training and inference times.
Probabilistic circuits (PCs) enable us to learn joint distributions over a set of random variables and to perform various probabilistic queries in a tractable fashion. Though the tractability property allows PCs to scale beyond non-tractable models such as Bayesian Networks, scaling training and inference of PCs to larger, real-world datasets remains challenging. To remedy the situation, we show how PCs can be learned across multiple machines by recursively partitioning a distributed dataset, thereby unveiling a deep connection between PCs and federated learning (FL). This leads to federated circuits (FCs) -- a novel and flexible federated learning (FL) framework that (1) allows one to scale PCs on distributed learning environments (2) train PCs faster and (3) unifies for the first time horizontal, vertical, and hybrid FL in one framework by re-framing FL as a density estimation problem over distributed datasets. We demonstrate FC's capability to scale PCs on various large-scale datasets. Also, we show FC's versatility in handling horizontal, vertical, and hybrid FL within a unified framework on multiple classification tasks.
We derive a novel generative model from iterative Gaussian posterior inference. By treating the generated sample as an unknown variable, we can formulate the sampling process in the language of Bayesian probability. Our model uses a sequence of prediction and posterior update steps to iteratively narrow down the unknown sample starting from a broad initial belief. In addition to a rigorous theoretical analysis, we establish a connection between our model and diffusion models and show that it includes Bayesian Flow Networks (BFNs) as a special case. In our experiments, we demonstrate that our model improves sample quality on ImageNet32 over both BFNs and the closely related Variational Diffusion Models, while achieving equivalent log-likelihoods on ImageNet32 and CIFAR10. Find our code at https://github.com/martenlienen/bsi.
Language models have recently been shown capable of performing regression tasks wherein numeric predictions are represented as decoded strings. In this work, we provide theoretical grounds for this capability and furthermore investigate the utility of causal auto-regressive sequence models when they are applied to any feature representation. We find that, despite being trained in the usual way - for next-token prediction via cross-entropy loss - decoding-based regression is as performant as traditional approaches for tabular regression tasks, while being flexible enough to capture arbitrary distributions, such as in the task of density estimation.
We present a novel approach to anomaly detection by integrating Generalized Hyperbolic (GH) processes into kernel-based methods. The GH distribution, known for its flexibility in modeling skewness, heavy tails, and kurtosis, helps to capture complex patterns in data that deviate from Gaussian assumptions. We propose a GH-based kernel function and utilize it within Kernel Density Estimation (KDE) and One-Class Support Vector Machines (OCSVM) to develop anomaly detection frameworks. Theoretical results confirmed the positive semi-definiteness and consistency of the GH-based kernel, ensuring its suitability for machine learning applications. Empirical evaluation on synthetic and real-world datasets showed that our method improves detection performance in scenarios involving heavy-tailed and asymmetric or imbalanced distributions. https://github.com/paulinebourigault/GHKernelAnomalyDetect
In transportation systems and autonomous vehicles, intelligent agents must understand the future motion of traffic participants to effectively plan motion trajectories. At the same time, the motion of traffic participants is inherently uncertain. In this paper, we propose TrajFlow, a generative framework for estimating the occupancy density of traffic participants. Our framework utilizes a causal encoder to extract semantically meaningful embeddings of the observed trajectory, as well as a normalizing flow to decode these embeddings and determine the most likely future location of traffic participants at some time point in the future. Our formulation differs from existing approaches because we model the marginal distribution of spatial locations instead of the joint distribution of unobserved trajectories. The advantages of a marginal formulation are numerous. First, we demonstrate that the marginal formulation produces higher accuracy on challenging trajectory forecasting benchmarks. Second, the marginal formulation allows for a fully continuous sampling of future locations. Finally, marginal densities are better suited for downstream tasks as they allow for the computation of per-agent motion trajectories and occupancy grids, the two most commonly used representations for motion forecasting. We present a novel architecture based entirely on neural differential equations as an implementation of this framework and provide ablations to demonstrate the advantages of a continuous implementation over a more traditional discrete neural network based approach. The code is available at https://github.com/kosieram21/TrajFlow .
Local Polynomial Regression (LPR) is a widely used nonparametric method for modeling complex relationships due to its flexibility and simplicity. It estimates a regression function by fitting low-degree polynomials to localized subsets of the data, weighted by proximity. However, traditional LPR is sensitive to outliers and high-leverage points, which can significantly affect estimation accuracy. This paper revisits the kernel function used to compute regression weights and proposes a novel framework that incorporates both predictor and response variables in the weighting mechanism. By introducing two positive definite kernels, the proposed method robustly estimates weights, mitigating the influence of outliers through localized density estimation. The method is implemented in Python and is publicly available at https://github.com/yaniv-shulman/rsklpr, demonstrating competitive performance in synthetic benchmark experiments. Compared to standard LPR, the proposed approach consistently improves robustness and accuracy, especially in heteroscedastic and noisy environments, without requiring multiple iterations. This advancement provides a promising extension to traditional LPR, opening new possibilities for robust regression applications.