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Probabilistic Methods · Gaussian Processes

Iskander Azangulov, Andrei Smolensky, Alexander Terenin, Viacheslav (Slava) Borovitskiy

Gaussian processes are arguably the most important class of spatiotemporal models within machine learning. They encode prior information about the modeled function and can be used for exact or approximate Bayesian learning. In many applications, particularly in physical sciences and engineering, but also in areas such as geostatistics and neuroscience, invariance to symmetries is one of the most fundamental forms of prior information one can consider. The invariance of a Gaussian process' covariance to such symmetries gives rise to the most natural generalization of the concept of stationarity to such spaces. In this work, we develop constructive and practical techniques for building stationary Gaussian processes on a very large class of non-Euclidean spaces arising in the context of symmetries. Our techniques make it possible to (i) calculate covariance kernels and (ii) sample from prior and posterior Gaussian processes defined on such spaces, both in a practical manner. This work is split into two parts, each involving different technical considerations: part I studies compact spaces, while part II studies non-compact spaces possessing certain structure. Our contributions make the non-Euclidean Gaussian process models we study compatible with well-understood computational techniques available in standard Gaussian process software packages, thereby making them accessible to practitioners.

Reinforcement Learning · Everything Else

Liangyu Zhang, Yang Peng, Jiadong Liang, Wenhao Yang, Zhihua Zhang

In this paper, we study distributional reinforcement learning from the perspective of statistical efficiency. We investigate distributional policy evaluation, aiming to estimate the complete return distribution (denoted \eta^\pi) attained by a given policy \pi. We use the certainty-equivalence method to construct our estimator \hat\eta^\pi_n, based on a generative model. In this circumstance we need a dataset of size \widetilde O(|\mathcal{S}||\mathcal{A}|{\varepsilon^{-2p}(1-\gamma)^{-(2p+2)}}) to guarantee the supremum p-Wasserstein metric between \hat\eta^\pi_n and \eta^\pi less than \varepsilon with high probability. This implies the distributional policy evaluation problem can be solved with sample efficiency. Also, we show that under different mild assumptions a dataset of size \widetilde O({|\mathcal{S}||\mathcal{A}|}{\varepsilon^{-2}(1-\gamma)^{-4}}) suffices to ensure the supremum Kolmogorov-Smirnov metric and supremum total variation metric between \hat\eta^\pi_n and \eta^\pi is below \varepsilon with high probability. Furthermore, we investigate the asymptotic behavior of \hat\eta^\pi_n. We demonstrate that the ``empirical process'' \sqrt{n}(\hat\eta^\pi_n-\eta^\pi) converges weakly to a Gaussian process in the space of bounded functionals on a Lipschitz function class \ell^\infty(\mathcal{F}_{W_1}), also in the space of bounded functionals on an indicator function class \ell^\infty(\mathcal{F}_{\textup{KS}}) and a bounded measurable function class \ell^\infty(\mathcal{F}_{\textup{TV}}) when some mild conditions hold.

General Machine Learning · Causality

Arshak Hovhannisyan, Armen Allahverdyan

Simpson's paradox poses a challenge in probabilistic inference and decision-making. Our study revisits the paradox by re-estimating its frequency with an unbiased data generation process and reaffirms that it is not an artifact of deficient data collection. Thus, it can lead to incorrect recommendations in fields as diverse as statistics, psychology, and artificial intelligence. We show that the paradox can be resolved by assuming a minimal — though not necessarily observed — common cause (or screening) variable for the involved random variables. In our approach, conditioning on this minimal common cause establishes the correct association between events, which coincides with the conditioning (i.e., fine-grained) option of the original Simpson paradox. This resolution applies to both discrete cases of binary variables and continuous settings modeled by Gaussian variables. For a non-minimal common cause, the resolution of the paradox is possible, but detailed knowledge of the common cause is required. Our findings extend traditional understandings of the paradox and offer practical guidance for resolving apparent contradictions in probabilistic inference, ultimately enhancing decision-making processes. This point is illustrated by several examples.

Computer Vision · 3D Rendering & Reconstruction

Yiming Wang, Lucy Chai, Xuan Luo, Michael Niemeyer, Manuel Lagunas, Stephen Lombardi, Siyu Tang, Tiancheng Sun

Recent advances in feed-forward 3D Gaussian Splatting have led to rapid improvements in efficient scene reconstruction from sparse views. However, most existing approaches construct Gaussian primitives directly aligned with the pixels in one or more of the input images. This leads to redundancies in the representation when input views overlap and constrains the position of the primitives to lie along the input rays without full flexibility in 3D space. Moreover, these pixel-aligned approaches do not naturally generalize to dynamic scenes, where effectively leveraging temporal information requires resolving both redundant and newly appearing content across frames. To address these limitations, we introduce a novel Fuse-and-Refine module that enhances existing feed-forward models by merging and refining the primitives in a canonical 3D space. At the core of our method is an efficient hybrid Splat-Voxel representation – from an initial set of pixel-aligned Gaussian primitives, we aggregate local features into a coarse-to-fine voxel hierarchy, and then use a sparse voxel transformer to process these voxel features and generate refined Gaussian primitives. By fusing and refining an arbitrary number of inputs into a consistent set of primitives, our representation effectively reduces redundancy and naturally adapts to temporal frames, enabling history-aware online reconstruction of dynamic scenes. Trained on large-scale static scene datasets, our model learns an effective global strategy to process around 200k primitives within 15ms and significantly enhances reconstruction quality compared to pixel-aligned reconstruction approaches. Without additional training, our model generalizes to video by fusing primitives across time, yielding a more temporally coherent result compared to baseline methods with graceful handling of occluded content. Our approach achieves state-of-the-art performance in both static and streaming scene reconstructions while running at interactive rates (15 fps with 350ms delay) on a single H100 GPU.

Social Aspects · Fairness, Equity, Justice and Safety

Xiaoyu Hu, Gengyu Xue, Zhenhua Lin, Yi Yu

Algorithmic fairness has become a central topic in machine learning, and mitigating disparities across different subpopulations has emerged as a rapidly growing research area. In this paper, we systematically study the classification of functional data under fairness constraints, ensuring the disparity level of the classifier is controlled below a pre-specified threshold. We propose a unified framework for fairness-aware functional classification, tackling an infinite-dimensional functional space, addressing key challenges from the absence of density ratios and intractability of posterior probabilities, and discussing unique phenomena in functional classification. We further design a post-processing algorithm Fair Functional Linear Discriminant Analysis classifier (Fair-FLDA), which targets at homoscedastic Gaussian processes and achieves fairness via group-wise thresholding. Under weak structural assumptions on eigenspace, theoretical guarantees on fairness and excess risk controls are established. As a byproduct, our results cover the excess risk control of the standard FLDA as a special case, which, to the best of our knowledge, is first time seen. Our theoretical findings are complemented by extensive numerical experiments on synthetic and real datasets, highlighting the practicality of our designed algorithm.

Xixi Wan, AIHUA ZHENG, Bo Jiang, Beibei Wang, Chenglong Li, Jin Tang

Multi-modal object Re-IDentification (ReID) has gained considerable attention with the goal of retrieving specific targets across cameras using heterogeneous visual data sources. At present, multi-modal object ReID faces two core challenges: (1) learning robust features under fine-grained local noise caused by occlusion, frame loss, and other disruptions; and (2) effectively integrating heterogeneous modalities to enhance multi-modal representation. To address the above challenges, we propose a robust approach named Uncertainty-Guided Graph model for multi-modal object ReID (UGG-ReID). UGG-ReID is designed to mitigate noise interference and facilitate effective multi-modal fusion by estimating both local and sample-level aleatoric uncertainty and explicitly modeling their dependencies. Specifically, we first propose the Gaussian patch-graph representation model that leverages uncertainty to quantify fine-grained local cues and capture their structural relationships. This process boosts the expressiveness of modal-specific information, ensuring that the generated embeddings are both more informative and robust. Subsequently, we design an uncertainty-guided mixture of experts strategy that dynamically routes samples to experts exhibiting low uncertainty. This strategy effectively suppresses noise-induced instability, leading to enhanced robustness. Meanwhile, we design an uncertainty-guided routing to strengthen the multi-modal interaction, improving the performance. UGG-ReID is comprehensively evaluated on five representative multi-modal object ReID datasets, encompassing diverse spectral modalities. Experimental results show that the proposed method achieves excellent performance on all datasets and is significantly better than current methods in terms of noise immunity. Our code is available at https://github.com/wanxixi11/UGG-ReID.

Probabilistic Methods · Gaussian Processes

Marshal Sinaga, Julien Martinelli, Samuel Kaski

Gaussian processes (GPs) are widely used for regression and optimization tasks such as Bayesian optimization (BO) due to their expressiveness and principled uncertainty estimates. However, in settings with large datasets corrupted by outliers, standard GPs and their sparse approximations struggle with computational tractability and robustness. We introduce Robust Computation-aware Gaussian Process (RCaGP), a novel GP model that jointly addresses these challenges by combining a principled treatment of approximation-induced uncertainty with robust generalized Bayesian updating. The key insight is that robustness and approximation-awareness are not orthogonal but intertwined: approximations can exacerbate the impact of outliers, and mitigating one without the other is insufficient. Unlike previous work that focuses narrowly on either robustness or approximation quality, RCaGP combines both in a principled and scalable framework, thus effectively managing both outliers and computational uncertainties introduced by approximations such as low-rank matrix multiplications. Our model ensures more conservative and reliable uncertainty estimates, a property we rigorously demonstrate. Additionally, we establish a robustness property and show that the mean function is key to preserving it, motivating a tailored model selection scheme for robust mean functions. Empirical results confirm that solving these challenges jointly leads to superior performance across both clean and outlier-contaminated settings, both on regression and high-throughput Bayesian optimization benchmarks.

Social Aspects · Fairness, Equity, Justice and Safety

Jindong Yang, Han Fang, Weiming Zhang, Nenghai Yu, Kejiang Chen

Diffusion models have advanced rapidly in recent years, producing high-fidelity images while raising concerns about intellectual property protection and the misuse of generative AI. Image watermarking for diffusion models, particularly Noise-as-Watermark (NaW) methods, encode watermark as specific standard Gaussian noise vector for image generation, embedding the infomation seamlessly while maintaining image quality. For detection, the generation process is inverted to recover the initial noise vector containing the watermark before extraction. However, existing NaW methods struggle to balance watermark robustness with generation diversity. Some methods achieve strong robustness by heavily constraining initial noise sampling, which degrades user experience, while others preserve diversity but prove too fragile for real-world deployment. To address this issue, we propose T2SMark, a two-stage watermarking scheme based on Tail-Truncated Sampling (TTS). Unlike prior methods that simply map bits to positive or negative values, TTS enhances robustness by embedding bits exclusively in the reliable tail regions while randomly sampling the central zone to preserve the latent distribution. Our two-stage framework then ensures sampling diversity by integrating a randomly generated session key into both encryption pipelines. We evaluate T2SMark on diffusion models with both U-Net and DiT backbones. Extensive experiments show that it achieves an optimal balance between robustness and diversity.

Probabilistic Methods

Amber Hu, Henry Smith, Scott Linderman

Latent stochastic differential equation (SDE) models are important tools for the unsupervised discovery of dynamical systems from data, with applications ranging from engineering to neuroscience. In these complex domains, exact posterior inference of the latent state path is typically intractable, motivating the use of approximate methods such as variational inference (VI). However, existing VI methods for inference in latent SDEs often suffer from slow convergence and numerical instability. Here, we propose SDE Inference via Natural Gradients (SING), a method that leverages natural gradient VI to efficiently exploit the underlying geometry of the model and variational posterior. SING enables fast and reliable inference in latent SDE models by approximating intractable integrals and parallelizing computations in time. We provide theoretical guarantees that SING will approximately optimize the intractable, continuous-time objective of interest. Moreover, we demonstrate that better state inference enables more accurate estimation of nonlinear drift functions using, for example, Gaussian process SDE models. SING outperforms prior methods in state inference and drift estimation on a variety of datasets, including a challenging application to modeling neural dynamics in freely behaving animals. Altogether, our results illustrate the potential of SING as a tool for accurate inference in complex dynamical systems, especially those characterized by limited prior knowledge and non-conjugate structure.

Reinforcement Learning · Batch Offline

Akifumi Wachi, Kohei Miyaguchi, Takumi Tanabe, Rei Sato, Youhei Akimoto

A longstanding goal in safe reinforcement learning (RL) is a method to ensure the safety of a policy throughout the entire process, from learning to operation. However, existing safe RL paradigms inherently struggle to achieve this objective. We propose a method, called Provably Lifetime Safe RL (PLS), that integrates offline safe RL with safe policy deployment to address this challenge. Our proposed method learns a policy offline using return-conditioned supervised learning and then deploys the resulting policy while cautiously optimizing a limited set of parameters, known as target returns, using Gaussian processes (GPs). Theoretically, we justify the use of GPs by analyzing the mathematical relationship between target and actual returns. We then prove that PLS finds near-optimal target returns while guaranteeing safety with high probability. Empirically, we demonstrate that PLS outperforms baselines both in safety and reward performance, thereby achieving the longstanding goal to obtain high rewards while ensuring the safety of a policy throughout the lifetime from learning to operation.

Computer Vision · 3D Rendering & Reconstruction

Hyunjin Kim, Haebeom Jung, Jaesik Park

We propose an adaptive sampling framework for 3D Gaussian Splatting (3DGS) that leverages comprehensive multi-view photometric error signals within a unified Metropolis-Hastings approach. Vanilla 3DGS heavily relies on heuristic-based density-control mechanisms (e.g., cloning, splitting, and pruning), which can lead to redundant computations or premature removal of beneficial Gaussians. Our framework overcomes these limitations by reformulating densification and pruning as a probabilistic sampling process, dynamically inserting and relocating Gaussians based on aggregated multi-view errors and opacity scores. Guided by Bayesian acceptance tests derived from these error-based importance scores, our method substantially reduces reliance on heuristics, offers greater flexibility, and adaptively infers Gaussian distributions without requiring predefined scene complexity. Experiments on benchmark datasets, including Mip-NeRF360, Tanks and Temples and Deep Blending, show that our approach reduces the number of Gaussians needed, achieving faster convergence while matching or modestly surpassing the view-synthesis quality of state-of-the-art models.

Probabilistic Methods

Jeffrey Alido, Tongyu Li, Yu Sun, Lei Tian

Conventional score-based diffusion models (DMs) may struggle with anisotropic Gaussian diffusion processes due to the required inversion of covariance matrices in the denoising score matching training objective \cite{vincent_connection_2011}. We propose Whitened Score (WS) diffusion models, a novel framework based on stochastic differential equations that learns the Whitened Score function instead of the standard score. This approach circumvents covariance inversion, extending score-based DMs by enabling stable training of DMs on arbitrary Gaussian forward noising processes. WS DMs establish equivalence with flow matching for arbitrary Gaussian noise, allow for tailored spectral inductive biases, and provide strong Bayesian priors for imaging inverse problems with structured noise. We experiment with a variety of computational imaging tasks using the CIFAR, CelebA ($64\times64$), and CelebA-HQ ($256\times256$) datasets and demonstrate that WS diffusion priors trained on anisotropic Gaussian noising processes consistently outperform conventional diffusion priors based on isotropic Gaussian noise.

Huiyang Shao, Xin Xia, Yuxi Ren, XING WANG, Xuefeng Xiao

Diffusion models have emerged as state‑of‑the‑art in image synthesis.However, it often suffer from semantic instability and slow iterative denoising. We introduce Latent Alignment Framework (LABridge), a novel Text–Image Latent Alignment Framework via an Ornstein–Uhlenbeck (OU) Process, which explicitly preserves and aligns textual and visual semantics in an aligned latent space. LABridge employs a Text-Image Alignment Encoder (TIAE) to encode text prompts into structured priors that are directly aligned with image latents. Instead of a homogeneous Gaussian, Mean-Conditioned OU process smoothly interpolates between these text‑conditioned priors and image latents, improving stability and reducing the number of denoising steps. Extensive experiments on standard text-to-image benchmarks show that LABridge achieves better text–image alignment metric and competitive FID scores compared to leading diffusion baselines. By unifying text and image representations through principled latent alignment, LABridge paves the way for more efficient, semantically consistent, and high‑fidelity text to image generation.

Optimization · Bayesian & Black-box Optimization

Xunzhao Yu

Meta-learning has been demonstrated to be useful to improve the sampling efficiency of Bayesian optimization (BO) and surrogate-assisted evolutionary algorithms (SAEAs) when solving expensive optimization problems (EOPs). Existing studies mainly focus on either combinations of existing meta-learning modeling methods with optimization algorithms, or the development of meta-learning acquisition functions for specific meta BO. However, the meta-learning models used in the literature are not designed for optimization purpose, and the generalization ability of meta-learning acquisition functions is limited. In this work, we develop a novel architecture of meta-learning model for optimization purpose and propose a generalized few-shot evolutionary optimization (FSEO) framework to solve EOPs. We focus on the scenario of expensive multi-objective EOPs (EMOPs) in the context of few-shot optimization as there are few studies on it and its high requirement on surrogate modeling performance. The surrogates in FSEO framework combines neural network with Gaussian Processes (GPs), their network parameters and some parameters of GPs represent task-independent experience and are meta-learned across related optimization tasks, the remaining GPs parameters are task-specific parameters that represent unique features of the target task. We demonstrate that our FSEO framework is able to improve the sampling efficiency of existing SAEAs on EMOPs.

Probabilistic Methods · Gaussian Processes

Pratik Rathore, Zachary Frangella, Sachin Garg, Shaghayegh Fazliani, Michal Derezinski, Madeleine Udell

Gaussian processes (GPs) play an essential role in biostatistics, scientific machine learning, and Bayesian optimization for their ability to provide probabilistic predictions and model uncertainty. However, GP inference struggles to scale to large datasets (which are common in modern applications), since it requires the solution of a linear system whose size scales quadratically with the number of samples in the dataset. We propose an approximate, distributed, accelerated sketch-and-project algorithm ($\texttt{ADASAP}$) for solving these linear systems, which improves scalability. We use the theory of determinantal point processes to show that the posterior mean induced by sketch-and-project rapidly converges to the true posterior mean. In particular, this yields the first efficient, condition number-free algorithm for estimating the posterior mean along the top spectral basis functions, showing that our approach is principled for GP inference. $\texttt{ADASAP}$ outperforms state-of-the-art solvers based on conjugate gradient and coordinate descent across several benchmark datasets and a large-scale Bayesian optimization task. Moreover, $\texttt{ADASAP}$ scales to a dataset with $> 3 \cdot 10^8$ samples, a feat which has not been accomplished in the literature.

Deep Learning · Theory

George Rapakoulias, Ali Reza Pedram, Fengjiao Liu, Lingjiong Zhu, Panagiotis Tsiotras

Schrodinger Bridges (SBs) are diffusion processes that steer, in finite time, a given initial distribution to another final one while minimizing a suitable cost functional. Although various methods for computing SBs have recently been proposed in the literature, most of these approaches require computationally expensive training schemes, even for solving low-dimensional problems. In this work, we propose an analytic parametrization of a set of feasible policies for steering the distribution of a dynamical system from one Gaussian Mixture Model (GMM) to another. Instead of relying on standard non-convex optimization techniques, the optimal policy within the set can be approximated as the solution of a low-dimensional linear program whose dimension scales linearly with the number of components in each mixture. The proposed method generalizes naturally to more general classes of dynamical systems, such as controllable linear time-varying systems, enabling efficient solutions to multi-marginal momentum SBs between GMMs, a challenging distribution interpolation problem. We showcase the potential of this approach in low-to-moderate dimensional problems such as image-to-image translation in the latent space of an autoencoder, learning of cellular dynamics using multi-marginal momentum SBs, and various other examples. The implementation is publicly available at https://github.com/georgeRapa/GMMflow.

Theory · Reinforcement Learning and Planning

Xiaoli Tang, Han Yu, Xiaoxiao Li

Auction-based Federated Learning (AFL) fosters collaboration among self-interested data consumers (DCs) and data owners (DOs). A major challenge in AFL pertains to how DCs select and bid for DOs. Existing methods are generally static, making them ill-suited for dynamic AFL markets. To address this issue, we propose the R}einforcement Learning-based Bidding Strategy for DCs in Auction-based Federated Learning (RLB-AFL). We incorporate historical states into a Deep Q-Network to capture sequential information critical for bidding decisions. To mitigate state space sparsity, where specific states rarely reoccur for each DC during auctions, we incorporate the Gaussian Mixture Model into RLB-AFL. This facilitates soft clustering on sequential states, reducing the state space dimensionality and easing exploration and action-value function approximation. In addition, we enhance the $\epsilon$-greedy policy to help the RLB-AFL agent balance exploitation and exploration, enabling it to be more adaptable in the AFL decision-making process. Extensive experiments under 6 widely used benchmark datasets demonstrate that RLB-AFL achieves superior performance compared to 8 state-of-the-art approaches. It outperforms the best baseline by 10.56% and 3.15% in terms of average total utility

Deep Learning · Generative Models and Autoencoders

Roi Benita, Miki Elad, Joseph Keshet

Diffusion models (DMs) have emerged as powerful tools for modeling complex data distributions and generating realistic new samples. Over the years, advanced architectures and sampling methods have been developed to make these models practically usable. However, certain synthesis process decisions still rely on heuristics without a solid theoretical foundation. In our work, we offer a novel analysis of the DM's inference process, introducing a comprehensive frequency response perspective. Specifically, by relying on Gaussianity assumption, we present the inference process as a closed-form spectral transfer function, capturing how the generated signal evolves in response to the initial noise. We demonstrate how the proposed analysis can be leveraged to design a noise schedule that aligns effectively with the characteristics of the data. The spectral perspective also provides insights into the underlying dynamics and sheds light on the relationship between spectral properties and noise schedule structure. Our results lead to scheduling curves that are dependent on the spectral content of the data, offering a theoretical justification for some of the heuristics taken by practitioners.

Social Aspects · Accountability, Transparency and Interpretability

Thummaluru Siddartha Reddy, Vempalli Naga Sai Saketh, Mahesh Chandran

Graph Neural Networks \texttt{(GNNs)} excel at jointly modeling node features and topology, yet their \emph{black-box} nature limits their adoption in real-world applications where interpretability is desired. Inspired by the success of interpretable Neural Additive Models \texttt{(NAM)} for tabular data, Graph Neural Additive Network \texttt{(GNAN)} extends the additive modeling approach to graph data to overcome limitations of GNNs. While being interpretable, \texttt{GNAN} representation learning overlooks the importance of local aggregation and more importantly suffers from parameter complexity. To mitigate the above challenges, we introduce Graph Neural Additive Model with Random Fourier Features (\texttt{G-NAMRFF}), a lightweight, self‐interpretable graph additive architecture. \texttt{G-NAMRFF} represents each node embedding as the sum of feature‐wise contributions where contributions are modeled via a \emph{Gaussian process} \texttt{(GP)} with a graph- and feature-aware kernel. Specifically, we construct a kernel using Radial Basis Function (\texttt{RBF}) with graph structure induced by Laplacian and learnable Finite Impulse Response (\texttt{FIR}) filter. We approximate the kernel using Random Fourier Features (\texttt{RFFs}) which transforms the \texttt{GP} prior to a Bayesian formulation, which are subsequently learnt using a single layer neural network with size equal to number of \texttt{RFF} features. \texttt{G-NAMRFF} is light weight with $168\times$ fewer parameters compared to \texttt{GNAN}. Despite its compact size, \texttt{G-NAMRFF} matches or outperforms state-of-the-art \texttt{GNNs} and \texttt{GNAN} on node and graph classification tasks, delivering real-time interpretability without sacrificing accuracy.

Probabilistic Methods

Guangting Yu, Shiwei Lan

Solving and learning partial differential equations (PDEs) lies at the core of physics-informed machine learning. Traditional numerical methods, such as finite difference and finite element approaches, are rooted in domain-specific techniques and often lack scalability. Recent advances have introduced neural networks and Gaussian processes (GPs) as flexible tools for automating PDE solving and incorporating physical knowledge into learning frameworks. While GPs offer tractable predictive distributions and a principled probabilistic foundation, they may be suboptimal in capturing complex behaviors such as sharp transitions or non-smooth dynamics. To address this limitation, we propose the use of the Q-exponential process (Q-EP), a recently developed generalization of GPs designed to better handle data with abrupt changes and to more accurately model derivative information. We advocate for Q-EP as a superior alternative to GPs in solving PDEs and associated inverse problems. Leveraging sparse variational inference, our method enables principled uncertainty quantification -- a capability not naturally afforded by neural network-based approaches. Through a series of experiments, including the Eikonal equation, Burgers’ equation, and an inverse Darcy flow problem, we demonstrate that the variational Q-EP method consistently yields more accurate solutions while providing meaningful uncertainty estimates.