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Probabilistic Methods · Everything Else

Peiman Mohseni, Nick Duffield, Raymond K. W. Wong

Many phenomena in science and engineering involve unknown latent functions observed through finite, irregularly sampled measurements. Neural processes (NPs) offer a powerful framework for probabilistic functional inference by bridging stochastic processes with deep learning. In many domains, these functions exhibit symmetries---most notably translation equivariance---that can be exploited to improve sample efficiency and generalization. Existing translation-equivariant NPs, however, have two key limitations: (i) they are constructed by stacking generic components with nonlinearities, obscuring the induced function class, thus limiting interpretability; and (ii) convolutional designs are based on localized receptive fields and require dense discretization, while attention-based methods avoid these issues but scale quadratically with the number of observations. We address these challenges through two contributions. First, we characterize continuous translation-equivariant operators through their Volterra expansions, representing them as sums of higher-order convolutions. This yields analytical transparency while remaining amenable to efficient approximation by first-order convolution operators. Second, we introduce set Fourier convolutions (SFConvs), a frequency-domain parameterization that operates directly on irregularly sampled sets. SFConvs achieve global receptive fields without spatial discretization and scale linearly in the number of observations. Building on these ideas, we propose two families of conditional NPs (CNPs): SFConvCNPs, constructed by stacking SFConv blocks with nonlinearies, and SFVConvCNPs, which integrate the Volterra formulation. Experiments on synthetic and real-world datasets demonstrate the efficacy of our methods compared to the state-of-the-art baselines.

General Machine Learning · Probabilistic Methods

Declan McNamara, Yicun Duan, Jeffrey Regier

Neural posterior estimation (NPE) is a likelihood-free amortized variational inference method that approximates projections of the posterior distribution. To date, NPE variational families have been either simple and interpretable (such as the Gaussian family) or highly flexible but black-box and potentially difficult to optimize (such as normalizing flows). In this work, we parameterize variational families via basis expansions of the latent variables. The log density of our variational distribution is a linear combination of latent basis functions (LBFs), which may be fixed a priori or adapted to the problem class of interest. Our training and inference procedures are computationally efficient even for problems with high-dimensional latent spaces, provided only a low-dimensional projection of the posterior is of interest, owing to NPE's automatic marginalization capabilities. In numerous inference problems, the proposed variational family exhibits better performance than existing variational families used with NPE, including mixtures of Gaussians (mixture density networks) and normalizing flows, as well as outperforming an existing basis expansion method for variational inference.

Computer Vision · Image and Video Generation

Tao Ren, Zishi Zhang, Jinyang Jiang, Zehao Li, Shentao Qin, Yi Zheng, Guanghao Li, Qianyou Sun, Yan Li, Jiafeng Liang 等

The probabilistic diffusion model (DM), generating content by inferencing through a recursive chain structure, has emerged as a powerful framework for visual generation. After pre-training on enormous data, the model needs to be properly aligned to meet requirements for downstream applications. How to efficiently align the foundation DM is a crucial task. Contemporary methods are either based on Reinforcement Learning (RL) or truncated Backpropagation (BP). However, RL and truncated BP suffer from low sample efficiency and biased gradient estimation, respectively, resulting in limited improvement or, even worse, complete training failure. To overcome the challenges, we propose the Recursive Likelihood Ratio (RLR) optimizer, a Half-Order (HO) fine-tuning paradigm for DM. The HO gradient estimator enables the computation graph rearrangement within the recursive diffusive chain, making the RLR's gradient estimator **an unbiased one with lower variance** than other methods. We theoretically investigate the bias, variance, and convergence of our method. Extensive experiments are conducted on image and video generation to validate the superiority of the RLR. Furthermore, we propose a novel prompt technique that is natural for the RLR to achieve a synergistic effect. The implementation is available at https://github.com/RTkenny/RLR-Optimizer.

Deep Learning · Generative Models and Autoencoders

Longze Chen, Lu Wang, Renke Shan, Ze Gong, Run Luo, Jiaming Li, Jing Luo, Qiyao Wang, Min Yang

Reward models are crucial for aligning large language models (LLMs) with human values and intentions. Existing approaches follow either Generative (GRMs) or Discriminative (DRMs) paradigms, yet both suffer from limitations: GRMs typically demand costly point-wise supervision, while DRMs produce uncalibrated relative scores that lack probabilistic interpretation. To address these challenges, we introduce a novel reward modeling paradigm: Probabilistic Reward Model (PRM). Instead of modeling reward as a deterministic scalar, our approach treats it as a random variable, learning a full probability distribution for the quality of each response. To make this paradigm practical, we present its closed-form, discrete realization: the **Ordinal Probabilistic Reward Model** (OPRM), which discretizes the quality score into a finite set of ordinal ratings. Building on OPRM, we propose a data-efficient training strategy called **Region Flooding Tuning** (RgFT). It enables rewards to better reflect absolute text quality by incorporating quality-level annotations, which guide the model to concentrate the probability mass within corresponding rating sub-regions. Experiments on various reward model benchmarks show that our method improves accuracy by **2.9% ～ 7.4%** compared to prior reward models, demonstrating strong performance and data efficiency. Analysis of the score distribution provides evidence that our method captures not only relative rankings but also absolute quality.

General Machine Learning · Probabilistic Methods

Zikai Xie, Linjiang Chen

Bayesian Optimization (BO) is a powerful tool for black-box optimization, but its application to high-dimensional permutation spaces is severely limited by the challenge of defining scalable representations. The current state-of-the-art BO approach for permutation spaces relies on an exhaustive $\Omega(n^2)$ pairwise comparison, inducing a dense representation that is impractical for large-scale permutations. To break this barrier, we introduce a novel framework for generating efficient permutation representations via kernel functions derived from sorting algorithms. Within this framework, the Mallows kernel can be viewed as a special instance derived from enumeration sort. Further, we introduce the \textbf{Merge Kernel} , which leverages the divide-and-conquer structure of merge sort to produce a compact, $\Theta(n\log n)$ to achieve the lowest possible complexity with no information loss and effectively capture permutation structure. Our central thesis is that the Merge Kernel performs competitively with the Mallows kernel in low-dimensional settings, but significantly outperforms it in both optimization performance and computational efficiency as the dimension $n$ grows. Extensive evaluations on various permutation optimization benchmarks confirm our hypothesis, demonstrating that the Merge Kernel provides a scalable and more effective solution for Bayesian optimization in high-dimensional permutation spaces, thereby unlocking the potential for tackling previously intractable problems such as large-scale feature ordering and combinatorial neural architecture search.

Theory · Probabilistic Methods

Zijing Ou, Chinmay Pani, Yingzhen Li

Discrete diffusion models have become highly effective across various domains. However, real-world applications often require the generative process to adhere to certain constraints. To this end, we propose a Sequential Monte Carlo (SMC) framework that enables scalable inference-time control of discrete diffusion models through principled importance weighting and optimal proposal construction. Specifically, our approach derives tractable importance weights for a range of intermediate targets and characterises the optimal proposal, for which we develop two practical approximations: a first-order gradient-based approximation and an amortised proposal trained to minimise the log-variance of the importance weights. Empirical results across synthetic tasks, language modelling, biology design, and text-to-image generation demonstrate that our framework enhances controllability and sample quality, highlighting the effectiveness of SMC as a versatile recipe for scaling discrete diffusion models at inference time.

Theory · Probabilistic Methods

Wei Guo, Molei Tao, Yongxin Chen

Given an unnormalized probability density $\pi\propto\mathrm{e}^{-V}$, estimating its normalizing constant $Z=\int_{\mathbb{R}^d}\mathrm{e}^{-V(x)}\mathrm{d}x$ or free energy $F=-\log Z$ is a crucial problem in Bayesian statistics, statistical mechanics, and machine learning. It is challenging especially in high dimensions or when $\pi$ is multimodal. To mitigate the high variance of conventional importance sampling estimators, annealing-based methods such as Jarzynski equality and annealed importance sampling are commonly adopted, yet their quantitative complexity guarantees remain largely unexplored. We take a first step toward a non-asymptotic analysis of annealed importance sampling. In particular, we derive an oracle complexity of $\widetilde{O}\left(\frac{d\beta^2{\mathcal{A}}^2}{\varepsilon^4}\right)$ for estimating $Z$ within $\varepsilon$ relative error with high probability, where $\beta$ is the smoothness of $V$ and $\mathcal{A}$ denotes the action of a curve of probability measures interpolating $\pi$ and a tractable reference distribution. Our analysis, leveraging Girsanov's theorem and optimal transport, does not explicitly require isoperimetric assumptions on the target distribution. Finally, to tackle the large action of the widely used geometric interpolation, we propose a new algorithm based on reverse diffusion samplers, establish a framework for analyzing its complexity, and empirically demonstrate its efficiency in tackling multimodality.

Social Aspects · Fairness, Equity, Justice and Safety

Zongxin Liu, Lijia Yu, Tao Lin, Zhiming Chi, Lijun Zhang

Object detection systems are essential in safety-critical applications, but they are vulnerable to object disappearance (OD) threats, in which valid objects become undetected under small input perturbations, creating serious risks. This paper addresses the problem of verifying the robustness of YOLO (You Only Look Once) networks against OD by proposing a three-step probabilistic verification framework: (1) estimating output ranges under a distribution of input perturbations, (2) formally verifying the Non-Maximum Suppression (NMS) process within these ranges, and (3) iteratively refining the results to reduce over-approximation. The framework scales to practical YOLO models. Both theoretical analysis and experimental results demonstrate that our method achieves comparable probabilistic guarantees and provides tighter Intersection-over-Union (IoU) lower bounds while requiring significantly fewer samples than existing methods.

General Machine Learning · Representation Learning

Yalan Qin, Guorui Feng

Multi-view clustering aims to segment the view-specific data into the corresponding clusters. There have been a large number of works for multi-view clustering in recent years. As representive methods in multi-view clustering, works built on the graph make use of a view-consistent and discriminative graph while utilizing graph partitioning for the final clustering results. Despite the achieved significant success, these methods usually construct full graphs and the efficiency is not well guaranteed for the multi-view datasets with large scales. To handle the large-scale data, multi-view clustering methods based on anchor have been developed by learning the anchor graph with smaller size. However, the existing works neglect the interpretability of multi-view clustering based on anchor from the probabilistic perspective. These methods also ignore analyzing the relationship between the input data and the final clustering results based on the assigned meaningful probability associations in a unified manner. In this work, we propose a novel method termed Unified and Efficient Multi-view Clustering from Probabilistic perspective(UEMCP). It aims to improve the explanation ability of multi-view clustering based on anchor from the probabilistic perspective in an end-to-end manner. It ensures the consistent inherent structures among these views by learning the common transition probability from data points to categories in one step. With the guidance of the common transition probability matrix from data points to categories, the soft label of data points can be achieved based on the common transition probability matrix from anchor points to categories in the learning framework. Experiments on different challenging multi-view datasets confirm the superiority of UEMCP compared with the representative ones.

General Machine Learning · Probabilistic Methods

Zijian Wang, Xiaofei Zhang, Xin Zhang, Yukun Liu, Qiong Zhang

Federated learning (FL) is increasingly adopted in domains like healthcare, where data privacy is paramount. A fundamental challenge in these systems is statistical heterogeneity—the fact that data distributions vary significantly across clients (e.g., different hospitals may treat distinct patient demographics). While current FL algorithms focus on aggregating model updates from these heterogeneous clients, the potential of the central server remains under-explored. This paper is motivated by a healthcare scenario: could a central server not only coordinate model training but also guide a new patient to the hospital best equipped for their specific condition? We generalize this idea to propose a novel paradigm for FL systems where the server actively guides the allocation of new tasks or queries to the most appropriate client. To enable this, we introduce a density ratio model and empirical likelihood-based framework that simultaneously addresses two goals: (1) learning effective local models on each client, and (2) finding the best matching client for a new query. Empirical results demonstrate the framework's effectiveness on benchmark datasets, showing improvements in both model accuracy and the precision of client guidance compared to standard FL approaches. This work opens a new direction for building more intelligent and resource-efficient FL systems that leverage heterogeneity as a feature, not just a bug.

Computer Vision · 3D Rendering & Reconstruction

Zike Wu, Qi Yan, Xuanyu Yi, Lele Wang, Renjie Liao

Real-time reconstruction of dynamic 3D scenes from uncalibrated video streams demands robust online methods that recover scene dynamics from sparse observations under strict latency and memory constraints. Yet most dynamic reconstruction methods rely on hours of per-scene optimization under full-sequence access, limiting practical deployment. In this work, we introduce **StreamSplat**, a fully feed-forward framework that instantly transforms uncalibrated video streams of arbitrary length into dynamic 3D Gaussian Splatting (3DGS) representations in an online manner. It is achieved via three key technical innovations: 1) a probabilistic sampling mechanism that robustly predicts 3D Gaussians from uncalibrated inputs; 2) a bidirectional deformation field that yields reliable associations across frames and mitigates long-term error accumulation; 3) an adaptive Gaussian fusion operation that propagates persistent Gaussians while handling emerging and vanishing ones. Extensive experiments on standard dynamic and static benchmarks demonstrate that StreamSplat achieves state-of-the-art reconstruction quality and dynamic scene modeling. Uniquely, our method supports the online reconstruction of arbitrarily long video streams with a $1200\times$ speedup over optimization-based methods. Our code and models are available at https://streamsplat3d.github.io/.

Yian Wang, Han Yang, Minghao Guo, Xiaowen Qiu, Johnson (Tsun-Hsuan) Wang, Wojciech Matusik, Joshua B Tenenbaum, Chuang Gan

Automatically generating interactive 3D environments is crucial for scaling up robotic data collection in simulation. While prior work has primarily focused on 3D asset placement, it often overlooks the physical relationships between objects (e.g., contact, support, balance, and containment), which are essential for creating complex and realistic manipulation scenarios such as tabletop arrangements, shelf organization, or box packing. Compared to classical 3D layout generation, producing complex physical scenes introduces additional challenges: (a) higher object density and complexity (e.g., a small shelf may hold dozens of books), (b) richer supporting relationships and compact spatial layouts, and (c) the need to accurately model both spatial placement and physical properties. To address these challenges, we propose PhyScensis, an LLM agent-based framework powered by a physics engine, to produce physically plausible scene configurations with high complexity. Specifically, our framework consists of three main components: an LLM agent iteratively proposes assets with spatial and physical predicates; a solver, equipped with a physics engine, realizes these predicates into a 3D scene; and feedback from the solver informs the agent to refine and enrich the configuration. Moreover, our framework preserves strong controllability over fine-grained textual descriptions and numerical parameters (e.g., relative positions, scene stability), enabled through probabilistic programming for stability and a complementary heuristic that jointly regulates stability and spatial relations. Experimental results show that our method outperforms prior approaches in scene complexity, visual quality, and physical accuracy, offering a unified pipeline for generating complex physical scene layouts for robotic manipulation.

Optimization · Sampling and Optimization

Daniel Severo, Brian Karrer, Niklas Nolte

Learning distributions over permutations is a fundamental problem in machine learning, with applications in ranking, combinatorial optimization, structured prediction, and data association. Existing methods rely on mixtures of parametric families or neural networks with expensive variational inference procedures. In this work, we propose a novel approach that leverages alternative representations for permutations, including Lehmer codes, Fisher-Yates draws, and Insertion-Vectors. These representations form a bijection with the symmetric group, allowing for unconstrained learning using conventional deep learning techniques, and can represent any probability distribution over permutations. Our approach enables a trade-off between expressivity of the model family and computational requirements. In the least expressive and most computationally efficient case our method subsumes previous families of well established probabilistic models over permutations, including Mallow's and the Repeated Insertion Model. Experiments indicate our method significantly outperforms current approaches on the jigsaw puzzle benchmark, a common task for permutation learning. However, we argue this benchmark is limited in its ability to assess learning probability distributions, as the target is a delta distribution (i.e., a single correct solution exists). We therefore propose two additional benchmarks: learning cyclic permutations and re-ranking movies based on user preference. We show that our method learns non-trivial distributions even in the least expressive mode, while traditional models fail to even generate valid permutations in this setting.

Theory · Probabilistic Methods

Yunrui Guan, Krishna Balasubramanian, Shiqian Ma

We study generative modeling on convex domains using flow matching and mirror maps, and identify two fundamental challenges. First, standard log-barrier mirror maps induce heavy-tailed dual distributions, leading to ill-posed dynamics. Second, coupling with Gaussian priors performs poorly when matching heavy-tailed targets. To address these issues, we propose Mirror Flow Matching based on a \emph{regularized mirror map} that controls dual tail behavior and guarantees finite moments, together with coupling to a Student-$t$ prior that aligns with heavy-tailed targets and stabilizes training. We provide theoretical guarantees, including spatial Lipschitzness and temporal regularity of the velocity field, Wasserstein convergence rates for flow matching with Student-$t$ priors and primal-space guarantees for constrained generation, under $\varepsilon$-accurate learned velocity fields. Empirically, our method outperforms baselines in synthetic convex-domain simulations and achieves competitive sample quality on real-world constrained generative tasks.

Xiangxin Zhou, Zichen Liu, Haonan Wang, Chao Du, Min Lin, Chongxuan Li, Liang Wang, Tianyu Pang

We introduce a **variational reasoning** framework for language models that treats thinking traces as latent variables and optimizes them through variational inference. Starting from the evidence lower bound (ELBO), we extend it to a multi-trace objective for tighter bounds and propose a forward-KL formulation that stabilizes the training of the variational posterior. We further show that rejection sampling finetuning and binary-reward RL, including GRPO, can be interpreted as local forward-KL objectives, where *an implicit weighting by model accuracy* naturally arises from the derivation and reveals a previously unnoticed bias toward easier questions. We empirically validate our method on the Qwen 2.5 and Qwen 3 model families across a wide range of reasoning tasks. Overall, our work provides a principled probabilistic perspective that unifies variational inference with RL-style methods and yields stable objectives for improving the reasoning ability of language models.

General Machine Learning · Probabilistic Methods

Deming Sheng, Ricardo Henao

Survival analysis plays a critical role in modeling time-to-event outcomes across various domains. Although recent advances have focused on improving _predictive accuracy_ and _concordance_, fine-grained _calibration_ remains comparatively underexplored. In this paper, we propose a survival modeling framework based on the Individually Calibrated Asymmetric Laplace Distribution (ICALD), which unifies _parametric_ and _nonparametric_ approaches based on the ALD. We begin by revisiting the probabilistic foundation of the widely used _pinball_ loss in _quantile regression_ and its reparameterization as the _asymmetry form_ of the ALD. This reparameterization enables a principled shift to _parametric_ modeling while preserving the flexibility of _nonparametric_ methods. Furthermore, we show theoretically that ICALD, with the _quantile regression_ loss is probably approximately individually calibrated. Then we design an extended ICALD framework that supports both _pre-calibration_ and _post-calibration_ strategies. Extensive experiments on 14 synthetic and 7 real-world datasets demonstrate that our method achieves competitive performance in terms of _predictive accuracy_, _concordance_, and _calibration_, while outperforming 12 existing baselines including recent _pre-calibration_ and _post-calibration_ methods.

General Machine Learning · Probabilistic Methods

Lorenzo Loconte, Adrián Javaloy, Antonio Vergari

Squared tensor networks (TNs) and their extension as computational graphs---squared circuits---have been used as expressive distribution estimators, yet supporting closed-form marginalization. However, the squaring operation introduces additional complexity when computing the partition function or marginalizing variables, which hinders their applicability in ML. To solve this issue, canonical forms of TNs are parameterized via unitary matrices to simplify the computation of marginals. However, these canonical forms do not apply to circuits, as they can represent factorizations that do not directly map to a known TN. Inspired by the ideas of orthogonality in canonical forms and determinism in circuits enabling tractable maximization, we show how to parameterize squared circuits to overcome their marginalization overhead. Our parameterizations unlock efficient marginalization even in factorizations different from TNs, but encoded as circuits, whose structure would otherwise make marginalization computationally hard. Finally, our experiments on distribution estimation show how our proposed conditions in squared circuits come with no expressiveness loss, while enabling more efficient learning.

General Machine Learning · Probabilistic Methods

David Berghaus, Patrick Seifner, Kostadin Cvejoski, César Ali Ojeda Marin, Ramses Sanchez

Modeling multi-type event sequences with marked temporal point processes (MTPPs) provides a principled framework for uncovering governing dynamical rules and predicting future events. Current neural approaches to MTPP inference typically require training separate, specialized models for each target system. We pursue a fundamentally different strategy: leveraging amortized inference and in-context learning, we pretrain a deep neural network to infer, *in-context*, the conditional intensity functions of event histories from a context consisting of sets of event sequences. Pretraining is performed on a large synthetic dataset of MTPPs sampled from a broad distribution over point processes. Once pretrained, our Foundation Inference Model for Point Processes (FIM-PP) can estimate MTPPs from real-world data without additional training, or be rapidly finetuned to specific target systems. Experiments show that FIM-PP matches the performance of specialized models on multi-event prediction across common benchmark datasets.

General Machine Learning · Everything Else

Michael Ibrahim, Nagi Gebraeel, Weijun Xie

We study the problem of federated clustering when the total number of clusters $K$ across clients is unknown, and the clients have heterogeneous but potentially overlapping cluster sets in their local data. To that end, we develop FedGEM: a federated generalized expectation-maximization algorithm for the training of mixture models with an unknown number of components. Our proposed algorithm relies on each of the clients performing EM steps locally, and constructing an uncertainty set around the maximizer associated with each local component. The central server utilizes the uncertainty sets to learn potential cluster overlaps between clients, and infer the global number of clusters via closed-form computations. We perform a thorough theoretical study of our algorithm, presenting probabilistic convergence guarantees under common assumptions. Subsequently, we study the specific setting of isotropic GMMs, providing tractable, low-complexity computations to be performed by each client during each iteration of the algorithm, as well as rigorously verifying assumptions required for algorithm convergence. We perform various numerical experiments, where we empirically demonstrate that our proposed method achieves comparable performance to centralized EM, and that it outperforms various existing federated clustering methods.

Deep Learning · Generative Models and Autoencoders

Kou Misaki, Takuya Akiba

We introduce _String Seed of Thought (SSoT)_, a novel prompting method for LLMs that improves _Probabilistic Instruction Following (PIF)_. We define PIF as a task requiring an LLM to select its answer from a predefined set of options, each associated with a specific probability, such that the empirical distribution of the generated answers aligns with the target distribution when prompted multiple times. While LLMs excel at tasks with single, deterministic answers, they often fail at PIF, exhibiting biases problematic for applications requiring non-deterministic behaviors, such as human-behavior simulation, content diversification, and multiplayer games. It also harms the diversity of generated responses, a crucial factor in test-time scaling, by causing the outputs to collapse into a limited set of answers. To address this, we propose SSoT, a simple prompting method that instructs an LLM to first output a random string to generate sufficient entropy. SSoT also instructs the LLM to extract randomness by manipulating this string to derive a final answer, thereby preserving diversity while adhering to specific constraints. We demonstrate that SSoT significantly improves the PIF performance of LLMs, approaching the ideal performance of a pseudo-random number generator. Notably, our experiments on NoveltyBench show SSoT's benefits extend beyond closed-set tasks to open-ended tasks by enhancing response diversity.