In this paper, we propose a general methodology for sampling from un-normalized densities defined on Riemannian manifolds, with a particular focus on multi-modal targets that remain challenging for existing sampling methods. Inspired by the framework of diffusion models developed for generative modeling, we introduce a sampling algorithm based on the simulation of a non-equilibrium deterministic dynamics that transports an easy-to-sample noise distribution toward the target. At the marginal level, the induced density path follows a prescribed stochastic interpolant between the noise and target distributions, specifically constructed to respect the underlying Riemannian geometry. In contrast to related generative modeling approaches that rely on machine learning, our method is entirely training-free. It instead builds on iterative posterior sampling procedures using only standard Monte Carlo techniques, thereby extending recent diffusion-based sampling methodologies beyond the Euclidean setting. We complement our approach with a rigorous theoretical analysis and demonstrate its effectiveness on a range of multi-modal sampling problems, including high-dimensional and heavy-tailed examples.
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）；School of Mathematics and Statistics, Fujian Normal University（福建师范大学数学与统计学学院）；Beijing Institute of Technology（北京理工大学）
DisRFM: Polar Riemannian Flow Matching for Structure-Preserving Graph Domain Adaptation
The Chinese University of Hong Kong（香港中文大学）；National University of Defense Technology（国防科学技术大学）；Hebei University of Technology（河北工业大学）；City University of Hong Kong（香港城市大学）
University of Johannesburg（约翰内斯堡大学）；University of Witwatersrand（威特沃特斯兰德大学）
Markov Chain Monte Carlo inference of target posterior distributions in machine learning is predominately conducted via Hamiltonian Monte Carlo and its variants. This is due to Hamiltonian Monte Carlo based samplers ability to suppress random-walk behaviour. As with other Markov Chain Monte Carlo methods, Hamiltonian Monte Carlo produces auto-correlated samples which results in high variance in the estimators, and low effective sample size rates in the generated samples. Adding antithetic sampling to Hamiltonian Monte Carlo has been previously shown to produce higher effective sample rates com…
Instituto Tecnologico Autonomo de Mexico（墨西哥自治技术学院）；Fox School of Business, Temple University（天普大学福克斯商学院）；Department of Mathematical Sciences, Durham University（杜伦大学数学科学系）
We provide a general framework for constructing probability distributions on Riemannian manifolds, taking advantage of area-preserving maps and isometries. Control over distributions' properties, such as parameters, symmetry and modality yield a family of flexible distributions that are straightforward to sample from, suitable for use within Monte Carlo algorithms and latent variable models, such as autoencoders. As an illustration, we empirically validate our approach by utilizing our proposed distributions within a variational autoencoder and a latent space network model. Finally, we take ad…
Fast and Robust Visuomotor Riemannian Flow Matching Policy
Bosch Center for Artificial Intelligence（博世人工智能中心）；Division of Robotics, Perception, and Learning, KTH Royal Institute of Technology（机器人、感知与学习 division，皇家理工学院）；Computer Science Department of the Technische Universität Darmstadt（达姆施塔特技术大学计算机科学系）
Jarzynski Reweighting and Sampling Dynamics for Training Energy-Based Models: Theoretical Analysis of Different Transition Kernels
Dipartimento di Scienze Matematiche, Politecnico di Torino, Italy（都灵理工大学数学科学系）；Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Italy（意大利国家核物理研究所都灵分部）；Laboratoire de Physique de l’École normale supérieure, ENS, Université PSL, France（法国高等科学研究所物理实验室）
Boltzmann generators for amorphous particle systems
Laboratoire de Physique de l’École normale supérieure ENS, Université PSL, CNRS, Sorbonne Université, Université de Paris（巴黎高等师范学院物理实验室，巴黎科学研究院，法国国家科学研究中心，索邦大学，巴黎大学）；CMAP, CNRS, École polytechnique, Institut Polytechnique de Paris（巴黎高等理工学院计算数学与应用物理实验室，法国国家科学研究中心，巴黎理工学院，巴黎理工研究院）；Dipartimento di Fisica, Università di Trieste（特伦蒂诺-上阿迪杰大学物理系）；Gulliver, CNRS UMR 7083, ESPCI Paris, PSL Research University（Gulliver，法国国家科学研究中心UMR 7083，巴黎高等师范学院，巴黎科学研究院）
Hamiltonian Monte Carlo (HMC) has become routinely used for sampling from posterior distributions. Its extension Riemann manifold HMC (RMHMC) modifies the proposal kernel through distortion of local distances by a Riemannian metric. The performance depends critically on the choice of metric, with the Fisher information providing the standard choice. In this article, we propose a new class of metrics aimed at improving HMC's performance on multi-modal target distributions. We refer to the proposed approach as geometrically tempered HMC (GTHMC) due to its connection to other tempering methods. W…
Deep generative models provide a systematic way to learn nonlinear data distributions, through a set of latent variables and a nonlinear "generator" function that maps latent points into the input space. The nonlinearity of the generator imply that the latent space gives a distorted view of the input space. Under mild conditions, we show that this distortion can be characterized by a stochastic Riemannian metric, and demonstrate that distances and interpolants are significantly improved under this metric. This in turn improves probability distributions, sampling algorithms and clustering in th…
The University of Tokyo（东京大学）；Hasso Plattner Institute for Digital Engineering（哈索·普拉特纳数字工程研究所）；Institute of Statistical Mathematics（统计数学研究所）；The Graduate University for Advanced Studies, SOKENDAI（综合研究大学院大学（SOKENDAI））；RIKEN Centre for Advanced Intelligence Project (AIP)（理化学研究所先进智能项目中心（AIP））；Hasso Plattner Institute for Digital Health at the Icahn School of Medicine at Mount Sinai（西奈山伊坎医学院哈索·普拉特纳数字健康研究所）
Many unsupervised representation learning methods belong to the class of similarity learning models. While various modality-specific approaches exist for different types of data, a core property of many methods is that representations of similar inputs are close under some similarity function. We propose EMDE (Efficient Manifold Density Estimator) - a framework utilizing arbitrary vector representations with the property of local similarity to succinctly represent smooth probability densities on Riemannian manifolds. Our approximate representation has the desirable properties of being fixed-si…
