Theory and Algorithms for Diffusion Processes on Riemannian Manifolds

2022/04/28 by Cheng, Xiang, Jingzhao Zhang, Suvrit Sra +2 · 1 citation
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #Differential Geometry (math.DG) #FOS: Mathematics #Morphological variations and asymmetry #Probability (math.PR) #Statistical Methods and Inference

paper · pdf · doi:10.48550/arxiv.2204.13665

openalex publication_date 2022/04/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study geometric stochastic differential equations (SDEs) and their approximations on Riemannian manifolds. In particular, we introduce a simple new construction of geometric SDEs, using which with bounded curvature. In particular, we provide the first (to our knowledge) non-asymptotic bound on the error of the geometric Euler-Murayama discretization. We then bound the distance between the exact SDE and a discrete geometric random walk, where the noise can be non-Gaussian; this analysis is useful for using geometric SDEs to model naturally occurring discrete non-Gaussian stochastic processes. Our results provide convenient tools for studying MCMC algorithms that adopt non-standard noise distributions.

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• Efficient Diffusion Models for Symmetric Manifolds

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