Theory and Algorithms for Diffusion Processes on Riemannian Manifolds

Xiang Cheng; Jingzhao Zhang; Suvrit Sra

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arXiv · 2204.13665

Theory and Algorithms for Diffusion Processes on Riemannian Manifolds

Xiang Cheng·Jingzhao Zhang·Suvrit Sra

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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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Xiang Cheng, Jingzhao Zhang, Suvrit Sra. 2022-04-28. Theory and Algorithms for Diffusion Processes on Riemannian Manifolds. https://arxiv.org/abs/2204.13665

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