Theory and Algorithms for Diffusion Processes on Riemannian Manifolds

TL;DR

This paper develops new theoretical tools for analyzing stochastic differential equations on Riemannian manifolds, including error bounds for discretizations and non-Gaussian noise, aiding the study of geometric MCMC algorithms.

Contribution

It introduces a simple construction of geometric SDEs on manifolds with bounded curvature and provides the first non-asymptotic error bounds for their discretizations and non-Gaussian noise models.

Findings

First non-asymptotic error bound for geometric Euler-Murayama discretization.

Bound on the distance between exact SDE and discrete geometric random walk.

Tools for analyzing MCMC algorithms with non-standard noise distributions.

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.