Efficient Random Walks on Riemannian Manifolds

TL;DR

This paper introduces an efficient algorithm for sampling Brownian motion on compact Riemannian manifolds by using approximate geodesic random walks based on retractions, which converge to true Brownian motion.

Contribution

The paper proposes a novel method using retractions to approximate geodesic walks, reducing computational cost while ensuring convergence to Brownian motion.

Findings

Approximate geodesic walks converge to true Brownian motion with second-order approximation.

The proposed algorithm is computationally efficient for sampling on compact Riemannian manifolds.

The method maintains theoretical convergence guarantees despite approximation.

Abstract

According to a version of Donsker's theorem, geodesic random walks on Riemannian manifolds converge to the respective Brownian motion. From a computational perspective, however, evaluating geodesics can be quite costly. We therefore introduce approximate geodesic random walks based on the concept of retractions. We show that these approximate walks converge in distribution to the correct Brownian motion as long as the geodesic equation is approximated up to second order. As a result we obtain an efficient algorithm for sampling Brownian motion on compact Riemannian manifolds.