A Dimension-Independent Bound on the Wasserstein Contraction Rate of a Geodesic Random Walk on the Sphere

TL;DR

This paper proves that a geodesic random walk on the sphere has a dimension-independent Wasserstein contraction rate, which has implications for the efficiency of certain MCMC sampling methods in high dimensions.

Contribution

It establishes a dimension-independent bound on the Wasserstein contraction rate for geodesic random walks on the sphere, advancing understanding of high-dimensional sampling algorithms.

Findings

Wasserstein contraction rate is bounded independently of dimension

Implications for high-dimensional MCMC sampling methods

Supports potential for dimension-independent sampling efficiency

Abstract

We theoretically analyze the properties of a geodesic random walk on the Euclidean $d$-sphere. Specifically, we prove that the random walk's transition kernel is Wasserstein contractive with a contraction rate which can be bounded from above independently of the dimension $d$. Our result is of particular interest due to its implications regarding the potential for dimension-independent performance of both geodesic slice sampling on the sphere and Gibbsian polar slice sampling, which are Markov chain Monte Carlo methods for approximate sampling from essentially arbitrary distributions on their respective state spaces.