Rapid Mixing of Geodesic Walks on Manifolds with Positive Curvature

TL;DR

This paper introduces a geodesic walk Markov chain for sampling uniformly on manifolds with positive curvature, proving rapid mixing times independent of dimension, and provides a practical algorithm for convex bodies.

Contribution

It establishes dimension-independent mixing time bounds for geodesic walks on positively curved manifolds and offers a new sampling algorithm for convex bodies.

Findings

Mixing time is bounded by the ratio of curvature bounds.

The algorithm is computationally tractable for convex bodies.

Sampling time depends only on curvature, not dimension.

Abstract

We introduce a Markov chain for sampling from the uniform distribution on a Riemannian manifold $\mathcal{M}$, which we call the $\textit{geodesic walk}$. We prove that the mixing time of this walk on any manifold with positive sectional curvature $C_{x}(u,v)$ bounded both above and below by $0 < \mathfrak{m}_{2} \leq C_{x}(u,v) \leq \mathfrak{M}_2 < \infty$ is $\mathcal{O}^*\left(\frac{\mathfrak{M}_2}{\mathfrak{m}_2}\right)$. In particular, this bound on the mixing time does not depend explicitly on the dimension of the manifold. In the special case that $\mathcal{M}$ is the boundary of a convex body, we give an explicit and computationally tractable algorithm for approximating the exact geodesic walk. As a consequence, we obtain an algorithm for sampling uniformly from the surface of a convex body that has running time bounded solely in terms of the curvature of the body.