Rapid mixing of geodesic walks on manifolds with positive curvature

Abstract

We introduce a Markov chain for sampling from the uniform distribution on a Riemannian manifold M, which we call the geodesic walk. We prove that the mixing time of this walk on any manifold with positive sectional curvature Cx (u, v) bounded both above and below by 0 < m2 ≤ Cx (u, v) ≤ M2 < ∞ is O∗ (Mm22 ). In particular, this bound on the mixing time does not depend explicitly on the dimension of the manifold. In the special case that 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.