Riemannian Proximal Sampler for High-accuracy Sampling on Manifolds

TL;DR

This paper introduces the Riemannian Proximal Sampler, a novel method for high-accuracy sampling on manifolds, with theoretical guarantees and practical implementations leveraging heat-kernel approximations.

Contribution

The paper develops the Riemannian Proximal Sampler with provable convergence guarantees and practical oracle implementations, advancing sampling techniques on Riemannian manifolds.

Findings

High-accuracy sampling guarantees with logarithmic iteration complexity.

Practical oracle implementations using heat-kernel truncation.

Effective numerical results demonstrating the method's potential.

Abstract

We introduce the Riemannian Proximal Sampler, a method for sampling from densities defined on Riemannian manifolds. The performance of this sampler critically depends on two key oracles: the Manifold Brownian Increments (MBI) oracle and the Riemannian Heat-kernel (RHK) oracle. We establish high-accuracy sampling guarantees for the Riemannian Proximal Sampler, showing that generating samples with $\varepsilon$-accuracy requires $O(\log(1/\varepsilon))$ iterations in Kullback-Leibler divergence assuming access to exact oracles and $O(\log^2(1/\varepsilon))$ iterations in the total variation metric assuming access to sufficiently accurate inexact oracles. Furthermore, we present practical implementations of these oracles by leveraging heat-kernel truncation and Varadhan's asymptotics. In the latter case, we interpret the Riemannian Proximal Sampler as a discretization of the entropy-regularized Riemannian Proximal Point Method on the associated Wasserstein space. We provide preliminary numerical results that illustrate the effectiveness of the proposed methodology.