Riemannian Proximal Sampler for High-accuracy Sampling on Manifolds
Yunrui Guan, Krishna Balasubramanian, Shiqian Ma
Abstract
We introduce the \textit{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 \textit{Manifold Brownian Increments (MBI)} oracle and the \textit{Riemannian Heat-kernel (RHK)} oracle. We establish high-accuracy sampling guarantees for the Riemannian Proximal Sampler, showing that generating samples with \(\varepsilon\)-accuracy requires \(\mathcal{O}(\log(1/\varepsilon))\) iterations in Kullback-Leibler divergence assuming access to exact oracles and \(\mathcal{O}(\log^2(1/\varepsilon))\) 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.
BibTeX
@inproceedings{
guan2025riemannian,
title={Riemannian Proximal Sampler for High-accuracy Sampling on Manifolds},
author={Yunrui Guan and Krishna Balasubramanian and Shiqian Ma},
booktitle={The Thirty-ninth Annual Conference on Neural Information Processing Systems},
year={2025},
url={https://openreview.net/forum?id=KxhCJc8BOg}
}