# Riemannian Proximal Sampler for High-accuracy Sampling on Manifolds

arXiv: https://arxiv.org/abs/2502.07265 · HTML: https://mlsanity.com/paper/2502.07265/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Feb 2025.

## Built on by (most cited first, 2 of 2)

- [Polynomial Convergence of Riemannian Diffusion Models](https://mlsanity.com/paper/2601.02499/) (2026)
- [Sharp Convergence and Sampling Trade-offs for Riemannian Diffusion under Nonnegative Ricci Curvature](https://mlsanity.com/paper/2609.36568/) (2026)

## Abstract (arXiv)

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(log2(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.