Riemannian Proximal Sampler for High-accuracy Sampling on Manifolds

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 ε-accuracy requires O(log(1/ε)) iterations in Kullback-Leibler divergence assuming access to exact oracles and O(log2(1/ε))iterations in the total variation metric assuming access to sufficiently accurate inexact oracles.