Riemannian Proximal Sampler for High-accuracy Sampling on Manifolds
This addresses a fundamental challenge in computational statistics and machine learning for applications involving manifold-constrained data, though it appears incremental as it builds on existing proximal and sampling methods.
The paper tackles the problem of sampling from densities on Riemannian manifolds by introducing the Riemannian Proximal Sampler, achieving high-accuracy guarantees with O(log(1/ε)) iterations for exact oracles and O(log²(1/ε)) for inexact ones in terms of KL divergence and total variation metrics.
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.