Riemannian Proximal Sampler for High-accuracy Sampling on Manifolds
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| author | Guan, Yunrui Balasubramanian, Krishnakumar Ma, Shiqian |
| author_facet | Guan, Yunrui Balasubramanian, Krishnakumar Ma, Shiqian |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_07265 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Riemannian Proximal Sampler for High-accuracy Sampling on Manifolds Guan, Yunrui Balasubramanian, Krishnakumar Ma, Shiqian Machine Learning Statistics Theory 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. |
| title | Riemannian Proximal Sampler for High-accuracy Sampling on Manifolds |
| topic | Machine Learning Statistics Theory |
| url | https://arxiv.org/abs/2502.07265 |