提出一种高精度流形采样方法，理论保证收敛快，适合复杂几何结构数据采样。
Riemannian Proximal Sampler for High-accuracy Sampling on Manifolds
- 基于黎曼近似点法构建采样器，依赖布朗增量与热核两个核心算子。
- 理论上在KL散度下仅需O(log(1/ε))次迭代即可达到ε精度，变体在总变差下为O(log²(1/ε))。
- 通过热核截断和渐近公式实现可计算的算子，适用于流形上概率分布采样任务。
我们提出黎曼近似点采样器（Riemannian Proximal Sampler），用于从定义在黎曼流形上的概率密度采样。该采样器性能依赖于两个关键算子：流形布朗增量（MBI）和黎曼热核（RHK）。我们建立了高精度采样保证：在使用精确算子时，达到ε-精度的样本只需O(log(1/ε))次迭代（以KL散度衡量）；使用足够精确的近似算子时，在总变差距离下需要O(log²(1/ε))次迭代。我们通过热核截断和Varadhan渐近公式实现了这两个算子的实用化。在此基础上，将采样器解释为熵正则化黎曼近似点法在相应Wasserstein空间中的离散化。初步数值实验验证了该方法的有效性。
原文摘要 · Abstract (English)
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.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。