We study sampling problems associated with non-convex potentials that meanwhile lack smoothness. In particular, we consider target distributions that satisfy either logarithmic-Sobolev inequality or Poincar\'e inequality. Rather than smooth, the potentials are assumed to be semi-smooth or the summation of multiple semi-smooth functions. We develop a sampling algorithm that resembles proximal algorithms in optimization for this challenging sampling task. Our algorithm is based on a special case of Gibbs sampling known as the alternating sampling framework (ASF). The key contribution of this work is a practical realization of the ASF based on rejection sampling in the non-convex and semi-smooth setting. This work extends the recent algorithm in \cite{LiaChe21,LiaChe22} for non-smooth/semi-smooth log-concave distribution to the setting with non-convex potentials. In almost all the cases of sampling considered in this work, our proximal sampling algorithm achieves better complexity than all existing methods.

翻译：我们研究了与非凸且同时缺乏光滑性的势能相关的采样问题。特别地，我们考虑满足对数-索博列夫不等式或庞加莱不等式的目标分布。假设势能函数并非光滑，而是半光滑或由多个半光滑函数求和而成。针对这一具有挑战性的采样任务，我们开发了一种类似于优化中邻近算法的采样算法。该算法基于吉布斯采样的特殊形式——交替采样框架（ASF）。本文的关键贡献在于，在非凸与半光滑设定下，基于拒绝采样实现了ASF的实用化方案。本研究将近期针对非光滑/半光滑对数凹分布提出的算法（见文献\cite{LiaChe21,LiaChe22}）扩展至非凸势能的情形。在本文考察的几乎所有采样实例中，我们的邻近采样算法均实现了优于现有方法的复杂度。