In this article we propose a novel method for sampling from Gibbs distributions of the form $\pi(x)\propto\exp(-U(x))$ with a potential $U(x)$. In particular, inspired by diffusion models we propose to consider a sequence $(\pi^{t_k})_k$ of approximations of the target density, for which $\pi^{t_k}\approx \pi$ for $k$ small and, on the other hand, $\pi^{t_k}$ exhibits favorable properties for sampling for $k$ large. This sequence is obtained by replacing parts of the potential $U$ by its Moreau envelopes. Sampling is performed in an Annealed Langevin type procedure, that is, sequentially sampling from $\pi^{t_k}$ for decreasing $k$, effectively guiding the samples from a simple starting density to the more complex target. In addition to a theoretical analysis we show experimental results supporting the efficacy of the method in terms of increased convergence speed and applicability to multi-modal densities $\pi$.

翻译：本文提出了一种从吉布斯分布 $\pi(x)\propto\exp(-U(x))$（其中 $U(x)$ 为势函数）中采样的新方法。具体而言，受扩散模型启发，我们提出考虑目标密度的一系列近似 $(\pi^{t_k})_k$，使得当 $k$ 较小时 $\pi^{t_k}\approx \pi$，而当 $k$ 较大时 $\pi^{t_k}$ 展现出更利于采样的特性。该序列是通过将势函数 $U$ 的部分替换为其莫罗包络而获得的。采样过程采用退火朗之万类型算法执行，即按 $k$ 递减的顺序依次从 $\pi^{t_k}$ 中采样，从而有效地将样本从简单的初始密度引导至更复杂的目标分布。除理论分析外，实验结果表明该方法在提升收敛速度及适用于多模态密度 $\pi$ 方面具有显著效能。