Diffusion at Absolute Zero: Langevin Sampling Using Successive Moreau Envelopes

In this article we propose a novel method for sampling from Gibbs distributions of the form π(x) ∝ exp(-U(x)) with a potential U(x). In particular, inspired by diffusion models we propose to consider a sequence ${\left( {{\pi ^{{t_k}}}} \right)_k}$ 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 π.