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
(πtk)k of approximations of the target density, for which
πtk≈π for
k small and, on the other hand,
πtk
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
πtk 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.