Diffusion at Absolute Zero: Langevin Sampling Using Successive Moreau Envelopes

A. Habring, A. Falk, M. Zach, T. Pock

SIAM Journal on Imaging Sciences, vol. 19, no. 1, pp. 35–77, 2026.

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We propose a method for sampling from Gibbs distributions of the form π(x) ∝ exp(-U(x)) that leverages a family (π^t)_t of approximations of the target density which is deliberately constructed such that π^t exhibits favorable properties for sampling when t is large and such that π^t approaches π as t approaches 0. This sequence is obtained by replacing (parts of) the potential U with its Moreau envelope. Through the sequential sampling from π^t for decreasing values of t by a Langevin algorithm with appropriate step size, the samples are guided from a simple starting density to the more complex target quickly. We prove that t ↦ π^t is Lipschitz continuous in the total variation distance and Hölder continuous in the Wasserstein-p distance, that the sampling algorithm is ergodic, and that it converges to the target density without assuming convexity or differentiability of the potential U. In addition to the theoretical analysis, we show experimental results that support the superiority of the method in terms of convergence speed and mode-coverage of multimodal densities over current algorithms. The experiments range from one-dimensional toy-problems to high-dimensional inverse imaging problems with learned potentials.

@ARTICLE(http://bigwww.epfl.ch/publications/habring2601.html,
AUTHOR="Habring, A. and Falk, A. and Zach, M. and Pock, T.",
TITLE="Diffusion at Absolute Zero: {L}angevin Sampling Using Successive
	{M}oreau Envelopes",
JOURNAL="{SIAM} Journal on Imaging Sciences",
YEAR="2026",
volume="19",
number="1",
pages="35--77",
month="",
note="")

DOI: 10.1137/25M1745830