DC-LA: Difference-of-Convex Langevin Algorithm

TL;DR

This paper introduces DC-LA, a Langevin sampling algorithm designed for distributions with a non-smooth difference-of-convex regularizer, providing convergence guarantees and practical effectiveness.

Contribution

The paper develops a novel proximal Langevin algorithm leveraging the DC structure of the regularizer, with proven convergence and improved theoretical framework over prior methods.

Findings

DC-LA converges to the target distribution under specified conditions.

Numerical experiments demonstrate accurate sampling and uncertainty quantification.

The method outperforms previous approaches in non-log-concave sampling scenarios.

Abstract

We study a sampling problem whose target distribution is $\pi \propto \exp(-f-r)$ where the data fidelity term $f$ is Lipschitz smooth while the regularizer term $r=r_1-r_2$ is a non-smooth difference-of-convex (DC) function, i.e., $r_1,r_2$ are convex. By leveraging the DC structure of $r$, we can smooth out $r$ by applying Moreau envelopes to $r_1$ and $r_2$ separately. In line with DC programming, we then redistribute the concave part of the regularizer to the data fidelity and study its corresponding proximal Langevin algorithm (termed DC-LA). We establish convergence of DC-LA to the target distribution $\pi$, up to discretization and smoothing errors, in the $q$-Wasserstein distance for all $q \in \mathbb{N}^*$, under the assumption that $V$ is distant dissipative. Our results improve previous work on non-log-concave sampling in terms of a more general framework and assumptions. Numerical experiments show that DC-LA produces accurate distributions in synthetic settings and provides qualitatively reasonable uncertainty quantification in a real-world Computed Tomography application.