Subgradient Langevin Methods for Sampling from Non-smooth Potentials

TL;DR

This paper introduces subgradient Langevin algorithms for sampling from complex distributions with non-smooth potentials, providing convergence guarantees and demonstrating practical effectiveness in high-dimensional Bayesian imaging tasks.

Contribution

It proposes two novel subgradient Langevin methods for non-smooth potentials, with convergence proofs and applicability to high-dimensional Bayesian imaging.

Findings

Convergence rates are established for both methods.

Numerical experiments confirm practical feasibility.

Methods perform well in high-dimensional Bayesian imaging.

Abstract

This paper is concerned with sampling from probability distributions $\pi$ on $\mathbb{R}^d$ admitting a density of the form $\pi(x) \propto e^{-U(x)}$, where $U(x)=F(x)+G(Kx)$ with $K$ being a linear operator and $G$ being non-differentiable. Two different methods are proposed, both employing a subgradient step with respect to $G\circ K$, but, depending on the regularity of $F$, either an explicit or an implicit gradient step with respect to $F$ can be implemented. For both methods, non-asymptotic convergence proofs are provided, with improved convergence results for more regular $F$. Further, numerical experiments are conducted for simple 2D examples, illustrating the convergence rates, and for examples of Bayesian imaging, showing the practical feasibility of the proposed methods for high dimensional data.