Subgradient Langevin Methods for Sampling from Non-smooth Potentials

This paper is concerned with sampling from probability distributions π on ℝd admitting a density of the form π(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 Gcirc 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.