DC-LA: Difference-of-Convex Langevin Algorithm

We study a sampling problem whose target distribution is π propto exp(-f-r) where the data fidelity term f is Lipschitz smooth while the regularizer term r=r1-r2 is a non-smooth difference-of-convex (DC) function, i.e., r1,r2 are convex. By leveraging the DC structure of r, we can smooth out r by applying Moreau envelopes to r1 and r2 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 π, up to discretization and smoothing errors, in the q-Wasserstein distance for all q ∈ mathbbN^*, 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.