Right Bregman proximal gradient with application to Poisson inverse problems *

Abstract: We introduce a novel Bregman proximal algorithm for convex composite optimization by applying the standard Bregman proximal gradient (BPG) method to a mirror-coordinate reparameterization of the objective. In primal variables, the resulting algorithm alternates between a preconditioned gradient step followed by a right Bregman proximal update. Our analysis relies on relative smoothness holding in the mirror coordinates instead of primal coordinates. Under this condition, we prove monotonic decrease of the objective in the general convex setting. We then specialize the method to Poisson inverse problems using weighted negative entropy as the potential. The resulting scheme recovers the classical Richardson-Lucy multiplicative updates and extends them to general convex regularizers. Building on a recent convergence analysis of multiplicative updates, we establish a sublinear convergence rate in function values for the Poisson setting. Finally, we demonstrate the performance of the regularized algorithm on several imaging inverse problems with Poisson-distributed observations. The code is publicly available at https://github.com/Tmodrzyk/MU-Bregman.