Bregman Plug-and-Play Priors

TL;DR

This paper introduces Bregman-based variants of plug-and-play and RED algorithms, extending their applicability beyond Euclidean norms to more general Bregman distances, with proven convergence and demonstrated effectiveness on Poisson inverse problems.

Contribution

It proposes a novel non-Euclidean framework for PnP and RED algorithms using Bregman distances, including new algorithms and theoretical convergence guarantees.

Findings

Convergence of PnP-BPGM is theoretically established.

Algorithms perform effectively on Poisson linear inverse problems.

Extension beyond Euclidean norms broadens applicability.

Abstract

The past few years have seen a surge of activity around integration of deep learning networks and optimization algorithms for solving inverse problems. Recent work on plug-and-play priors (PnP), regularization by denoising (RED), and deep unfolding has shown the state-of-the-art performance of such integration in a variety of applications. However, the current paradigm for designing such algorithms is inherently Euclidean, due to the usage of the quadratic norm within the projection and proximal operators. We propose to broaden this perspective by considering a non-Euclidean setting based on the more general Bregman distance. Our new Bregman Proximal Gradient Method variant of PnP (PnP-BPGM) and Bregman Steepest Descent variant of RED (RED-BSD) replace the traditional updates in PnP and RED from the quadratic norms to more general Bregman distance. We present a theoretical convergence result for PnP-BPGM and demonstrate the effectiveness of our algorithms on Poisson linear inverse problems.