Learning Firmly Nonexpansive Operators

TL;DR

This paper introduces a data-driven method for learning firmly nonexpansive operators, enhancing Plug-and-Play algorithms with theoretical guarantees and practical applications in image denoising.

Contribution

It develops a mathematical framework for learning firmly nonexpansive operators via risk minimization, ensuring convergence and interpretability within PnP methods.

Findings

Convergence of empirical to expected risk minimization via Gamma-convergence.

Construction of piecewise affine, firmly nonexpansive operators within convex envelopes.

Successful application to image denoising with a novel PnP Chambolle-Pock method.

Abstract

This paper proposes a data-driven approach for constructing firmly nonexpansive operators. We demonstrate its applicability in Plug-and-Play (PnP) methods, where classical algorithms such as Forward-Backward splitting, Chambolle-Pock primal-dual iteration, Douglas-Rachford iteration or alternating directions method of multipliers (ADMM), are modified by replacing one proximal map by a learned firmly nonexpansive operator. We provide sound mathematical background to the problem of learning such an operator via expected and empirical risk minimization. We prove that, as the number of training points increases, the empirical risk minimization problem converges (in the sense of Gamma-convergence) to the expected risk minimization problem. Further, we derive a solution strategy that ensures firmly nonexpansive and piecewise affine operators within the convex envelope of the training set. We show that this operator converges to the best empirical solution as the number of points in the envelope increases in an appropriate way. Finally, the experimental section details practical implementations of the method and presents an application in image denoising, where we consider a novel, interpretable PnP Chambolle-Pock primal-dual iteration.