Plug-and-Play Methods Provably Converge with Properly Trained Denoisers

TL;DR

This paper proves convergence of plug-and-play algorithms with properly trained denoisers, introducing spectral normalization to ensure theoretical guarantees and validating results through experiments.

Contribution

It provides the first theoretical convergence guarantees for PnP methods with deep denoisers by training them to satisfy a Lipschitz condition.

Findings

Convergence of PnP-FBS and PnP-ADMM established under Lipschitz condition.

Spectral normalization effectively trains denoisers to meet theoretical criteria.

Experimental results confirm the theoretical predictions.

Abstract

Plug-and-play (PnP) is a non-convex framework that integrates modern denoising priors, such as BM3D or deep learning-based denoisers, into ADMM or other proximal algorithms. An advantage of PnP is that one can use pre-trained denoisers when there is not sufficient data for end-to-end training. Although PnP has been recently studied extensively with great empirical success, theoretical analysis addressing even the most basic question of convergence has been insufficient. In this paper, we theoretically establish convergence of PnP-FBS and PnP-ADMM, without using diminishing stepsizes, under a certain Lipschitz condition on the denoisers. We then propose real spectral normalization, a technique for training deep learning-based denoisers to satisfy the proposed Lipschitz condition. Finally, we present experimental results validating the theory.