On the Contractivity of Plug-and-Play Operators

TL;DR

This paper investigates the convergence properties of plug-and-play algorithms, demonstrating linear convergence under certain conditions for symmetric and kernel denoisers, and validating results through experiments.

Contribution

It extends convergence analysis of PnP algorithms to PnP-ADMM and relaxes assumptions on the forward model, providing theoretical guarantees.

Findings

PnP-ISTA and PnP-ADMM exhibit linear convergence with symmetric denoisers.

Convergence is proven for kernel denoisers in image inpainting.

Experimental results support the theoretical convergence claims.

Abstract

In plug-and-play (PnP) regularization, the proximal operator in algorithms such as ISTA and ADMM is replaced by a powerful denoiser. This formal substitution works surprisingly well in practice. In fact, PnP has been shown to give state-of-the-art results for various imaging applications. The empirical success of PnP has motivated researchers to understand its theoretical underpinnings and, in particular, its convergence. It was shown in prior work that for kernel denoisers such as the nonlocal means, PnP-ISTA provably converges under some strong assumptions on the forward model. The present work is motivated by the following questions: Can we relax the assumptions on the forward model? Can the convergence analysis be extended to PnP-ADMM? Can we estimate the convergence rate? In this letter, we resolve these questions using the contraction mapping theorem: (i) for symmetric denoisers, we show that (under mild conditions) PnP-ISTA and PnP-ADMM exhibit linear convergence; and (ii) for kernel denoisers, we show that PnP-ISTA and PnP-ADMM converge linearly for image inpainting. We validate our theoretical findings using reconstruction experiments.