A relaxed proximal gradient descent algorithm for convergent plug-and-play with proximal denoiser

TL;DR

This paper introduces a relaxed proximal gradient descent algorithm for Plug-and-Play methods, enabling convergence with a broader range of parameters and improving image restoration accuracy.

Contribution

It proposes a new relaxed PGD algorithm for PnP that guarantees convergence under less restrictive conditions on the denoiser and parameters.

Findings

Converges with a wider range of regularization parameters.

Allows more accurate image restoration.

Ensures convergence with a relaxed denoiser.

Abstract

This paper presents a new convergent Plug-and-Play (PnP) algorithm. PnP methods are efficient iterative algorithms for solving image inverse problems formulated as the minimization of the sum of a data-fidelity term and a regularization term. PnP methods perform regularization by plugging a pre-trained denoiser in a proximal algorithm, such as Proximal Gradient Descent (PGD). To ensure convergence of PnP schemes, many works study specific parametrizations of deep denoisers. However, existing results require either unverifiable or suboptimal hypotheses on the denoiser, or assume restrictive conditions on the parameters of the inverse problem. Observing that these limitations can be due to the proximal algorithm in use, we study a relaxed version of the PGD algorithm for minimizing the sum of a convex function and a weakly convex one. When plugged with a relaxed proximal denoiser, we show that the proposed PnP-$\alpha$PGD algorithm converges for a wider range of regularization parameters, thus allowing more accurate image restoration.