Linear Convergence of Plug-and-Play Algorithms with Kernel Denoisers

TL;DR

This paper proves global linear convergence of Plug-and-Play algorithms using kernel denoisers, including symmetric and nonsymmetric cases, with quantitative bounds and numerical validation for various image reconstruction tasks.

Contribution

It develops a unified framework for convergence analysis of PnP algorithms with kernel denoisers, extending prior work to nonsymmetric kernels and providing explicit convergence rate bounds.

Findings

Established global linear convergence for symmetric and nonsymmetric kernel denoisers.

Derived quantitative bounds on the convergence rate for specific inverse problems.

Validated theoretical results with numerical experiments.

Abstract

The use of denoisers for image reconstruction has shown significant potential, especially for the Plug-and-Play (PnP) framework. In PnP, a powerful denoiser is used as an implicit regularizer in proximal algorithms such as ISTA and ADMM. The focus of this work is on the convergence of PnP iterates for linear inverse problems using kernel denoisers. It was shown in prior work that the update operator in standard PnP is contractive for symmetric kernel denoisers under appropriate conditions on the denoiser and the linear forward operator. Consequently, we could establish global linear convergence of the iterates using the contraction mapping theorem. In this work, we develop a unified framework to establish global linear convergence for symmetric and nonsymmetric kernel denoisers. Additionally, we derive quantitative bounds on the contraction factor (convergence rate) for inpainting, deblurring, and superresolution. We present numerical results to validate our theoretical findings.