On the Strong Convexity of PnP Regularization Using Linear Denoisers

TL;DR

This paper proves that PnP algorithms using linear denoisers correspond to strongly convex optimization problems, ensuring convergence and stability in image reconstruction tasks.

Contribution

It establishes the strong convexity of the associated optimization problem for linear denoisers in PnP methods, linking denoising to convex optimization guarantees.

Findings

Strong convexity of the PnP regularization problem with linear denoisers.

Certification of convergence for PnP algorithms based on classical proximal methods.

Applicability to linear inverse problems in image reconstruction.

Abstract

In the Plug-and-Play (PnP) method, a denoiser is used as a regularizer within classical proximal algorithms for image reconstruction. It is known that a broad class of linear denoisers can be expressed as the proximal operator of a convex regularizer. Consequently, the associated PnP algorithm can be linked to a convex optimization problem $\mathcal{P}$. For such a linear denoiser, we prove that $\mathcal{P}$ exhibits strong convexity for linear inverse problems. Specifically, we show that the strong convexity of $\mathcal{P}$ can be used to certify objective and iterative convergence of any PnP algorithm derived from classical proximal methods.