Recovery Analysis for Plug-and-Play Priors using the Restricted Eigenvalue Condition

TL;DR

This paper provides the first theoretical recovery guarantees for plug-and-play priors and RED methods in inverse problems, assuming solutions are near neural network fixed points, and demonstrates superior empirical performance in compressive sensing.

Contribution

It establishes the first theoretical recovery analysis for PnP/RED methods under the restricted eigenvalue condition, linking fixed-point solutions to recovery guarantees.

Findings

PnP/RED achieve better recovery than state-of-the-art methods in compressive sensing.

Numerical results support the effectiveness of pre-trained artifact removal networks.

Theoretical analysis connects fixed points of neural networks to recovery guarantees.

Abstract

The plug-and-play priors (PnP) and regularization by denoising (RED) methods have become widely used for solving inverse problems by leveraging pre-trained deep denoisers as image priors. While the empirical imaging performance and the theoretical convergence properties of these algorithms have been widely investigated, their recovery properties have not previously been theoretically analyzed. We address this gap by showing how to establish theoretical recovery guarantees for PnP/RED by assuming that the solution of these methods lies near the fixed-points of a deep neural network. We also present numerical results comparing the recovery performance of PnP/RED in compressive sensing against that of recent compressive sensing algorithms based on generative models. Our numerical results suggest that PnP with a pre-trained artifact removal network provides significantly better results compared to the existing state-of-the-art methods.