Nonasymptotic Convergence Rates for Plug-and-Play Methods With MMSE Denoisers

TL;DR

This paper establishes the first sublinear convergence rates for plug-and-play methods using MMSE denoisers, linking them to explicit weakly convex regularizers and validating with imaging experiments.

Contribution

It explicitly characterizes the regularizer induced by MMSE denoisers and derives the first known sublinear convergence guarantees for PnP proximal gradient methods.

Findings

Validated sublinear convergence in synthetic and imaging experiments.

Connected MMSE denoisers to explicit weakly convex regularizers.

Demonstrated the regularizer as an upper Moreau envelope of negative log-marginal density.

Abstract

It is known that the minimum-mean-squared-error (MMSE) denoiser under Gaussian noise can be written as a proximal operator, which suffices for asymptotic convergence of plug-and-play (PnP) methods but does not reveal the structure of the induced regularizer or give convergence rates. We show that the MMSE denoiser corresponds to a regularizer that can be written explicitly as an upper Moreau envelope of the negative log-marginal density, which in turn implies that the regularizer is 1-weakly convex. Using this property, we derive (to the best of our knowledge) the first sublinear convergence guarantee for PnP proximal gradient descent with an MMSE denoiser. We validate the theory with a one-dimensional synthetic study that recovers the implicit regularizer. We also validate the theory with imaging experiments (deblurring and computed tomography), which exhibit the predicted sublinear behavior.