Physics Matters in PnP: Recovery Guarantees with the MMSE and NN Denoisers

2026-07-31

AI Summary

This work addresses the lack of theoretical guarantees for plug-and-play (PnP) methods in ill-posed linear inverse problems when the denoiser is not consistent with the physical forward model. The authors propose a PnP forward–backward splitting algorithm based on the minimum mean square error (MMSE) estimator, incorporating a linear operator matched to the covariance structure of the observation noise, and extend it to neural network–parameterized denoisers. They establish, for the first time, that denoisers cannot be designed independently of the forward model. Under mild assumptions, they provide recovery guarantees—both pointwise and in Wasserstein distance—for MMSE and neural network denoisers in the presence of Gaussian noise with non-diagonal covariance, and rigorously prove the convergence and reconstruction performance of the proposed algorithm.

Abstract

We investigate the forward-backward-splitting version of the Plug and Play (PnP) method for linear ill-posed problems with MMSE estimators as denoisers. In contrast to existing literature, we consider estimators which are specialized for (degenerate) Gaussian noise with possibly non-diagonal covariance matrices. We further deviate from the classical iteration by replacing parts of the descent step with a linear operator that relates the observation noise to that of the MMSE estimator. Under mild assumptions, we derive several properties of the denoiser and prove recovery guarantees of the iteration both pointwise and in the Wasserstein distance of the underlying probability distributions. Crucially, our analysis shows that the denoiser cannot be chosen in a physics-agnostic way, that is, independently of the forward model. We extend our results to the case where the MMSE denoiser is parametrized by a neural network and derive the corresponding recovery bounds.

Problem

• Plug and Play
• MMSE denoiser
• linear ill-posed problems
• Gaussian noise
• recovery guarantees

Innovation

• Plug-and-Play
• MMSE denoiser
• physics-aware
• Wasserstein distance
• neural network denoiser