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

2026/07/31 by Tobias Wolf, Jalal Fadili, Jin Guo +1
Mathematics · Computer Science · #math.OC #cs.IT #math.IT

paper · pdf

arxiv created 2026/07/31 · arxiv updated 2026/08/03

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.

Citations

• Nonasymptotic Convergence Rates for Plug-and-Play Methods With MMSE Denoisers
• MAP Estimation with Denoisers: Convergence Rates and Guarantees
• Implicit Regularization of the Deep Inverse Prior Trained with Inertia
• Guided Diffusion Sampling on Function Spaces with Applications to PDEs
• Stein's unbiased risk estimate and Hyvärinen's score matching
• Learning Firmly Nonexpansive Operators
• Consistent Diffusion Meets Tweedie: Training Exact Ambient Diffusion Models with Noisy Data
• Recovery Guarantees of Unsupervised Neural Networks for Inverse Problems trained with Gradient Descent
• Weakly Convex Regularisers for Inverse Problems: Convergence of Critical Points and Primal-Dual Optimisation
• Tweedie Moment Projected Diffusions For Inverse Problems
• Convergence and Recovery Guarantees of Unsupervised Neural Networks for Inverse Problems
• Convergent regularization in inverse problems and linear plug-and-play denoisers
• Diffusion with Forward Models: Solving Stochastic Inverse Problems Without Direct Supervision
• Convergence Guarantees of Overparametrized Wide Deep Inverse Prior
• Plug-and-Play image reconstruction is a convergent regularization method
• Diffusion Posterior Sampling for General Noisy Inverse Problems
• Averaged Deep Denoisers for Image Regularization
• Learned reconstruction methods with convergence guarantees
• Deep neural networks can stably solve high-dimensional, noisy, non-linear inverse problems
• Improving Diffusion Models for Inverse Problems using Manifold Constraints
• Proximal Denoiser for Convergent Plug-and-Play Optimization with Nonconvex Regularization
• Solving Inverse Problems in Medical Imaging with Score-Based Generative Models
• Gradient Step Denoiser for convergent Plug-and-Play
• Regularising Inverse Problems with Generative Machine Learning Models
• End-to-end reconstruction meets data-driven regularization for inverse problems
• Recovery Analysis for Plug-and-Play Priors using the Restricted Eigenvalue Condition
• A Generative Variational Model for Inverse Problems in Imaging
• Learning local regularization for variational image restoration
• Plug-and-Play Image Restoration with Deep Denoiser Prior
• Denoising Diffusion Probabilistic Models
• Scalable Plug-and-Play ADMM with Convergence Guarantees
• A General Derivative Identity for the Conditional Mean Estimator in\n Gaussian Noise and Some Applications
• Deep Learning Techniques for Inverse Problems in Imaging
• Total Deep Variation for Linear Inverse Problems
• Algorithm Unrolling: Interpretable, Efficient Deep Learning for Signal\n and Image Processing
• Algorithm Unrolling: Interpretable, Efficient Deep Learning for Signal and Image Processing
• Unsupervised Deep Basis Pursuit: Learning inverse problems without ground-truth data
• Generative Modeling by Estimating Gradients of the Data Distribution
• Deep null space learning for inverse problems: convergence analysis and rates
• Adversarial Regularizers in Inverse Problems
• Lipschitz regularity of deep neural networks: analysis and efficient estimation
• NETT: solving inverse problems with deep neural networks
• Modern Regularization Methods for Inverse Problems
• Deep Image Prior
• Optimal approximation of piecewise smooth functions using deep ReLU\n neural networks
• Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance
• Solving ill-posed inverse problems using iterative deep neural networks
• Deep Convolutional Neural Network for Inverse Problems in Imaging
• The Little Engine that Could: Regularization by Denoising (RED)
• Low Complexity Regularization of Linear Inverse Problems
• A Connection Between Score Matching and Denoising Autoencoders
• On the mean speed of convergence of empirical and occupation measures in Wasserstein distance
• Estimation in Gaussian Noise: Properties of the Minimum Mean-Square Error
• Estimation in Gaussian Noise: Properties of the Minimum Mean-Square Error
• Mutual Information and Minimum Mean-square Error in Gaussian Channels
• Nonlinear total variation based noise removal algorithms
• Fractal Geometry: Mathematical Foundations and Applications.
• Gradient of mutual information in linear vector Gaussian channels