mlsanityData to 7 Oct 2026

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

arXiv 2510.27211 ↗First posted 31 Oct 2025math.OC eess.SP stat.MLcited by 2

Not on the map. Why?

Abstract (arXiv)

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.

Key facts. Cited by 2 papers in the map (data through 7 Oct 2026). Builds on 37 of its 74 references that are in the map. First posted on arXiv 31 Oct 2025.

Builds on

37 of 74 references.

References · 37

  1. 2016Beyond a Gaussian Denoiser: Residual Learning of Deep CNN for Image Denoisingcited by 1,352
  2. 2014On the Number of Linear Regions of Deep Neural Networkscited by 378
  3. 2017Compressed Sensing using Generative Modelscited by 337
  4. 2016On the Expressive Power of Deep Neural Networkscited by 302
  5. 2020Deep Learning Techniques for Inverse Problems in Imagingcited by 290
  6. 2016Plug-and-Play ADMM for Image Restoration: Fixed Point Convergence and Applicationscited by 244
  7. 2018Sorting out Lipschitz function approximationcited by 211
  8. 2019Plug-and-Play Methods Provably Converge with Properly Trained Denoiserscited by 197
  9. 2021Gradient Step Denoiser for convergent Plug-and-Playcited by 107
  10. 2022Plug-and-Play Methods for Integrating Physical and Learned Models in Computational Imagingcited by 94
  11. 2019Preventing Gradient Attenuation in Lipschitz Constrained Convolutional Networkscited by 86
  12. 2022Proximal Denoiser for Convergent Plug-and-Play Optimization with Nonconvex Regularizationcited by 79
25 more
  1. 2019Plug and play methods for magnetic resonance imaging (long version)cited by 75
  2. 2018An Online Plug-and-Play Algorithm for Regularized Image Reconstructioncited by 70
  3. 2018Image Restoration using Total Variation Regularized Deep Image Priorcited by 65
  4. 2020Learning Maximally Monotone Operators for Image Recoverycited by 62
  5. 2022A Neural-Network-Based Convex Regularizer for Inverse Problemscited by 40
  6. 2020Provable Convergence of Plug-and-Play Priors with MMSE denoiserscited by 39
  7. 2024Denoising: A Powerful Building-Block for Imaging, Inverse Problems, and Machine Learningcited by 35
  8. 2018A characterization of proximity operatorscited by 32
  9. 2016Image Restoration Using Convolutional Auto-encoders with Symmetric Skip Connectionscited by 30
  10. 2023Learning Weakly Convex Regularizers for Convergent Image-Reconstruction Algorithmscited by 30
  11. 2023What's in a Prior? Learned Proximal Networks for Inverse Problemscited by 24
  12. 2023Deep Learning Meets Sparse Regularization: A Signal Processing Perspectivecited by 21
  13. 2021Fixed-Point and Objective Convergence of Plug-and-Play Algorithmscited by 19
  14. 2020Variable Smoothing for Weakly Convex Composite Functionscited by 12
  15. 2020Plug-and-play ISTA converges with kernel denoiserscited by 12
  16. 2024Weakly Convex Regularisers for Inverse Problems: Convergence of Critical Points and Primal-Dual Optimisationcited by 12
  17. 2025Learning Regularization Functionals for Inverse Problems: A Comparative Studycited by 12
  18. 2021On Plug-and-Play Regularization using Linear Denoiserscited by 9
  19. 2025DEALing with Image Reconstruction: Deep Attentive Least Squarescited by 6
  20. 2023Convergent plug-and-play with proximal denoiser and unconstrained regularization parametercited by 5
  21. 2023Convergence of Nonconvex PnP-ADMM with MMSE Denoiserscited by 3
  22. 2024Learning Firmly Nonexpansive Operatorscited by 3
  23. 2024Learning of Patch-Based Smooth-Plus-Sparse Models for Image Reconstructioncited by 3
  24. 2025Multivariate Fields of Experts for Convergent Image Reconstructioncited by 2
  25. 2023On the Contractivity of Plug-and-Play Operatorscited by 1

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2026: 2.

2 papers.

  1. 2026Lasso Universality Under Linearly Dependent Covariates in the Sparse Regimecited by 1
  2. 2026Plug-and-Play Methods Provably Converge Even with Improperly Trained Denoisers: Convergence by Architectural Designcited by 0