# Convergence of Nonconvex PnP-ADMM with MMSE Denoisers

arXiv: https://arxiv.org/abs/2311.18810 · HTML: https://mlsanity.com/paper/2311.18810/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 24 references that are in the map.
- First posted on arXiv 30 Nov 2023.

## Builds on (most cited first, 14 of 14)

- [Plug-and-Play Image Restoration with Deep Denoiser Prior](https://mlsanity.com/paper/2008.13751/) (2020)
- [Learning Deep CNN Denoiser Prior for Image Restoration](https://mlsanity.com/paper/1704.03264/) (2017)
- [Plug-and-Play ADMM for Image Restoration: Fixed Point Convergence and Applications](https://mlsanity.com/paper/1605.01710/) (2016)
- [Plug-and-Play Methods Provably Converge with Properly Trained Denoisers](https://mlsanity.com/paper/1905.05406/) (2019)
- [Regularization by Denoising: Clarifications and New Interpretations](https://mlsanity.com/paper/1806.02296/) (2018)
- [Proximal Denoiser for Convergent Plug-and-Play Optimization with Nonconvex Regularization](https://mlsanity.com/paper/2201.13256/) (2022)
- [Plug-and-Play Priors for Bright Field Electron Tomography and Sparse Interpolation](https://mlsanity.com/paper/1512.07331/) (2015)
- [Plug and play methods for magnetic resonance imaging (long version)](https://mlsanity.com/paper/1903.08616/) (2019)
- [Scalable Plug-and-Play ADMM with Convergence Guarantees](https://mlsanity.com/paper/2006.03224/) (2020)
- [Provable Convergence of Plug-and-Play Priors with MMSE denoisers](https://mlsanity.com/paper/2005.07685/) (2020)
- [A Simple Proximal Stochastic Gradient Method for Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/1802.04477/) (2018)
- [Structured Nonconvex and Nonsmooth Optimization: Algorithms and Iteration Complexity Analysis](https://mlsanity.com/paper/1605.02408/) (2016)
- [Convergence of multi-block Bregman ADMM for nonconvex composite problems](https://mlsanity.com/paper/1505.03063/) (2015)
- [Multi-Block Nonconvex Nonsmooth Proximal ADMM: Convergence and Rates under Kurdyka-Lojasiewicz Property](https://mlsanity.com/paper/2009.04014/) (2020)

## Built on by (most cited first, 3 of 3)

- [Nonasymptotic Convergence Rates for Plug-and-Play Methods With MMSE Denoisers](https://mlsanity.com/paper/2510.27211/) (2025)
- [Beyond sparse denoising in frames: minimax estimation with a scattering transform](https://mlsanity.com/paper/2510.19612/) (2025)
- [Plug-and-Play Diffusion Meets ADMM: Dual-Variable Coupling for Robust Medical Image Reconstruction](https://mlsanity.com/paper/2602.23214/) (2026)

## Abstract (arXiv)

Plug-and-Play Alternating Direction Method of Multipliers (PnP-ADMM) is a widely-used algorithm for solving inverse problems by integrating physical measurement models and convolutional neural network (CNN) priors. PnP-ADMM has been theoretically proven to converge for convex data-fidelity terms and nonexpansive CNNs. It has however been observed that PnP-ADMM often empirically converges even for expansive CNNs. This paper presents a theoretical explanation for the observed stability of PnP-ADMM based on the interpretation of the CNN prior as a minimum mean-squared error (MMSE) denoiser. Our explanation parallels a similar argument recently made for the iterative shrinkage/thresholding algorithm variant of PnP (PnP-ISTA) and relies on the connection between MMSE denoisers and proximal operators. We also numerically evaluate the performance gap between PnP-ADMM using a nonexpansive DnCNN denoiser and expansive DRUNet denoiser, thus motivating the use of expansive CNNs.