# Learning Maximally Monotone Operators for Image Recovery

arXiv: https://arxiv.org/abs/2012.13247 · HTML: https://mlsanity.com/paper/2012.13247/

## Key facts

- Cited by 62 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Dec 2020.

## Built on by (most cited first, 30 of 62)

- [Gradient Step Denoiser for convergent Plug-and-Play](https://mlsanity.com/paper/2110.03220/) (2021)
- [Bayesian imaging using Plug & Play priors: when Langevin meets Tweedie](https://mlsanity.com/paper/2103.04715/) (2021)
- [PnP-Flow: Plug-and-Play Image Restoration with Flow Matching](https://mlsanity.com/paper/2410.02423/) (2024)
- [Proximal Denoiser for Convergent Plug-and-Play Optimization with Nonconvex Regularization](https://mlsanity.com/paper/2201.13256/) (2022)
- [Learned reconstruction methods with convergence guarantees](https://mlsanity.com/paper/2206.05431/) (2022)
- [A Neural-Network-Based Convex Regularizer for Inverse Problems](https://mlsanity.com/paper/2211.12461/) (2022)
- [End-to-end reconstruction meets data-driven regularization for inverse problems](https://mlsanity.com/paper/2106.03538/) (2021)
- [Equivariant plug-and-play image reconstruction](https://mlsanity.com/paper/2312.01831/) (2023)
- [Solving Inverse Problems by Joint Posterior Maximization with Autoencoding Prior](https://mlsanity.com/paper/2103.01648/) (2021)
- [Generalized Normalizing Flows via Markov Chains](https://mlsanity.com/paper/2111.12506/) (2021)
- [WaveFit: An Iterative and Non-autoregressive Neural Vocoder based on Fixed-Point Iteration](https://mlsanity.com/paper/2210.01029/) (2022)
- [Plug-and-Play image restoration with Stochastic deNOising REgularization](https://mlsanity.com/paper/2402.01779/) (2024)
- [Provably Convergent Plug-and-Play Quasi-Newton Methods](https://mlsanity.com/paper/2303.07271/) (2023)
- [Learning Regularization Functionals for Inverse Problems: A Comparative Study](https://mlsanity.com/paper/2510.01755/) (2025)
- [On Maximum-a-Posteriori estimation with Plug & Play priors and stochastic gradient descent](https://mlsanity.com/paper/2201.06133/) (2022)
- [A relaxed proximal gradient descent algorithm for convergent plug-and-play with proximal denoiser](https://mlsanity.com/paper/2301.13731/) (2023)
- [Designing Stable Neural Networks using Convex Analysis and ODEs](https://mlsanity.com/paper/2306.17332/) (2023)
- [Memory-efficient model-based deep learning with convergence and robustness guarantees](https://mlsanity.com/paper/2206.04797/) (2022)
- [Unfolded proximal neural networks for robust image Gaussian denoising](https://mlsanity.com/paper/2308.03139/) (2023)
- [Invertible Monotone Operators for Normalizing Flows](https://mlsanity.com/paper/2210.08176/) (2022)
- [Inverse problem regularization with hierarchical variational autoencoders](https://mlsanity.com/paper/2303.11217/) (2023)
- [Model-corrected learned primal-dual models for fast limited-view photoacoustic tomography](https://mlsanity.com/paper/2304.01963/) (2023)
- [Learning pseudo-contractive denoisers for inverse problems](https://mlsanity.com/paper/2402.05637/) (2024)
- [FiRe: Fixed-points of Restoration Priors for Solving Inverse Problems](https://mlsanity.com/paper/2411.18970/) (2024)
- [Convexity in ReLU Neural Networks: beyond ICNNs?](https://mlsanity.com/paper/2501.03017/) (2025)
- [DEALing with Image Reconstruction: Deep Attentive Least Squares](https://mlsanity.com/paper/2502.04079/) (2025)
- [Proximal Residual Flows for Bayesian Inverse Problems](https://mlsanity.com/paper/2211.17158/) (2022)
- [Convergent plug-and-play with proximal denoiser and unconstrained regularization parameter](https://mlsanity.com/paper/2311.01216/) (2023)
- [Unrolled Variational Bayesian Algorithm for Image Blind Deconvolution](https://mlsanity.com/paper/2110.07202/) (2021)
- [Deep network series for large-scale high-dynamic range imaging](https://mlsanity.com/paper/2210.16060/) (2022)

## Abstract (arXiv)

We introduce a new paradigm for solving regularized variational problems. These are typically formulated to address ill-posed inverse problems encountered in signal and image processing. The objective function is traditionally defined by adding a regularization function to a data fit term, which is subsequently minimized by using iterative optimization algorithms. Recently, several works have proposed to replace the operator related to the regularization by a more sophisticated denoiser. These approaches, known as plug-and-play (PnP) methods, have shown excellent performance. Although it has been noticed that, under some Lipschitz properties on the denoisers, the convergence of the resulting algorithm is guaranteed, little is known about characterizing the asymptotically delivered solution. In the current article, we propose to address this limitation. More specifically, instead of employing a functional regularization, we perform an operator regularization, where a maximally monotone operator (MMO) is learned in a supervised manner. This formulation is flexible as it allows the solution to be characterized through a broad range of variational inequalities, and it includes convex regularizations as special cases. From an algorithmic standpoint, the proposed approach consists in replacing the resolvent of the MMO by a neural network (NN). We present a universal approximation theorem proving that nonexpansive NNs are suitable models for the resolvent of a wide class of MMOs. The proposed approach thus provides a sound theoretical framework for analyzing the asymptotic behavior of first-order PnP algorithms. In addition, we propose a numerical strategy to train NNs corresponding to resolvents of MMOs. We apply our approach to image restoration problems and demonstrate its validity in terms of both convergence and quality.