# Convolutional Proximal Neural Networks and Plug-and-Play Algorithms

arXiv: https://arxiv.org/abs/2011.02281 · HTML: https://mlsanity.com/paper/2011.02281/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Nov 2020.

## Built on by (most cited first, 30 of 36)

- [Gradient Step Denoiser for convergent Plug-and-Play](https://mlsanity.com/paper/2110.03220/) (2021)
- [PnP-Flow: Plug-and-Play Image Restoration with Flow Matching](https://mlsanity.com/paper/2410.02423/) (2024)
- [Image Denoising: The Deep Learning Revolution and Beyond -- A Survey Paper --](https://mlsanity.com/paper/2301.03362/) (2023)
- [A Neural-Network-Based Convex Regularizer for Inverse Problems](https://mlsanity.com/paper/2211.12461/) (2022)
- [Improving Lipschitz-Constrained Neural Networks by Learning Activation Functions](https://mlsanity.com/paper/2210.16222/) (2022)
- [Generalized Normalizing Flows via Markov Chains](https://mlsanity.com/paper/2111.12506/) (2021)
- [PatchNR: Learning from Very Few Images by Patch Normalizing Flow Regularization](https://mlsanity.com/paper/2205.12021/) (2022)
- [Approximation of Lipschitz Functions using Deep Spline Neural Networks](https://mlsanity.com/paper/2204.06233/) (2022)
- [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)
- [Wasserstein Patch Prior for Image Superresolution](https://mlsanity.com/paper/2109.12880/) (2021)
- [A relaxed proximal gradient descent algorithm for convergent plug-and-play with proximal denoiser](https://mlsanity.com/paper/2301.13731/) (2023)
- [Generalization Error Bounds for Iterative Recovery Algorithms Unfolded as Neural Networks](https://mlsanity.com/paper/2112.04364/) (2021)
- [Prior Mismatch and Adaptation in PnP-ADMM with a Nonconvex Convergence Analysis](https://mlsanity.com/paper/2310.00133/) (2023)
- [WPPNets and WPPFlows: The Power of Wasserstein Patch Priors for Superresolution](https://mlsanity.com/paper/2201.08157/) (2022)
- [Dynamical systems' based neural networks](https://mlsanity.com/paper/2210.02373/) (2022)
- [Designing Stable Neural Networks using Convex Analysis and ODEs](https://mlsanity.com/paper/2306.17332/) (2023)
- [Bayesian Inversion for Nonlinear Imaging Models using Deep Generative Priors](https://mlsanity.com/paper/2203.10078/) (2022)
- [An Adaptive Orthogonal Convolution Scheme for Efficient and Flexible CNN Architectures](https://mlsanity.com/paper/2501.07930/) (2025)
- [Approximation theory for 1-Lipschitz ResNets](https://mlsanity.com/paper/2505.12003/) (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)
- [Stability of Data-Dependent Ridge-Regularization for Inverse Problems](https://mlsanity.com/paper/2406.12289/) (2024)
- [MAP Estimation with Denoisers: Convergence Rates and Guarantees](https://mlsanity.com/paper/2507.15397/) (2025)
- [NF-ULA: Langevin Monte Carlo with Normalizing Flow Prior for Imaging Inverse Problems](https://mlsanity.com/paper/2304.08342/) (2023)
- [How Much Training Data is Memorized in Overparameterized Autoencoders? An Inverse Problem Perspective on Memorization Evaluation](https://mlsanity.com/paper/2310.02897/) (2023)
- [Parseval Convolution Operators and Neural Networks](https://mlsanity.com/paper/2408.09981/) (2024)
- [Invertible ResNets for Inverse Imaging Problems: Competitive Performance with Provable Regularization Properties](https://mlsanity.com/paper/2409.13482/) (2024)
- [Reconstruct Anything Model: a lightweight general model for computational imaging](https://mlsanity.com/paper/2503.08915/) (2025)

## Abstract (arXiv)

In this paper, we introduce convolutional proximal neural networks (cPNNs), which are by construction averaged operators. For filters of full length, we propose a stochastic gradient descent algorithm on a submanifold of the Stiefel manifold to train cPNNs. In case of filters with limited length, we design algorithms for minimizing functionals that approximate the orthogonality constraints imposed on the operators by penalizing the least squares distance to the identity operator. Then, we investigate how scaled cPNNs with a prescribed Lipschitz constant can be used for denoising signals and images, where the achieved quality depends on the Lipschitz constant. Finally, we apply cPNN based denoisers within a Plug-and-Play (PnP) framework and provide convergence results for the corresponding PnP forward-backward splitting algorithm based on an oracle construction.