# Learning Firmly Nonexpansive Operators

arXiv: https://arxiv.org/abs/2407.14156 · HTML: https://mlsanity.com/paper/2407.14156/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Jul 2024.

## 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)
- [Analysis and Synthesis Denoisers for Forward-Backward Plug-and-Play Algorithms](https://mlsanity.com/paper/2411.13276/) (2024)
- [Provably Contractive and High-Quality Denoisers for Convergent Restoration](https://mlsanity.com/paper/2603.26168/) (2026)

## Abstract (arXiv)

This paper proposes a data-driven approach for constructing firmly nonexpansive operators. We demonstrate its applicability in Plug-and-Play (PnP) methods, where classical algorithms such as Forward-Backward splitting, Chambolle-Pock primal-dual iteration, Douglas-Rachford iteration or alternating directions method of multipliers (ADMM), are modified by replacing one proximal map by a learned firmly nonexpansive operator. We provide sound mathematical background to the problem of learning such an operator via expected and empirical risk minimization. We prove that, as the number of training points increases, the empirical risk minimization problem converges (in the sense of Gamma-convergence) to the expected risk minimization problem. Further, we derive a solution strategy that ensures firmly nonexpansive and piecewise affine operators within the convex envelope of the training set. We show that this operator converges to the best empirical solution as the number of points in the envelope increases in an appropriate way. Finally, the experimental section details practical implementations of the method and presents an application in image denoising, where we consider a novel, interpretable PnP Chambolle-Pock primal-dual iteration.