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Learning Truly Monotone Operators with Applications to Nonlinear Inverse Problems

Year: 2025

Type: article

Abstract: . This article introduces a novel approach to learning monotone neural networks (NNs) through a newly defined penalization loss. The proposed method is particularly effective in solving classes of variational problems, specifically monotone inclusion problems, commonly encountered in image processing tasks. The forward-backward-forward (FBF) algorithm is employed to address these problems, offering a solution even when the Lipschitz constant of the NN is unknown. Notably, the FBF algorithm provides convergence guarantees under the condition that the learned operator is monotone. Building on plug-and-play methodologies, our objective is to apply these newly learned operators to solving nonlinear inverse problems. To achieve this, we initially formulate the problem as a variational inclusion problem. Subsequently, we train a monotone NN to approximate an operator that may not inherently be monotone. Leveraging the FBF algorithm, we then show simulation examples where the nonlinear inverse problem is successfully solved.

Source: SIAM Journal on Imaging Sciences

DOI: 10.1137/24m1643347

Authors: Younes Belkouchi, Jean-Christophe Pesquet, Audrey Repetti, Hugues Talbot