Learning truly monotone operators with applications to nonlinear inverse problems
This work addresses nonlinear inverse problems in image processing by enabling the use of learned monotone operators, though it appears incremental as it builds on plug-and-play methodologies.
The authors tackled the problem of learning monotone neural networks for solving nonlinear inverse problems by introducing a penalization loss and employing the Forward-Backward-Forward algorithm, achieving successful simulation examples with convergence guarantees when the operator is monotone.
This article introduces a novel approach to learning monotone neural networks 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 neural network 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 non-linear inverse problems. To achieve this, we initially formulate the problem as a variational inclusion problem. Subsequently, we train a monotone neural network to approximate an operator that may not inherently be monotone. Leveraging the FBF algorithm, we then show simulation examples where the non-linear inverse problem is successfully solved.