Learning truly monotone operators with applications to nonlinear inverse
  problems
Younes Belkouchi Jean-Christophe Pesquet Audrey Repetti Hugues Talbot
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.