Learning truly monotone operators with applications to nonlinear inverse problems

2024-03-30

arXiv.org

AI Summary

This work addresses nonlinear inverse problems in image processing by proposing the first end-to-end monotone neural operator learning framework with provable convergence. Instead of relying on pre-specified Lipschitz constants or plug-and-play architectures, the method formulates the inverse problem as a monotone variational inclusion and introduces a monotonicity-constrained loss function. Training and inference are performed via the Forward-Backward-Forward (FBF) algorithm—a fixed-point iteration that requires no Lipschitz continuity prior. Theoretically, the framework guarantees strict convergence of the iterative scheme under mild assumptions. Experimentally, it achieves significant improvements in reconstruction accuracy and stability on deblurring and super-resolution tasks. Notably, this is the first approach to unify provably monotone approximation with end-to-end joint optimization, bridging theoretical guarantees with practical deep learning performance.

Technology Category

• Computer Vision: Learning & Optimization for CV
• Search and Optimization: Non-convex Optimization
• Machine Learning: Learning with Manifolds

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

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.

Problem

• Learning monotone neural networks for nonlinear inverse problems
• Solving variational inclusion problems in image processing
• Using FBF algorithm for convergence with unknown Lipschitz constant

Innovation

• Penalization loss for monotone neural networks
• Forward-Backward-Forward algorithm for convergence
• Plug-and-play methods for nonlinear inverse problems