Learning truly monotone operators with applications to nonlinear inverse problems

TL;DR

This work addresses nonlinear inverse problems where the forward operator is unknown by learning a monotone neural network to approximate a monotone operator within a plug-and-play Forward-Backward-Forward (FBF) framework. It introduces a differentiable penalization scheme that enforces monotonicity via the Jacobian, enabling convergence guarantees for the PnP-FBF iterations without requiring knowledge of the Lipschitz constant. The authors compare multiple learned-forward models, including a monotone and a least-squares variant, and demonstrate improved reconstruction performance on nonlinear deconvolution-like imaging problems while maintaining theoretical convergence. The approach broadens the applicability of monotone operator learning to nonlinear inverse problems and offers a scalable path toward stable, convergent deep learning-based solvers in imaging and related domains.

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.

Figures (8)

• Figure 1: Blurring kernels used to model linear operators $(L_k)_{1\leqslant k \leqslant 5}$, used in model \ref{['e:generalF']}.
• Figure 2: Example of an original image $\overline{x}$, the observation of this image through \ref{['e:generalF']} with $K=5$ and $\sigma=0$, and the linearized observation of $\overline{x}$ through \ref{['e:generalFlinear']}.
• Figure 3: Examples of output images obtained with the different versions of the measurement operator, with $\delta = 1$. First and third rows, left to right: true unknown operator $F$, true unknown linear approximation ${F}^{\rm lin}$, learned linear approximation $F_{\theta}^{\rm{lin}}$. Second and fourth rows, left to right: learned non-monotone approximation $F_{\theta}^{\rm{nom}}$, proposed learned monotone approximation $F_{\theta}^{\rm{mon}}$, and proposed relaxed monotone approximation $\widetilde{F}^{\rm{mon}}_{\theta}$. All results are shown when training models without noise (i.e., $\sigma_{\rm train}=0$).
• Figure 4: Examples of output images obtained with the different versions of the measurement operator, with $\delta = 0.6$. First and third rows, left to right: true unknown operator $F$, true unknown linear approximation ${F}^{\rm lin}$, learned linear approximation $F_{\theta}^{\rm{lin}}$. Second and fourth rows, left to right: learned non-monotone approximation $F_{\theta}^{\rm{nom}}$, proposed learned monotone approximation $F_{\theta}^{\rm{mon}}$, and proposed relaxed monotone approximation $\widetilde{F}^{\rm{mon}}_{\theta}$. All results are shown when training models without noise (i.e., $\sigma_{\rm train}=0$).
• Figure 5: Results for low noise level $\sigma=0.01$: Restoration results for $K=5$ and $\delta = 1$, for two images. For each image and method, we provide (PSNR, SSIM) values between the solution and $\overline{x}$. Last row shows the convergence profiles associated with the reconstruction of each image, for the three considered models.

Theorems & Definitions (14)

• Definition 2.3: Armijo-Goldstein rule
• Proposition 2.4
• Proposition 2.6
• proof
• Lemma 3.2
• Proposition 3.3
• proof
• Remark 3.4
• Remark 3.5
• Proposition 4.1