Learning Firmly Nonexpansive Operators

TL;DR

The paper tackles the challenge of integrating learned denoisers into Plug-and-Play algorithms with theoretical guarantees by learning firmly nonexpansive operators. It develops a rigorous framework that combines Gamma-convergence of empirical risk to the expected risk with a constructive discretization using simplicial partitions to yield piecewise affine, 1-Lipschitz operators, and proves a density result ensuring the discretized class approximates the continuous one. A concrete ADMM-based learning procedure enforces local Lipschitz constraints while producing operators suitable for use in a convergent Chambolle--Pock PnP scheme via Moreau's identity. The authors demonstrate the approach in image denoising, showing competitive performance against classical regularizers and providing extensive analyses of the learned operators, including a conservativity test suggesting proximity-operator-like behavior. Overall, the work offers a principled path from data to stable, interpretable denoisers within variational splitting frameworks, with potential to extend to broader Lipschitz-constrained learning tasks and nonconvex regularizers.

Abstract

This paper proposes a data-driven approach for constructing firmly nonexpansive operators. We demonstrate its applicability in Plug-and-Play (PnP) methods, where classical algorithms such as Forward-Backward splitting, Chambolle-Pock primal-dual iteration, Douglas-Rachford iteration or alternating directions method of multipliers (ADMM), are modified by replacing one proximal map by a learned firmly nonexpansive operator. We provide sound mathematical background to the problem of learning such an operator via expected and empirical risk minimization. We prove that, as the number of training points increases, the empirical risk minimization problem converges (in the sense of Gamma-convergence) to the expected risk minimization problem. Further, we derive a solution strategy that ensures firmly nonexpansive and piecewise affine operators within the convex envelope of the training set. We show that this operator converges to the best empirical solution as the number of points in the envelope increases in an appropriate way. Finally, the experimental section details practical implementations of the method and presents an application in image denoising, where we consider a novel, interpretable PnP Chambolle-Pock primal-dual iteration.

Figures (8)

• Figure 1: Clean data images. In order to construct the data set, we make use of Gaussian noise with noise level $\eta = 10$.
• Figure 2: Clean version of the test images for experiments 1 and 2, respectively.
• Figure 3: Butterfly images. Results of the experiment performed using a noisy image with Gaussian noise with noise level $\eta=30$.
• Figure 4: Circles. Obtained results when using a noisy image with Gaussian noise, with noise level $\eta=30$. We compare the performance of our method with classical regularization methods, and also when considering different training sets.
• Figure 5: On the left: Lipschitz constants for the learned operator (Learned 1) in every element of the triangulation. On the right: convergence behavior of the primal and dual residuals for the PnP Chambolle-Pock method described in \ref{['eq:PnP_CP']}, employing the operator learned from butterfly images (Learned 1).

Theorems & Definitions (40)

• Lemma 2.1
• Proposition 2.2
• Proof 1
• Theorem 2.3
• Proof 2
• Theorem 2.4
• Proof 3
• Lemma 2.5
• Proof 4
• Corollary 2.6