Stability Bounds for the Unfolded Forward-Backward Algorithm

TL;DR

This paper analyzes the robustness of a neural network architecture based on unrolled forward-backward algorithms for inverse problems, providing theoretical bounds and numerical illustrations of its stability against input perturbations.

Contribution

It introduces a novel robustness analysis for an unfolded forward-backward neural network, including bias perturbations, with derived Lipschitz bounds and numerical validation.

Findings

The network's robustness is characterized by Lipschitz bounds.

Bias perturbations significantly affect the network's stability.

Numerical results confirm the theoretical bounds.

Abstract

We consider a neural network architecture designed to solve inverse problems where the degradation operator is linear and known. This architecture is constructed by unrolling a forward-backward algorithm derived from the minimization of an objective function that combines a data-fidelity term, a Tikhonov-type regularization term, and a potentially nonsmooth convex penalty. The robustness of this inversion method to input perturbations is analyzed theoretically. Ensuring robustness complies with the principles of inverse problem theory, as it ensures both the continuity of the inversion method and the resilience to small noise - a critical property given the known vulnerability of deep neural networks to adversarial perturbations. A key novelty of our work lies in examining the robustness of the proposed network to perturbations in its bias, which represents the observed data in the inverse problem. Additionally, we provide numerical illustrations of the analytical Lipschitz bounds derived in our analysis.