Weakly Convex Regularisers for Inverse Problems: Convergence of Critical Points and Primal-Dual Optimisation

TL;DR

This paper introduces a framework for convergence analysis of weakly convex regularisers in inverse problems, establishing theoretical guarantees for primal-dual methods and demonstrating improved learned regularisation techniques in CT reconstruction.

Contribution

It generalizes regularisation convergence to critical points using weakly convex regularisers and proves convergence of primal-dual algorithms with applications to neural network regularisers.

Findings

Proves convergence of primal-dual methods for weakly convex regularisers.

Establishes universal approximation for input weakly convex neural networks.

Demonstrates improved CT reconstruction performance with learned adversarial regularisers.

Abstract

Variational regularisation is the primary method for solving inverse problems, and recently there has been considerable work leveraging deeply learned regularisation for enhanced performance. However, few results exist addressing the convergence of such regularisation, particularly within the context of critical points as opposed to global minimisers. In this paper, we present a generalised formulation of convergent regularisation in terms of critical points, and show that this is achieved by a class of weakly convex regularisers. We prove convergence of the primal-dual hybrid gradient method for the associated variational problem, and, given a Kurdyka-Lojasiewicz condition, an $\mathcal{O}(\log{k}/k)$ ergodic convergence rate. Finally, applying this theory to learned regularisation, we prove universal approximation for input weakly convex neural networks (IWCNN), and show empirically that IWCNNs can lead to improved performance of learned adversarial regularisers for computed tomography (CT) reconstruction.