Weakly Convex Regularisers for Inverse Problems: Convergence of Critical Points and Primal-Dual Optimisation
This work addresses a theoretical gap in variational regularisation for inverse problems, offering convergence guarantees that could enhance reliability in applications like medical imaging, though it is incremental by building on existing regularisation frameworks.
The paper tackles the lack of convergence results for learned regularisation in inverse problems by proposing a generalised formulation using critical points and weakly convex regularisers, proving convergence and an O(log k/k) rate for primal-dual optimisation, and demonstrating improved performance in CT reconstruction with input weakly convex neural networks.
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.