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 minima. 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.

翻译：变分正则化是解决反问题的主要方法，近年来已有大量研究利用深度学习的正则化手段提升性能。然而，关于此类正则化收敛性的结果尚不多见，尤其在涉及临界点而非全局极小值的背景下。本文提出了一种基于临界点的广义收敛正则化框架，并证明该类正则化可通过一类弱凸正则化器实现。我们证明了原始-对偶混合梯度法在相应变分问题中的收敛性，并在满足Kurdyka-Lojasiewicz条件时，给出$\mathcal{O}(\log{k}/k)$的遍历收敛速率。最后，将这一理论应用于学习型正则化，我们证明了输入弱凸神经网络（IWCNN）的通用逼近性质，并通过实验表明IWCNN可提升计算断层重建（CT）中学习型对抗正则化器的性能。