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

التفاصيل البيبلوغرافية
العنوان: Weakly Convex Regularisers for Inverse Problems: Convergence of Critical Points and Primal-Dual Optimisation
المؤلفون: Shumaylov, Zakhar, Budd, Jeremy, Mukherjee, Subhadip, Schönlieb, Carola-Bibiane
سنة النشر: 2024
المجموعة: Computer Science
Mathematics
Statistics
مصطلحات موضوعية: Mathematics - Optimization and Control, Computer Science - Computer Vision and Pattern Recognition, Computer Science - Machine Learning, Statistics - Machine Learning
الوصف: 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.
Comment: 26 pages, 4 figures; https://openreview.net/forum?id=E8FpcUyPuS
نوع الوثيقة: Working Paper
URL الوصول: http://arxiv.org/abs/2402.01052
رقم الأكسشن: edsarx.2402.01052
قاعدة البيانات: arXiv