Provably Convergent Data-Driven Convex-Nonconvex Regularization

Zakhar Shumaylov, Jeremy Budd, Subhadip Mukherjee, Carola-Bibiane Schönlieb

Keywords: Inverse problems, data-driven regularization, variational imaging, input-convex neural networks, weak convexity, convergent regularisation

TL;DR: We turn the full variational problem convex by using a learned regulariser that is weakly convex over the data space and convex over the model space

Abstract: An emerging new paradigm for solving inverse problems is via the use of deep learning to learn a regularizer from data. This leads to high-quality results, but often at the cost of provable guarantees. In this work, we show how well-posedness and convergent regularisation arises within the convex-nonconvex (CNC) framework for inverse problems. We introduce a novel input weakly convex neural network (IWCNN) construction to adapt the method of learned adversarial regularization to the CNC framework. Empirically we show that our method overcomes numerical issues of previous adversarial methods.

Submission Number: 25