Provably Convergent Data-Driven Convex-Nonconvex Regularization

TL;DR

This paper introduces a provably convergent framework for data-driven inverse problem solving using a novel weakly convex neural network, ensuring well-posedness and overcoming previous numerical issues.

Contribution

It proposes a new convex-nonconvex framework with an innovative IWCNN construction, providing theoretical guarantees for learned regularization methods.

Findings

The method achieves provable convergence.

It overcomes numerical issues of previous adversarial regularization.

The approach ensures well-posedness in inverse problems.

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 regularization 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.