Learning convex regularizers satisfying the variational source condition for inverse problems

TL;DR

This paper introduces a new data-driven convex regularizer for inverse problems, trained with adversarial methods and variational source conditions, providing theoretical convergence guarantees.

Contribution

It extends adversarial convex regularization by incorporating variational source conditions, enabling convergence rate estimates for learned regularizers.

Findings

The proposed ACR-SC performs comparably to ACR in practice.

ACR-SC provides a quantitative convergence rate estimate.

The method successfully combines deep learning with classical regularization theory.

Abstract

Variational regularization has remained one of the most successful approaches for reconstruction in imaging inverse problems for several decades. With the emergence and astonishing success of deep learning in recent years, a considerable amount of research has gone into data-driven modeling of the regularizer in the variational setting. Our work extends a recently proposed method, referred to as adversarial convex regularization (ACR), that seeks to learn data-driven convex regularizers via adversarial training in an attempt to combine the power of data with the classical convex regularization theory. Specifically, we leverage the variational source condition (SC) during training to enforce that the ground-truth images minimize the variational loss corresponding to the learned convex regularizer. This is achieved by adding an appropriate penalty term to the ACR training objective. The resulting regularizer (abbreviated as ACR-SC) performs on par with the ACR, but unlike ACR, comes with a quantitative convergence rate estimate.