Explainable Learning Based Regularization of Inverse Problems

TL;DR

This paper investigates explainable learning-based regularizers for inverse problems, analyzing their convergence, robustness, and structure, and demonstrating their effectiveness through theoretical insights and numerical experiments.

Contribution

It advances the theoretical understanding of spectral architectures and adversarial robustness in explainable learning-based regularization for inverse problems.

Findings

Spectral architectures have convergence rates influenced by data smoothness.

Adversarial training acts as a convergent regularization method.

Extensions to frame systems and CNN-type architectures are feasible and effective.

Abstract

Machine learning techniques for the solution of inverse problems have become an attractive approach in the last decade, while their theoretical foundations are still in their infancy. In this chapter we want to pursue the study of regularization properties, robustness, convergence rates, and structure of regularizers for inverse problems obtained from different learning paradigms. For this sake we study simple architectures that are explainable in the sense that they allow for a theoretical analysis also in the infinite-dimensional limit. In particular we will advance the study of spectral architectures with new results on convergence rates highlighting the role of the smoothness in the training data set, and a study of adversarial robustness. We can show that adversarial training is actually a convergent regularization method. Moreover, we discuss extensions to frame systems and CNN-type architectures for variational regularizers, where we obtain some results on their structure by carefully designed numerical experiments.