Stability of Data-Dependent Ridge-Regularization for Inverse Problems

TL;DR

This paper introduces a data-dependent, pixel-based ridge regularizer for inverse problems, providing theoretical stability guarantees and demonstrating high-quality reconstructions in biomedical imaging and materials science with limited training data.

Contribution

It proposes a novel data-dependent regularizer with proven solution existence and stability, linking it to a maximum-a-posteriori framework for inverse problems.

Findings

High-quality reconstructions with limited training data

Theoretical stability and existence of solutions

Effective in biomedical imaging and material sciences

Abstract

Theoretical guarantees for the robust solution of inverse problems have important implications for applications. To achieve both guarantees and high reconstruction quality, we propose learning a pixel-based ridge regularizer with a data-dependent and spatially varying regularization strength. For this architecture, we establish the existence of solutions to the associated variational problem and the stability of its solution operator. Further, we prove that the reconstruction forms a maximum-a-posteriori approach. Simulations for biomedical imaging and material sciences demonstrate that the approach yields high-quality reconstructions even if only a small instance-specific training set is available.