Convergence of non-linear diagonal frame filtering for regularizing inverse problems

TL;DR

This paper extends the convergence analysis of diagonal frame filtering in inverse problems from linear to general non-linear filters, connecting it with variational regularization and plug-and-play methods.

Contribution

It generalizes spectral filtering convergence from linear to non-linear filters and links non-linear diagonal frame filtering with variational regularization.

Findings

Established convergence of non-linear spectral filtering

Connected non-linear filtering with variational regularization

Explored benefits of non-linear filtering in plug-and-play regularization

Abstract

Inverse problems are key issues in several scientific areas, including signal processing and medical imaging. Since inverse problems typically suffer from instability with respect to data perturbations, a variety of regularization techniques have been proposed. In particular, the use of filtered diagonal frame decompositions has proven to be effective and computationally efficient. However, existing convergence analysis applies only to linear filters and a few non-linear filters such as soft thresholding. In this paper, we analyze filtered diagonal frame decompositions with general non-linear filters. In particular, our results generalize SVD-based spectral filtering from linear to non-linear filters as a special case. As a first approach, we establish a connection between non-linear diagonal frame filtering and variational regularization, allowing us to use results from variational regularization to derive the convergence of non-linear spectral filtering. In the second approach, as our main theoretical results, we relax the assumptions involved in the variational case while still deriving convergence. Furthermore, we discuss connections between non-linear filtering and plug-and-play regularization and explore potential benefits of this relationship.