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

Inverse problems are core issues in several scientific areas, including signal processing and medical imaging. As 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, the existing convergence analysis applies only to linear filters and a few non-linear filters such as soft thresholding. In this paper, we analyze the filtered diagonal frame decomposition with general non-linear filters. In particular, our results generalize SVD-based spectral filtering from linear to non-linear filters as a special case. We present three strategies to demonstrate convergence. The first two strategies relate non-linear diagonal frame filtering to variational regularization and plug-and-play regularization, respectively. The third strategy allows us to relax the assumptions involved and still obtain a full convergence analysis.