  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.
