Nonlinear eigenvalue problems for seminorms and applications

Abstract

The aim of this paper is to discuss recent progress in nonlinear eigenvalue problems for seminorms (absolutely one-homogeneous convex functionals), which find many applications in data science, inverse problems, and image processing. We provide a unified viewpoint on the notion of nonlinear singular vectors and eigenvectors for homogeneous nonlinear operators respectively functionals. We further discuss in particular ground states, i.e., the first eigenvector or eigenfunction. Moreover, we review a recent approach to the analysis of eigenvectors based on duality, which has implications to the possible computation of spectral decompositions, i.e., signal dependent linear expansions in a system of eigenvectors.

Moreover, we discuss some relevant implications such as the refined analysis of variational regularization methods and their bias, as well as the analysis of some iteration methods and time-continuous flows. Finally, we provide more direct applications of the nonlinear eigenvalue problems such as nonlinear spectral clustering on graphs.