Solution Paths of Variational Regularization Methods for Inverse Problems

Abstract: We consider a family of variational regularization functionals for a generic inverse problem, where the data fidelity and regularization term are given by powers of a Hilbert norm and an absolutely one-homogeneous functional, respectively, and the regularization parameter is interpreted as artificial time. We investigate the small and large time behavior of the associated solution paths and, in particular, prove finite extinction time for a large class of functionals. Depending on the powers, we also show that the solution paths are of bounded variation or even Lipschitz continuous. In addition, it will turn out that the models are "almost" mutually equivalent in terms of the minimizers they admit. Finally, we apply our results to define and compare two different non-linear spectral representations of data and show that only one of it is able to decompose a linear combination of non-linear eigenfunctions into the individual eigenfunctions. For that purpose, we will also briefly address piecewise affine solution paths.