Solution Paths of Variational Regularization Methods for Inverse Problems

TL;DR

This paper analyzes the behavior of solution paths in variational regularization methods for inverse problems, revealing finite extinction times, regularity properties, and spectral decomposition capabilities.

Contribution

It introduces a comprehensive analysis of solution path properties, including finite extinction time and regularity, and compares spectral representations in variational regularization.

Findings

Finite extinction time for a large class of functionals

Solution paths are of bounded variation or Lipschitz continuous

Only one spectral representation can decompose linear combinations of eigenfunctions

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.