# Nonlinear Spectral Decompositions by Gradient Flows of One-Homogeneous Functionals

## TL;DR

This paper develops a comprehensive theory of nonlinear spectral decompositions using gradient flows of one-homogeneous functionals in infinite-dimensional Hilbert spaces, linking eigenvalue problems, variational methods, and spectral analysis.

## Contribution

It introduces a geometric characterization of eigenvectors, connects gradient flows to spectral decompositions, and generalizes existing results to broader settings.

## Key findings

- Eigenvectors characterized via dual unit ball and minimal norm subgradients
- Gradient flows decompose initial conditions into eigenvectors
- Spectral decompositions verified for various functionals like total variation

## Abstract

This paper establishes a theory of nonlinear spectral decompositions by considering the eigenvalue problem related to an absolutely one-homogeneous functional in an infinite-dimensional Hilbert space. This approach is both motivated by works for the total variation, where interesting results on the eigenvalue problem and the relation to the total variation flow have been proven previously, and by recent results on finite-dimensional polyhedral semi-norms, where gradient flows can yield spectral decompositions into eigenvectors. We provide a geometric characterization of eigenvectors via a dual unit ball and prove them to be subgradients of minimal norm. This establishes the connection to gradient flows, whose time evolution is a decomposition of the initial condition into subgradients of minimal norm. If these are eigenvectors, this implies an interesting orthogonality relation and the equivalence of the gradient flow to a variational regularization method and an inverse scale space flow. Indeed we verify that all scenarios where these equivalences were known before by other arguments - such as one-dimensional total variation, multidimensional generalizations to vector fields, or certain polyhedral semi-norms - yield spectral decompositions, and we provide further examples. We also investigate extinction times and extinction profiles, which we characterize as eigenvectors in a very general setting, generalizing several results from literature.