# Solution Paths of Variational Regularization Methods for Inverse Problems

arXiv: https://arxiv.org/abs/1808.01783 · HTML: https://mlsanity.com/paper/1808.01783/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Aug 2018.

## Built on by (most cited first, 4 of 4)

- [CLIP: Cheap Lipschitz Training of Neural Networks](https://mlsanity.com/paper/2103.12531/) (2021)
- [The Geometry of Adversarial Training in Binary Classification](https://mlsanity.com/paper/2111.13613/) (2021)
- [Gradient flows and randomised thresholding: sparse inversion and classification](https://mlsanity.com/paper/2203.11555/) (2022)
- [Optimal Linear Baseline Models for Scientific Machine Learning](https://mlsanity.com/paper/2508.05831/) (2025)

## Abstract (arXiv)

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.