Sparsity of solutions for variational inverse problems with finite-dimensional data

TL;DR

This paper characterizes sparse solutions for variational inverse problems with finite-dimensional data, showing they can be represented as linear combinations of extremal points, and applies this to total variation and Radon norm regularizers.

Contribution

It provides a theoretical framework for the sparsity of solutions in variational problems with finite-dimensional data, extending previous results to weaker hypotheses.

Findings

Existence of sparse minimizers as linear combinations of extremal points

Application to total variation and Radon norm regularizers

Theoretical justification of the staircase effect

Abstract

In this paper we characterize sparse solutions for variational problems of the form $\min_{u\in X} \phi(u) + F(\mathcal{A} u)$, where $X$ is a locally convex space, $\mathcal{A}$ is a linear continuous operator that maps into a finite dimensional Hilbert space and $\phi$ is a seminorm. More precisely, we prove that there exists a minimizer that is `sparse' in the sense that it is represented as a linear combination of the extremal points of the unit ball associated with the regularizer $\phi$ (possibly translated by an element in the null space of $\phi$). We apply this result to relevant regularizers such as the total variation seminorm and the Radon norm of a scalar linear differential operator. In the first example, we provide a theoretical justification of the so-called staircase effect and in the second one, we recover the result in [31] under weaker hypotheses.