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

Abstract

In this paper we characterize sparse solutions for variational problems of the form ${\min }_{u\in X}\varphi (u)+F(\mathcal{A}u)$, where $X$ is a locally convex space, $\mathcal{A}$ is a linear continuous operator that maps into a finite dimensional Hilbert space and $\varphi$ 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 $\varphi$ (possibly translated by an element in the null space of $\varphi$). 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.