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

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.