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

Abstract

AbstractIn this paper we characterize sparse solutions for variational problems of the form $$\min _{u\in X} \phi (u) + F(\mathcal {A}u)$$minu∈Xϕ(u)+F(Au), where X is a locally convex space, $$\mathcal {A}$$A is a linear continuous operator that maps into a finite dimensional Hilbert space and $$\phi $$ϕ 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 $$\phi $$ϕ (possibly translated by an element in the null space of $$\phi $$ϕ). 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 Unser et al. (SIAM Rev 59(4):769–793, 2017) under weaker hypotheses.