The aim of the first part of this talk is to provide a characterization for sparse solutions of abstract variational inverse problems with finite dimensional data. We consider the minimization of functionals that are the sum of two terms: a convex regularizer and a finite dimensional soft constraint. It was observed for specific examples that minimizers of variational problems of this type are sparse in a suitable sense. We formalise this fact proving the existence of a minimizer that is represented as a finite linear combination of extremal points of the unit ball of the regularizer. This finding provides a natural notion of sparsity for abstract variational inverse problems.
We apply this abstract result to relevant examples as TV denoising and higher order scalar regularizers. Then, we consider the framework of dynamic inverse problems with the Benamou-Brenier energy as a regularizer. Using the classical theory of Optimal Transport, we provide a characterisation for sparse solutions in this specific case. Then, in the last part of the talk, we show how to construct a variant of the Alternating Descent Conditional Gradient Method that relies on the structure of sparse solutions for dynamic inverse problems.