Exact solutions of infinite dimensional total-variation regularized problems

Axel Flinth (1) , Pierre Weiss (2, 3, 4, 5)

1 TU - Technical University of Berlin / Technische Universität Berlin
2 CNRS - Centre National de la Recherche Scientifique
3 ITAV - Institut des Technologies Avancées en sciences du Vivant
4 IMT - Institut de Mathématiques de Toulouse UMR5219
5 PRES Université de Toulouse

We study the solutions of infinite dimensional linear inverse problems over Banach spaces. The regularizer is defined as the total variation of a linear mapping of the function to recover, while the data fitting term is a near arbitrary convex function. The first contribution is about the solu-tion's structure: we show that under suitable assumptions, there always exist an m-sparse solution, where m is the number of linear measurements of the signal. Our second contribution is about the computation of the solution. While most existing works first discretize the problem, we show that exacts solutions of the infinite dimensional problem can be obtained by solving two consecutive finite dimensional convex programs. These results extend recent advances in the understanding of total-variation reg-ularized problems.

Domaines: Optimisation et contrôle [math.OC]; Analyse numérique [math.NA]

Axel Flinth, Pierre Weiss. Exact solutions of infinite dimensional total-variation regularized problems. Information and Inference, 2018, ⟨10.1093/imaiai/iay016⟩. ⟨hal-01572196v2⟩