Séance Séminaire

Séminaire de Probabilités et Statistique

On the structure of solutions of convex regularization

We establish a general principle which states that regularizing an inverse problem with a convex function yields solutions which are convex combinations of a small number of atoms. These atoms are identified with the extreme points and elements of the extreme rays of the regularizer level sets. As a side result, we characterize the minimizers of the total gradient variation, which was still an unresolved problem. This can be viewed to be in the same vein of the representer theorem in machine learning.