Convex Regularization and Representer Theorems

Abstract

We establish a result which states that regularizing an inverse problem with the gauge of a convex set $C$ yields solutions which are linear combinations of a few extreme points or elements of the extreme rays of $C$. These can be understood as the \textit{atoms} of the regularizer. We then explicit that general principle by using a few popular applications. In particular, we relate it to the common wisdom that total gradient variation minimization favors the reconstruction of piecewise constant images.

in Proceedings of iTWIST'18, Paper-ID: 30, Marseille, France, November, 21-23, 2018