Section: New Results

Representer theorems in variational problems

C. Boyer, A. Chambolle, Y. De Castro, V. Duval, F. De Gournay, P. Weiss

In [29], we have established 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. An extension to a broader class of quasi-convex regularizers is also discussed. As a side result, we characterize the minimizers of the total gradient variation, describing the solutions of total variation problem as a superposition of indicator functions of simply connected sets. That result provides an explanation of the so-called staircasing phenomenon.