On Representer Theorems and Convex Regularization

TL;DR

This paper presents a unifying principle linking convex regularization in inverse problems to convex combinations of a limited set of atoms, extending to quasi-convex regularizers and characterizing total gradient variation minimizers.

Contribution

It establishes a general representer theorem for convex regularizers, extending to quasi-convex functions and characterizes minimizers of total gradient variation.

Findings

Solutions are convex combinations of a small number of atoms.

Extension of the principle to quasi-convex regularizers.

Characterization of total gradient variation minimizers.

Abstract

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. 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, which was still an unresolved problem.