On Representer Theorems and Convex Regularization

We establish a general principle which states that regularizing an inverse problem with a convex function yields solutions that 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 previously an unresolved problem.