# Convex Regularization and Representer Theorems

## TL;DR

This paper proves that convex regularization via gauge functions results in solutions composed of a few fundamental elements, explaining why certain regularizers favor specific structures like piecewise constant images.

## Contribution

It establishes a general representer theorem for convex regularizers based on gauge functions, linking solutions to extreme points or rays of the regularizer set.

## Key findings

- Solutions are linear combinations of a few extreme points or rays of the convex set.
- Application to total gradient variation explains its effectiveness in reconstructing piecewise constant images.
- Provides a unified framework connecting regularization and the structure of solutions.

## 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.