Is the 1-norm the best convex sparse regularization?

TL;DR

This paper investigates why the 1-norm is the most effective convex regularization for sparse recovery, demonstrating its optimality among convex methods in low-dimensional settings.

Contribution

The paper introduces notions of 'best' convex regularization and proves the 1-norm's optimality for sparse recovery within this framework.

Findings

The 1-norm is shown to be an optimal convex regularizer for sparse recovery.

Other convex regularizations do not outperform the 1-norm in this context.

Theoretical framework supports the 1-norm's supremacy in low-dimensional recovery tasks.

Abstract

The 1-norm is a good convex regularization for the recovery of sparse vectors from under-determined linear measurements. No other convex regularization seems to surpass its sparse recovery performance. How can this be explained? To answer this question, we define several notions of "best" (convex) regulariza-tion in the context of general low-dimensional recovery and show that indeed the 1-norm is an optimal convex sparse regularization within this framework.