The Capra-subdifferential of the l0 pseudonorm

TL;DR

This paper explores the Capra-subdifferential of the nonconvex l0 pseudonorm, providing explicit formulas and visualizations when using lp norms with p > 1, advancing the understanding of its convexity properties.

Contribution

It introduces explicit formulations for the Capra subdifferential of l0 with lp norms, demonstrating its Capra-convexity under certain conditions.

Findings

Explicit formulas for Capra subdifferential of l0 with lp norms.

Visualization of Capra subdifferential for Euclidean norm.

L0 pseudonorm is Capra-convex under specific norm choices.

Abstract

The l0 pseudonorm counts the nonzero coordinates of a vector. It is often used in optimization problems to enforce the sparsity of the solution. However, this function is nonconvex and noncontinuous, and optimization problems formulated with l0 in the objective function or in the constraints are hard to solve in general. Recently, a new family of coupling functions - called Capra (constant along primal rays) - has proved to induce relevant generalized Fenchel-Moreau conjugacies to handle the l0 pseudonorm. In particular, under a suitable choice of source norm on the Euclidean space used in the definition of the Capra coupling - the function l0 is Capra-subdifferentiable, hence is Capra-convex. In this article, we give explicit formulations for the Capra subdifferential of l0, when the source norm is a lp norm with p larger that 1. We illustrate our results with graphical visualizations of the Capra subdifferential of l0 for the Euclidean source norm.