A Suitable Conjugacy for the l0 Pseudonorm

TL;DR

This paper introduces a new conjugacy based on the Caprac coupling that makes the l0 pseudonorm behave like a convex function under this framework, revealing new properties and dual norm expressions.

Contribution

It proposes a novel conjugacy induced by the Caprac coupling, demonstrating that the l0 pseudonorm is Caprac-convex and providing explicit conjugate expressions.

Findings

l0 pseudonorm equals its biconjugate under Caprac conjugacy

On the sphere, l0 pseudonorm coincides with a convex lsc function

Conjugates expressed via 2-k-symmetric gauge and k-support norms

Abstract

The so-called l0 pseudonorm on R d counts the number of nonzero components of a vector. It is well-known that the l0 pseudonorm is not convex, as its Fenchel biconjugate is zero. In this paper, we introduce a suitable conjugacy, induced by a novel coupling, Caprac, having the property of being constant along primal rays, like the l0 pseudonorm. The Caprac coupling belongs to the class of one-sided linear couplings, that we introduce. We show that they induce conjugacies that share nice properties with the classic Fenchel conjugacy. For the Caprac conjugacy, induced by the coupling Caprac, we prove that the l0 pseudonorm is equal to its biconjugate: hence, the l0 pseudonorm is Caprac-convex in the sense of generalized convexity. As a corollary, we show that the l0 pseudonorm coincides, on the sphere, with a convex lsc function. We also provide expressions for conjugates in terms of two families of dual norms, the 2-k-symmetric gauge norms and the k-support norms.