Hidden Convexity in the l₀ Pseudonorm

The so-called $\ell_0$ pseudonorm on $\mathbb{R}^d$ counts the number of nonzero components of a vector. It is well-known that the $\ell_0$ pseudonorm is not convex, as its Fenchel biconjugate is zero. In this paper, we introduce a suitable conjugacy, induced by a novel coupling, ECapra, that has the property of being constant along primal rays like the $\ell_0$ pseudonorm. The coupling ECapra 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 ECapra conjugacy, induced by the coupling ECapra, we relate the ECapra conjugate and biconjugate of the $\ell_0$ pseudonorm, the characteristic functions of its level sets and the sequence of so-called top-k norms. In particular, we prove that the $\ell_0$ pseudonorm is equal to its biconjugate: hence, the $\ell_0$ pseudonorm is ECapra-convex in the sense of generalized convexity. As a corollary, we show that there exists a proper convex lower semicontinuous function on $\mathbb{R}^d$ such that this function and the $\ell_0$ pseudonorm coincide on the Euclidian unit sphere. This hidden convexity property is somewhat surprising as the $\ell_0$ pseudonorm is a highly nonconvex function of combinatorial nature. We provide different expressions for this proper convex lower semicontinuous function, and we give explicit formulas in the two-dimensional case.