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Journal of Convex Analysis 28 (2021), No. 1, 203--236
Copyright Heldermann Verlag 2021

Hidden Convexity in the l0 Pseudonorm

Jean-Philippe Chancelier
CERMICS, Ecole des Ponts, Marne-la-Vallée, France

Michel De Lara
CERMICS, Ecole des Ponts, Marne-la-Vallée, France

The so-called l0 pseudonorm on Rd 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, ECapra, that has the property of being constant along primal rays like the l0 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 l0 pseudonorm, the characteristic functions of its level sets and the sequence of so-called top-k norms. In particular, we prove that the l0 pseudonorm is equal to its biconjugate: hence, the l0 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 Rd such that this function and the l0 pseudonorm coincide on the Euclidian unit sphere. This hidden convexity property is somewhat surprising as the l0 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.

Keywords: l-0-pseudonorm, coupling, Fenchel-Moreau conjugacy, top-k norms, k-support norms, hidden convexity.

MSC: 46N10, 49N15, 46B99, 52A41, 90C46.

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