Capra-Convexity, Convex Factorization and Variational Formulations for the ℓ0 Pseudonorm

Autor:	Michel De Lara, Jean-Philippe Chancelier
Rok vydání:	2021
Předmět:	Statistics and Probability Unit sphere Numerical Analysis Pure mathematics Applied Mathematics Mathematics::General Topology Monotonic function Function (mathematics) Convexity Cardinality Factorization Norm (mathematics) Geometry and Topology Analysis Dual norm Mathematics
Zdroj:	Set-Valued and Variational Analysis. 30:597-619
ISSN:	1877-0541 1877-0533
DOI:	10.1007/s11228-021-00606-z
Popis:	The so-called l0 pseudonorm, or cardinality function, counts the number of nonzero components of a vector. In this paper, we analyze the l0 pseudonorm by means of so-called Capra (constant along primal rays) conjugacies, for which the underlying source norm and its dual norm are both orthant-strictly monotonic (a notion that we formally introduce and that encompasses the lp norms, but for the extreme ones). We obtain three main results. First, we show that the l0 pseudonorm is equal to its Capra-biconjugate, that is, is a Capra-convex function. Second, we deduce an unexpected consequence, that we call convex factorization: the l0 pseudonorm coincides, on the unit sphere of the source norm, with a proper convex lower semicontinuous function. Third, we establish a variational formulation for the l0 pseudonorm by means of generalized top-k dual~norms and k-support dual~norms (that we formally introduce).
Databáze:	OpenAIRE
Externí odkaz:	https://explore.openaire.eu/search/publication?articleId=doi_________::fb48c49f1e126cae10106b875b90423a https://doi.org/10.1007/s11228-021-00606-z Zobrazit plný text záznamu Full text from SpringerLink