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The so-called l0 pseudonorm, or cardinality function, counts the number ofnonzero components of a vector. In this paper, we analyze the l0 pseudonorm bymeans of so-called Capra (constant along primal rays) conjugacies, for whichthe underlying source norm and its dual norm are both orthant-strictlymonotonic (a notion that we formally introduce and that encompasses the lpnorms, but for the extreme ones). We obtain three main results. First, we showthat the l0 pseudonorm is equal to its Capra-biconjugate, that is, is aCapra-convex function. Second, we deduce an unexpected consequence, that wecall convex factorization: the l0 pseudonorm coincides, on the unit sphere ofthe source norm, with a proper convex lower semicontinuous function. Third, weestablish a variational formulation for the l0 pseudonorm by means ofgeneralized top-k dual~norms and k-support dual~norms (that we formallyintroduce).

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