On the Building Blocks of Sparsity Measures

Understanding the mathematics and the innate machinery of sparsity measures is instrumental in the proper usage of such information measures in various application arenas, ranging from information collection and sensing, to communications and signal processing. In this letter, the structure of sparsity measures is investigated. Specifically, it is shown that sparsity measures satisfying proper sparsity axioms may only be constructed by vector norms. Moreover, the asymptotic behavior of sparsity measures is studied. Owing to their mathematical structure, our numerical results illustrate a convergence of sparsity measures, as the number of input samples grows large.