Application of concave/Schur-concave functions to the learning of overcomplete dictionaries and sparse representations

Given a very overcomplete m/spl times/n dictionary of representation vectors a/sub i/, A=[a/sub 1/,...,a/sub n/], n/spl Gt/m, an environmentally generated signal, y, can be succinctly represented within the dictionary by obtaining a sparse solution, x, to the linear inverse problem Ax/spl ap/y using various previously proposed methodologies. In particular, sparse solutions can be found by an appropriately regularized minimization of the error e=y-Ax. In this paper we briefly discuss our investigations into the use of concave/Schur-concave functions as regularizing sparsity measures, and their application to the problem of obtaining sparse representations, x, of environmentally generated signals y, and the problem of learning environmentally adapted overcomplete dictionaries.