Sparse Representations in Unions of Bases
Résumé
The purpose of this paper is to generalize a result by Donoho, Huo, Elad and Bruckstein on sparse representations of signals in a union of two orthonormal bases. We consider general (redundant) dictionaries in finite dimension, and derive sufficient conditions on a signal for having unique sparse representations in such dictionaries. The special case where the dictionary is given by a union of several orthonormal bases is studied in more detail. In particular, it is proved that the result of Donoho and Huo, concerning the replacement of a combinatorial optimization problem with a linear programming problem when searching for sparse representations, has an analog for dictionaries that may be highly redundant.