Sparse representations in unions of bases

The purpose of this correspondence is to generalize a result by Donoho and Huo and Elad and Bruckstein on sparse representations of signals in a union of two orthonormal bases for R^N. We consider general (redundant) dictionaries for R^N, and derive sufficient conditions for having unique sparse representations of signals in such dictionaries. The special case where the dictionary is given by the union of L≥2 orthonormal bases for R^N is studied in more detail. In particular, it is proved that the result of Donoho and Huo, concerning the replacement of the ℓ⁰ optimization problem with a linear programming problem when searching for sparse representations, has an analog for dictionaries that may be highly redundant.