Sparse representations in unions of bases

Authors:
R. Gribonval;M. Nielsen
Affiliations:
IRISA/INRIA, France;-
Venue:
IEEE Transactions on Information Theory
Year:
2003

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Abstract

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 RN. We consider general (redundant) dictionaries for RN, 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 RN 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.