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On sparse representations in arbitrary redundant bases

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1 Author(s)
Fuchs, J.-J. ; IRISA/Univ. de Rennes, France

The purpose of this contribution is to generalize some recent results on sparse representations of signals in redundant bases. The question that is considered is the following: given a matrix A of dimension (n,m) with m>n and a vector b=Ax, find a sufficient condition for b to have a unique sparsest representation x as a linear combination of columns of A. Answers to this question are known when A is the concatenation of two unitary matrices and either an extensive combinatorial search is performed or a linear program is solved. We consider arbitrary A matrices and give a sufficient condition for the unique sparsest solution to be the unique solution to both a linear program or a parametrized quadratic program. The proof is elementary and the possibility of using a quadratic program opens perspectives to the case where b=Ax+e with e a vector of noise or modeling errors.

Published in:

Information Theory, IEEE Transactions on  (Volume:50 ,  Issue: 6 )