An Open Problem on Sparse Representations in Unions of Bases

We consider sparse representations of signals from redundant dictionaries which are unions several orthonormal bases. The spark introduced by Donoho and Elad plays an important role in representations. However, numerical computations sparks generally combinatorial. For bases, two lower bounds on the via mutual coherence were established previous work. constructively prove that both them tight. Our main results give positive answers to Gribonval Nielsen’s open problem Constructive proofs rely a family mutually unbiased bases first appears quantum information theory.