We are concerned with the problem of recovering sparse vectors exactly by two commonly used methods, orthogonal matching pursuit (OMP) and basis pursuit (BP). Suppose that overcomplete dictionaries are unions of several orthonormal bases. Gribonval and Nielsen have provided a sufficient mutual coherence condition for BP to recover every sparse vector exactly. Then Tropp proved that it is a sharp sufficient condition for OMP by existence proofs. We establish here the sharpness of Gribonval and Nielsen’s sufficient condition for both OMP and BP by constructing several families of counterexamples. Our main tools are Hadamard matrices and mutually unbiased bases from quantum information theory.
