New results on sparse representations in unions of orthonormal bases

The problem of sparse representation has significant applications in signal processing. The spark of a dictionary plays a crucial role in the study of sparse representation. Donoho and Elad initially explored the spark, and they provided a general lower bound. When the dictionary is a union of several orthonormal bases, Gribonval and Nielsen presented an improved lower bound for spark. In this paper, we introduce a new construction of dictionary, achieving the spark bound given by Gribonval and Nielsen. More precisely, let q be a power of 2, we show that for any positive integer t, there exists a dictionary in $R^{q^{2t}}$, which is a union of $q+1$ orthonormal bases, such that the spark of the dictionary attains Gribonval-Nielsen's bound. Our result extends previously best known result from $t=1,2$ to arbitrarily positive integer t, and our construction is technically different from previous ones. Their method is more combinatorial, while ours is algebraic, which is more general.