Abstract
In this paper, we use the block orthogonal matching pursuit (BOMP) algorithm to recover block sparse signals x from measurements y = Ax + v, where v is an ℓ2-bounded noise vector (i.e., kvk2 ≤ ǫ for some constant ǫ). We investigate some sufficient conditions based on the block restricted isometry property (block-RIP) for exact (when v = 0) and stable (when v , 0) recovery of block sparse signals x. First, on the one hand, we show that if A satisfies the block-RIP with δK+1 < 1/√K + 1, then every block K-sparse signal x can be exactly or stably recovered by BOMP in K iterations. On the other hand, we show that, for any K ≥ 1 and 1/√K + 1 ≤ δ < 1, there exists a matrix A satisfying the block-RIP with δK+1 = δ and a block K-sparse signal x such that BOMP may fail to recover x in K iterations. Then, we study some sufficient conditions for recovering block α-strongly-decaying K-sparse signals. We show that if A satisfies the block-RIP with δK+1 < √2/2, then every α-strongly-decaying block K-sparse signal can be exactly or stably recovered by BOMP in K iterations under some conditions on α. Our newly found sufficient condition on the block-RIP of A is less restrictive than that for ℓ1 minimization for this special class of sparse signals. Furthermore, for any K ≥ 1, α > 1 and √2/2 ≤ δ < 1, the recovery of x may fail in K iterations for a sensingmatrix A which satisfies the block-RIP with δK+1 = δ. Finally, we study some sufficient conditions for partial recovery of block sparse signals. Specifically, if A satisfies the block-RIP with δK+1 < √2/2, then BOMP is guaranteed to recover some blocks of x if these blocks satisfy a sufficient condition. We further show that this condition is also sharp.