Sharp sufficient conditions for stable recovery of block sparse signals by block orthogonal matching pursuit

摘要

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 l(2)-bounded noise vector (i.e., parallel to v parallel to(2) <= epsilon for some constant epsilon). We investigate some sufficient conditions based on the block restricted isometry property (block-RIP) for exact (when v = 0) and stable (when v not equal 0) recovery of block sparse signals x. First, on the one hand, we show that if A satisfies the block-RIP with delta(K+1) < 1/root 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 INK 1/root K+1 <= delta < 1, there exists a matrix A satisfying the block-RIP with delta(K+1) = delta 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 alpha-strongly-decaying K-sparse signals. We show that if A satisfies the block-RIP with delta(K+1) < root 2/2, then every alpha-strongly-decaying block K-sparse signal can be exactly or stably recovered by BOMP in K iterations under some conditions on alpha. Our newly found sufficient condition on the block-RIP of A is less restrictive than that for l(1) minimization for this special class of sparse signals. Furthermore, for any K >= 1, alpha > 1 and root 2/2 <= delta < 1, the recovery of x may fail in K iterations for a sensing matrix A which satisfies the block-RIP with delta(K+1) = delta. Finally, we study some sufficient conditions for partial recovery of block sparse signals. Specifically, if A satisfies the block-RIP with delta(K+1) < root 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. (C) 2018 Elsevier Inc. All rights reserved.