Sharp Sufficient Conditions for Stable Recovery of Block Sparse Signals by Block Orthogonal Matching Pursuit

Abstract: In this paper, we use the block orthogonal matching pursuit (BOMP) algorithm to recover block sparse signals $\x$ from measurements $\y=\A\x+\v$, where $\v$ is an $\ell_2$-bounded noise vector (i.e., $\|\v\|_2\leq \epsilon$ 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 $\v\neq\0$) recovery of block sparse signals $\x$. First, on the one hand, we show that if $\A$ satisfies the block-RIP with $\delta_{K+1}<1/\sqrt{K+1}$, 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 $K\geq 1$ and $1/\sqrt{K+1}\leq \delta<1$, 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 $\delta_{K+1}<\sqrt{2}/2$, 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 $\ell_1$ minimization for this special class of sparse signals. Furthermore, for any $K\geq 1$, $\alpha>1$ and $\sqrt{2}/2\leq \delta<1$, 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 $\delta_{K+1}<\sqrt{2}/2$, then BOMP is guaranteed to recover some blocks of $\x$ if these blocks satisfy a sufficient condition.