An Optimal Condition for the Block Orthogonal Matching Pursuit Algorithm

Abstract

Recovery of the support of a block $K$ -sparse signal ${{x}}$ from a linear model ${{y}}= {A} {{x}} + {v}$ , where ${A}$ is a sensing matrix and ${v}$ is a noise vector, arises from many applications. The block orthogonal matching pursuit (BOMP) algorithm is a popular block sparse recovery algorithm and has received much attention in the recent decade. It was proved by Eldar et al. that the BOMP can recover the positions $\Omega $ of the nonzero blocks of any block $K$ -sparse vector ${{x}}$ with a block length $d$ in the noisy case (under certain condition on ${{x}}$ and ${v}$ ) and can exactly recover ${{x}}$ in the noiseless case in $K$ iterations if the block mutual coherence $\mu ({A})$ and sub-coherence $\nu ({A})$ of ${A}$ satisfy $(2K-1)d\mu ({A}) +(d-1) \nu ({A}) In this paper, we first improve and develop sufficient conditions of recovering $\Omega $ with the BOMP algorithm under the $\ell _{2}$ -bounded and $\ell _{\infty }$ -bounded noises, respectively. Then, we show that for any given positive integers $K$ and $d$ , there always exist a block $K$ -sparse vector ${{x}}$ with the block length $d$ , and a sensing matrix ${A}$ with $(2K-1)d\mu ({A}) + (d-1) \nu ({A})=1 $ such that the BOMP is not able to recover ${{x}}$ from ${{y}}= {A} {{x}} $ in $K$ iterations. This indicates that the condition proposed by Eldar et al. is sharp in terms of the condition on ${A}$ .