An Optimal Condition for the Block Orthogonal Matching Pursuit Algorithm

Recovery of the support of a block K-sparse signal x from a linear model y = Ax + 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 Ω 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 μ(A) and sub-coherence ν(A) of A satisfy (2K - 1)dμ(A) + (d - 1)ν(A) <; 1. In this paper, we first improve and develop sufficient conditions of recovering Ω with the BOMP algorithm under the ℓ
₂
-bounded and ℓ
_∞
-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)dμ(A) + (d - 1)ν(A) = 1 such that the BOMP is not able to recoverx fromy = Ax in K iterations. This indicates that the condition proposed by Eldar et al. is sharp in terms of the condition on A.