A perturbation analysis based on group sparse representation with orthogonal matching pursuit

Abstract

Abstract In this paper, by exploiting orthogonal projection matrix and block Schur complement, we extend the study to a complete perturbation model. Based on the block-restricted isometry property (BRIP), we establish some sufficient conditions for recovering the support of the block 𝐾-sparse signals via block orthogonal matching pursuit (BOMP) algorithm. Under some constraints on the minimum magnitude of the nonzero elements of the block 𝐾-sparse signals, we prove that the support of the block 𝐾-sparse signals can be exactly recovered by the BOMP algorithm in the case of ℓ 2 \ell_{2} and ℓ 2 / ℓ ∞ \ell_{2}/\ell_{\infty} bounded total noise if 𝑨 satisfies the BRIP of order K + 1 K+1 with δ K + 1 < 1 K + 1 ⁢ ( 1 + ϵ A ( K + 1 ) ) 2 + 1 ( 1 + ϵ A ( K + 1 ) ) 2 - 1 . \delta_{K+1}<\frac{1}{\sqrt{K+1}(1+\epsilon_{\boldsymbol{A}}^{(K+1)})^{2}}+\frac{1}{(1+\epsilon_{\boldsymbol{A}}^{(K+1)})^{2}}-1. In addition, we also show that this is a sharp condition for exactly recovering any block 𝐾-sparse signal with the BOMP algorithm. Moreover, we also give the reconstruction upper bound of the error between the recovered block-sparse signal and the original block-sparse signal. In the noiseless and perturbed case, we also prove that the BOMP algorithm can exactly recover the block 𝐾-sparse signal under some constraints on the block 𝐾-sparse signal and δ K + 1 < 2 + 2 2 ⁢ ( 1 + ϵ A ( K + 1 ) ) 2 - 1 . \delta_{K+1}<\frac{2+\sqrt{2}}{2(1+\epsilon_{\boldsymbol{A}}^{(K+1)})^{2}}-1. Finally, we compare the actual performance of perturbed OMP and perturbed BOMP algorithm in the numerical study. We also present some numerical experiments to verify the main theorem by using the completely perturbed BOMP algorithm.