New RIP Bounds for Recovery of Sparse Signals With Partial Support Information via Weighted ${\ell_{p}}$ -Minimization

Abstract

In this paper, we consider the recovery of $k$ -sparse signals using the weighted $\ell _{p}$ ( $0 ) minimization when some partial prior information on the support is available. First, we present a unified analysis of restricted isometry constant $\delta _{tk}$ with $d ( $d \geqslant 1$ is determined by the prior support information) for sparse signal recovery by the weighted $\ell _{p}$ ( $0 ) minimization in both noiseless and noisy settings. This result fills a vacancy on $\delta _{tk}$ with $t , compared with previous works on $\delta _{(a+1)k}$ ( $a>1$ ). Second, we provide a sufficient condition on $\delta _{tk}$ with $1 for the recovery of sparse signals using the $\ell _{p}$ ( $0 ) minimization, which extends the existing optimal result on $\delta _{2k}$ in the literature. Last, various numerical examples are presented to demonstrate the better performance of the weighted $\ell _{p}$ ( $0 ) minimization is achieved when the accuracy of prior information on the support is at least 50%, compared with that of the $\ell _{p}$ ( $0 ) minimization.