Stable and Robust Recovery of Approximately $k$-Sparse Signals with Partial Support Information in Noise Settings via Weighted $ℓ_p\ (0 < p ≤ 1)$ Minimization

Abstract

In the existing work, the recovery of strictly $k$-sparse signals with partial support information was derived in the $\u2113_2$ bounded noise setting. In this paper, the recovery of approximately $k$-sparse signals with partial support information in two noise settings is investigated via weighted $\u2113_p \ (0 < p \u2264 1)$ minimization method. The restricted isometry constant (RIC) condition $\u03b4_{tk} <\frac{1}{p\u03b7^{ \frac{2}{p}\u22121} +1}$ on the measurement matrix for some $t \u2208 [1+\frac{ 2\u2212p}{ 2+p} \u03c3, 2]$ is proved to be sufficient to guarantee the stable and robust recovery of signals under sparsity defect in noisy cases. Herein, $\u03c3 \u2208 [0, 1]$ is a parameter related to the prior support information of the original signal, and $\u03b7 \u2265 0$ is determined by $p,$ $t$ and $\u03c3.$ The new results not only improve the recent work in [17], but also include the optimal results by weighted $\u2113_1$ minimization or by standard $\u2113_p$ minimization as special cases.