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A complete characterization for the restricted isometry constant (RIC) bounds on $\delta _{{{ tk}}}$ for all $ {t}>0$ is an important problem on recovery of sparse signals with prior support information via weighted $\ell _{{p}}$ -minimization ($0 < {p} \leqslant 1$). In this paper, new bounds on the restricted isometry constants $\delta _{{{ tk}}}$ ($0 < {t} < \frac {4}{3}{d}$), where $d$ is a key constant determined by prior support information, are established to guarantee the sparse signal recovery via the weighted $\ell _{{p}}$ minimization in both noiseless and noisy settings. This result fills a vacancy on $\delta _{{{ tk}}}$ with $0 < {t} < \frac {4}{3}{d}$ , compared with previous works on $\delta _{{{ tk}}}$ (${t} \geqslant \frac {4}3{d}$). We show that, when the accuracy of prior support estimate is at least 50%, the new recovery condition in terms of $\delta _{{{ tk}}}$ ($0 < {t} < \frac {4}{3}{d}$) via weighted $\ell _{1}$ minimization is weaker than the condition required by classical $\ell _{1}$ minimization without weighting. Our weighted $\ell _{1}$ minimization gives better recovery error bounds in noisy setting. Similarly, the new recovery condition in terms of $\delta _{{{ tk}}}$ ($0 < {t} < \frac {4}{3}{d}$) is extended to weighted $\ell _{{p}}$ ($0 < {p} < 1$) minimization, and it is also weaker than the condition obtained by standard non-convex $\ell _{{p}}$ ($0 < {p} < 1$) minimization without weighting. Numerical illustrations are provided to demonstrate our new theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the recovery of $k$ -sparse signals using the weighted $\ell _{p}$ ($0< p \leqslant 1$) minimization when some partial prior information on the support is available. First, we present a unified analysis of restricted isometry constant $\delta _{tk}$ with $d< t \leqslant 2d$ ($d \geqslant 1$ is determined by the prior support information) for sparse signal recovery by the weighted $\ell _{p}$ ($0< p \leqslant 1$) minimization in both noiseless and noisy settings. This result fills a vacancy on $\delta _{tk}$ with $t< 2$ , compared with previous works on $\delta _{(a+1)k}$ ($a>1$). Second, we provide a sufficient condition on $\delta _{tk}$ with $1< t \leqslant 2$ for the recovery of sparse signals using the $\ell _{p}$ ($0< p \leqslant 1$) 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 $\ell _{p}$ ($0< p \leqslant 1$) minimization is achieved when the accuracy of prior information on the support is at least 50%, compared with that of the $\ell _{p}$ ($0< p \leqslant 1$) minimization. [ABSTRACT FROM AUTHOR]