An Improved Sufficient Condition for Weighted $\ell _{r}-\ell _{1}$ Minimization

The weighted$\ell _{r}-\ell _{1}$minimization with weight$\alpha$has been extensively employed to robustly estimate a high-dimensional sparse signal$x$coded by the underdetermined linear measurements$y=Ax+z$, where$A$and$z$are the measurement matrix and noise, respectively. In this paper, we demonstrate that if the restricted isometry constant (RIC)$\delta _{s}$of$A$fulfills\begin{align*} \delta _{s}< 1/\left(1+3t/\sqrt{5}\right), \end{align*}where$t$relies on sparsity level$s$for known model parameters$\alpha$and$r$, then any sparse signal$x$are ensured to be robustly reconstructed through solving the weighted$\ell _{r}-\ell _{1}$minimization in the noisy situation. The gained condition is testified to be much better that the state-of-art ones.