Given the measurement matrix $A$ and the observation signal $y$, the central purpose of compressed sensing is to find the most sparse solution of the underdetermined linear system $y=Ax+z$, where $x$ is the $s$-sparse signal to be recovered and $z$ is the noise vector. Zhou and Yu \cite{Zhou and Yu 2019} recently proposed a novel non-convex weighted $\ell_r-\ell_1$ minimization method for effective sparse recovery. In this paper, we reveal that based on $(y,A)$, any $s$-sparse signal can be robustly reconstructed via this method provided that the mutual coherence $\mu$ of $A$ fulfills $\mu<1/(s-1+2^{1/r-1}s^{1/r})$. To our best of knowledge, this is the first mutual coherence based sufficient condition for such approach.
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