A New Sufficient Condition for Non-Convex Sparse Recovery via Weighted $\ell _{r}\!-\!\ell _{1}$ Minimization

In this letter, we discuss the reconstruction of sparse signals from undersampled data, which belongs to the core content of compressed sensing. A new sufficient condition in terms of the restricted isometry constant (RIC) and restricted orthogonality constant (ROC) is first established for the performance guarantee of recently proposed non-convex weighted $\ell _{r}-\ell _{1}$ minimization in recovering (approximately) sparse signals that may be polluted by noise. To be specific, it is shown that if the RIC $\delta _{s}$ and ROC $\theta _{s,s}$ of measurement matrix obey \begin{equation*} \delta _{s}+\nu (s)\theta _{s,s}< 1, \end{equation*} where $\nu (s)$ depends on $s$ for given quantities, then any $s$-sparse signals in noiseless setting are guaranteed to be recovered accurately via solving the constrained weighted $\ell _{r}-\ell _{1}$ minimization optimization problem and any (approximately) $s$-sparse signals can be estimated robustly in the noisy case. In addition, we provide several pivotal remarks which indicate the recovery guarantee is much less restricted than the existing one. The results obtained contribute to proving the fidelity of the excellent weighted $\ell _{r}-\ell _{1}$ minimization method.