An Improved Sufficient Condition for Sparse Signal Recovery With Minimization of L1-L2

The $\ell _{1}-\ell _{2}$-minimization is widely used to stably recover a $K$-sparse signal ${\boldsymbol{x}}$ from its low dimensional measurements ${\boldsymbol{y}}=\boldsymbol{A}{\boldsymbol{x}}+\boldsymbol{v}$, where $\boldsymbol{A}$ is a measurement matrix and $\boldsymbol{v}$ is a noise vector. In this paper, we show that if the mutual coherence $\mu$ of $\boldsymbol{A}$ satisfies $\mu < \frac{4K-1- \sqrt{8K+1}}{\text{8}\;K^{2}-8\;K}$, then any $K$-sparse signal ${\boldsymbol{x}}$ can be stably recovered via the $\ell _{1}-\ell _{2}$-minimization. As far as we know, this is the best mutual coherence based sufficient condition of stably recovering $K$-sparse signals with the $\ell _{1}-\ell _{2}$-minimization.