Sparse Signal Recovery with Minimization of 1-Norm Minus 2-Norm

The key aim of compressed sensing is to stably recover a $K$ -sparse signals ${\boldsymbol{x}}$ from a linear model ${\boldsymbol{y}}=\boldsymbol{A}{\boldsymbol{x}}+\boldsymbol{v}$ , where $\boldsymbol{v}$ is a noise vector. Minimization of $\Vert {\boldsymbol{x}}\Vert _1-\Vert {\boldsymbol{x}}\Vert _2$ is a recently proposed effective recovery method. In this paper, we show that if the mutual coherence $\mu$ of $\boldsymbol{A}$ satisfies $\mu < \frac{1}{3K}$ , then this method can stably recover any $K$ -sparse signal ${\boldsymbol{x}}$ based on ${\boldsymbol{y}}$ and $\boldsymbol{A}$ . As far as we know, this is the first sufficient condition based on mutual coherence for such method.