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.