An Improved Sufficient Condition for Sparse Signal Recovery With Minimization of L1-L2
Zihao He, Hongyu He, Xiaoli Liu, Jinming Wen
Abstract
The <inline-formula><tex-math notation="LaTeX">$\ell _{1}-\ell _{2}$</tex-math></inline-formula>-minimization is widely used to stably recover a <inline-formula><tex-math notation="LaTeX">$K$</tex-math></inline-formula>-sparse signal <inline-formula><tex-math notation="LaTeX">${\boldsymbol{x}}$</tex-math></inline-formula> from its low dimensional measurements <inline-formula><tex-math notation="LaTeX">${\boldsymbol{y}}=\boldsymbol{A}{\boldsymbol{x}}+\boldsymbol{v}$</tex-math></inline-formula>, where <inline-formula><tex-math notation="LaTeX">$\boldsymbol{A}$</tex-math></inline-formula> is a measurement matrix and <inline-formula><tex-math notation="LaTeX">$\boldsymbol{v}$</tex-math></inline-formula> is a noise vector. In this paper, we show that if the mutual coherence <inline-formula><tex-math notation="LaTeX">$\mu$</tex-math></inline-formula> of <inline-formula><tex-math notation="LaTeX">$\boldsymbol{A}$</tex-math></inline-formula> satisfies <inline-formula><tex-math notation="LaTeX">$\mu < \frac{4K-1- \sqrt{8K+1}}{\text{8}\;K^{2}-8\;K}$</tex-math></inline-formula>, then any <inline-formula><tex-math notation="LaTeX">$K$</tex-math></inline-formula>-sparse signal <inline-formula><tex-math notation="LaTeX">${\boldsymbol{x}}$</tex-math></inline-formula> can be stably recovered via the <inline-formula><tex-math notation="LaTeX">$\ell _{1}-\ell _{2}$</tex-math></inline-formula>-minimization. As far as we know, this is the best mutual coherence based sufficient condition of stably recovering <inline-formula><tex-math notation="LaTeX">$K$</tex-math></inline-formula>-sparse signals with the <inline-formula><tex-math notation="LaTeX">$\ell _{1}-\ell _{2}$</tex-math></inline-formula>-minimization.