Unconstrained <inline-formula><tex-math id="M1">\begin{document}$ \ell_1 $\end{document}</tex-math></inline-formula>-<inline-formula><tex-math id="M2">\begin{document}$ \ell_2 $\end{document}</tex-math></inline-formula> minimization for sparse recovery via mutual coherence
Pengbo Geng, Wengu Chen
Abstract
The paradigm of compressed sensing is to exactly or stably recover any sparse signal $ x\in \mathbb{R}^n $ from a small number of linear measurements $ b = Ax+e $, where $ A\in\mathbb{R}^{m\times n} $ with $ m\ll n $ and $ e\in \mathbb{R}^m $ denotes the measurement noise. $ \ell_1 $-$ \ell_2 $ minimization has recently become an effective signal recovery method. In this paper, a mutual coherence based signal recovery guarantee by the unconstrained $ \ell_1 $-$ \ell_2 $ minimization model is given to achieve the stable recovery of any sparse signal $ x $ in the presence of the Dantzig Selector (DS) type noise or the $ \ell_2 $ bounded noise, respectively. To the best of our knowledge, this is the first mutual coherence based sufficient condition to achieve sparse signal recovery via the unconstrained $ \ell_1 $-$ \ell_2 $ minimization.