k-Sparse Vector Recovery via \(\ell _1-\alpha \ell _2\) Local Minimization

This paper studies the \(\ell _1-\alpha \ell _2\) ${\ell }_1-\alpha {\ell }_2$ local minimization model for \(\alpha \in (0,2]\) $\alpha \in (0,2]$ , which is the first time to consider the case of \(\alpha >1\) $\alpha >1$ . We obtain the necessary and sufficient conditions for a fixed sparse signal to be recovered from this model. Based on this condition, we also obtain the necessary and sufficient conditions for any k-sparse signal to be recovered from \(\ell _1-\alpha \ell _2\) ${\ell }_1-\alpha {\ell }_2$ local minimization model with \(0<\alpha <1\) $0<\alpha <1$ , \(\alpha =1\) $\alpha =1$ and \(1<\alpha \le 2\) $1<\alpha \le 2$ . The experimental data show that the size of \(\alpha \) $\alpha$ is positively correlated with the success rate of signal recovery.