k-Sparse Vector Recovery via ℓℓLocal Minimization

Shaohua Xie, Jia Li, Kaihao Liang

Abstract: This paper studies the \(\ell _1-\alpha \ell _2\) local minimization model for \(\alpha \in (0,2]\), which is the first time to consider the case of \(\alpha >1\). 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 \(\ell _1-\alpha \ell _2\) local minimization model with \(0<\alpha <1\), \(\alpha =1\) and \(1<\alpha \le 2\). The experimental data show that the size of \(\alpha \) is positively correlated with the success rate of signal recovery.