k-sparse vector recovery via Truncated $\ell _1-\ell _2$ local minimization

Abstract

This article mainly solves the following model,

$$\begin{aligned} min\ \Vert x_{\Gamma ^C_{x,t}}\Vert _1-\Vert x_{\Gamma ^C_{x,t}}\Vert _2\qquad subject \quad to \qquad Ax=y, \end{aligned}$$

where $\Gamma _{x,t}\subset [n]$ represents the index of the maximum number of t elements in x after taking the absolute value. We call this model Truncated $\ell _1-\ell _2$ model. We mainly deal with the recovery of unknown signals under the condition of $|supp(x)|>t$, $\sigma _t(x)>\sigma _{t+1}(x)$, where $\sigma _t(x)$ represents the t largest number in |x|. Firstly, we give the necessary and sufficient condition for recovering the fixed unknown signal satisfying the above two conditions via Truncated $\ell _1-\ell _2$ local minimization. Then, according to this condition, we give the necessary and sufficient conditions to recovering for all unknown signals satisfying the above two conditions via Truncated $\ell _1-\ell _2$ local minimization. Compared with N. Bi’s recent proposed condition in Bi and Tang (Appl Comput Harmon Anal 56:337–350, 2022), we will show that our condition is weaker and the detail of such discussion is in Remark 3 of the manuscript. Then, we give the algorithm of Truncated $\ell _1-\ell _2$ model. According to this algorithm, we do data experiments and the data experiments show that the recovery rate of Truncated $\ell _1-\ell _2$ is better than that of model $\ell _1-\ell _2$.