In this paper, we focus on ${\ell }_1-{\ell }_2$ minimization model, i.e., investigating the nonconvex model:$\min {\Vert x\Vert }_1-{\Vert x\Vert }_2\mathit{\text{s.t.}}Ax=y$ and provide a null space property of the measurement matrix A such that a vector x can be recovered from Ax via ${\ell }_1-{\ell }_2$ minimization. The ${\ell }_1-{\ell }_2$ minimization model was first proposed by E.Esser, et al (2013) [8]. As a nonconvex model, it is well known that global minimizer and local minimizer are usually inconsistent. In this paper, we present a necessary and sufficient condition for the measurement matrix A such that (1) a vector x can be recovered from Ax via ${\ell }_1-{\ell }_2$ local minimization (Theorem 4); (2) any k-sparse vector x can be recovered from Ax via ${\ell }_1-{\ell }_2$ local minimization (Theorem 5); (3) any k-sparse vector x can be recovered from Ax via ${\ell }_1-{\ell }_2$ global minimization (Theorem 6).

在本文中，我们专注于 ${\ell }_1-{\ell }_2$ 最小化模型，即研究非凸模型：$\mathrm{分钟}{\Vert X\Vert }_1-{\Vert X\Vert }_2\mathit{\text{英石}}\mathrm{一种}X=是$并提供测量矩阵的零空间属性甲使得向量X可从被回收斧经由${\ell }_1-{\ell }_2$最小化。这${\ell }_1-{\ell }_2$最小化模型首先由 E.Esser 等人 (2013) [8] 提出。作为非凸模型，众所周知，全局最小化器和局部最小化器通常是不一致的。在本文中，我们提出了一个充分必要条件测量矩阵甲使得（1）的向量X可从被回收斧经由${\ell }_1-{\ell }_2$局部最小化（定理 4）；（2）任何ķ -sparse矢量X可以从回收的斧经由${\ell }_1-{\ell }_2$局部最小化（定理 5）；（3）任何ķ -sparse矢量X可以从回收的斧经由${\ell }_1-{\ell }_2$ 全局最小化（定理 6）。