Ning Bi, Wai-Shing Tang

In this paper, we focus on ${\ell}_{1}-{\ell}_{2}$ minimization model, i.e., investigating the nonconvex model:$\mathrm{min}{\Vert x\Vert}_{1}-{\Vert x\Vert}_{2}\phantom{\rule{1em}{0ex}}\mathit{\text{s.t.}}\phantom{\rule{1em}{0ex}}Ax=y$ 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).