The key aim of compressed sensing is to stably recover a k-sparse signals \({\textbf{x}}\) from a linear model \(\mathbf {y=Ax+v}\) , where \({\textbf{v}}\) is a noise vector. In this paper, we consider \(\ell _{p}\) - \(\ell _{q}\) minimization for sparse signal recovery, where \(0<p\le 1\) and \(1<q\le 2\) . Specifically, we consider sparse signal recovery conditions under \(\ell _{2}\) -bound noise and \(\ell _{\infty }\) -bound noise, respectively. First, we derive a new sufficient condition of stable recovery of \({\textbf{x}}\) based on RIP (restricted isometry property) of order 2k via \(\ell _{p}\) - \(\ell _{q}\) minimization. Second, we derive the high order RIP with tk for some \(t\ge 3\) to guarantee signal recovery via \(\ell _{p}\) - \(\ell _{q}\) minimization.
