Recovery of signals under the high order RIP condition via prior support information

In this paper we study the recovery conditions of weighted l1 minimization for signal reconstruction from incomplete linear measurements when partial prior support information is available. We obtain that a high order RIP condition can guarantee stable and robust recovery of signals in bounded l2 and Dantzig selector noise settings. Meanwhile, we not only prove that the sufficient recovery condition of weighted l1 minimization method is weaker than that of standard l1 minimization method, but also prove that weighted l1 minimization method provides better upper bounds on the reconstruction error in terms of the measurement noise and the compressibility of the signal, provided that the accuracy of prior support estimate is at least 50%. Furthermore, the condition is proved sharp.