Sufficient Conditions for CS-recovery
Hiroshi Inoue *
Graduate School of Mathematics, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan.
*Author to whom correspondence should be addressed.
Abstract
In this paper we define the k-restrictly norm constant rk(A) of a matrix A to be used in compressed
sensing and give better error estimations on recovering compressive signals with noise using the
matrix A~ _ A
rk(A) . Furthermore, we define the notion of k-restricted invertibility of A, which is
equivalent to that A~ _ A=rk(A) obeys the RIP of order k. And by using the Q. Mo and S. Li
idea and T. Cai and A. Zhang idea, we establish the sufficient condition for the restricted isometry
constant _~k (k _ s) of A~ under the assumption that A is k-restrictly invertible. In particular, if
~_s < 0:5 and ~_2s < 0:828, then an unknown compressive signal with noise can be recovered.
Keywords: Compressed sensing, Restricted norm constants, Restricted invertible, Restricted isometry constants, Restricted isometry property, Sparse approximation, Sparse signal recovery.