Improving the Bound on the RIP Constant in Generalized Orthogonal Matching Pursuit

The generalized Orthogonal Matching Pursuit (gOMP) is a recently proposed compressive sensing greedy recovery algorithm which generalizes the OMP algorithm by selecting N(>= 1) atoms in each iteration. In this letter, we demonstrate that the gOMP can successfully reconstruct a K-sparse signal from a compressed measurement y = Phi x by a maximum of K iterations if the sensing matrix Phi satisfies the Restricted Isometry Property (RIP) of order NK, with the RIP constant delta(NK) satisfying delta(NK) < root N/root K+2 root N. The proposed bound is an improvement over the existing bound on delta(NK). We also show that by increasing the RIP order just by one (i.e., NK+1 from NK), it is possible to refine the bound further to delta(NK+1) < root N/root K+root N, which is consistent (for N = 1) with the near optimal bound on delta(K+1) in OMP.