A sharp bound on RIC in generalized orthogonal matching pursuit

Generalized orthogonal matching pursuit (gOMP) algorithm has received much attention in recent years as a natural extension of orthogonal matching pursuit. It is used to recover sparse signals in compressive sensing. In this paper, a new bound is obtained for the exact reconstruction of every K-sparse signal via the gOMP algorithm in the noiseless case. That is, if the restricted isometry constant (RIC) δNK+1 of the sensing matrix A satisfies δNK+1 < 1 √ K N + 1 , then the gOMP can perfectly recover every K-sparse signal x from y = Ax. Furthermore, the bound is proved to be sharp in the following sense. For any given positive integer K, we construct a matrix A with the RIC δNK+1 = 1 √ K N + 1 such that the gOMP may fail to recover some K-sparse signal x. In the noise case, an extra condition on the minimum magnitude of the nonzero components of every K−sparse signal combining with the above bound on RIC of the sensing matrix A is sufficient to recover the true support of every K-sparse signal by the gOMP. W. Chen is with Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China, e-mail: [email protected]. H. Ge is with Graduate School, China Academy of Engineering Physics, Beijing, 100088, China, email:[email protected]. This work was supported by the NSF of China (Nos.11271050, 11371183) .