A sharp bound on RIC in generalized orthogonal matching pursuit

Abstract: 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) $\delta_{NK+1}$ of the sensing matrix $A$ satisfies \begin{eqnarray*} \delta_{NK+1}<\frac{1}{\sqrt{\frac{K}{N}+1}}, \end{eqnarray*} 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 \begin{eqnarray*} \delta_{NK+1}=\frac{1}{\sqrt{\frac{K}{N}+1}} \end{eqnarray*} 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.