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# A sharp bound on RIC in generalized orthogonal matching pursuit

Published:2017-04-17
Printed: Mar 2018
• Wengu Chen,
Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China
• Huanmin Ge,
Generalized orthogonal matching pursuit (gOMP) algorithm has received much attention in recent years as a natural extension of orthogonal matching pursuit (OMP). 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 $\delta_{NK+1}\lt \frac{1}{\sqrt{\frac{K}{N}+1}}$, then the gOMP can perfectly recover every $K$-sparse signal $x$ from $y=Ax$. Furthermore, the bound is proved to be sharp. In the noisy case, the above bound on RIC combining with an extra condition on the minimum magnitude of the nonzero components of $K$-sparse signals can guarantee that the gOMP selects all of support indices of the $K$-sparse signals.