A Novel Sufficient Condition for Generalized Orthogonal Matching Pursuit

Generalized orthogonal matching pursuit (gOMP), also called orthogonal multi-matching pursuit, is an extension of OMP in the sense that $N\geq1$ indices are identified per iteration. In this paper, we show that if the restricted isometry constant (RIC) $δ_{NK+1}$ of a sensing matrix $\A$ satisfies $δ_{NK+1} < 1/\sqrt {K/N+1}$, then under a condition on the signal-to-noise ratio, gOMP identifies at least one index in the support of any $K$-sparse signal $\x$ from $\y=\A\x+\v$ at each iteration, where $\v$ is a noise vector. Surprisingly, this condition does not require $N\leq K$ which is needed in Wang, \textit{et al} 2012 and Liu, \textit{et al} 2012. Thus, $N$ can have more choices. When $N=1$, it reduces to be a sufficient condition for OMP, which is less restrictive than that proposed in Wang 2015. Moreover, in the noise-free case, it is a sufficient condition for accurately recovering $\x$ in $K$ iterations which is less restrictive than the best known one. In particular, it reduces to the sharp condition proposed in Mo 2015 when $N=1$.