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

Abstract

Generalized Orthogonal Matching Pursuit (gOMP) is a natural extension of OMP algorithm where unlike OMP, it may select N (≥ 1) atoms in each iteration. In this paper, we demonstrate that gOMP can successfully reconstruct a K-sparse signal from a compressed measurement y = Φx by K th iteration if the sensing matrix Φ satisfies restricted isometry property (RIP) of order N K where δNK < √ N √ K+2 √ N . Our bound offers an improvement over the very recent result shown in [1]. Moreover, we present another bound for gOMP of order N K + 1 with δNK+1 < √ N √ K+ √ N which exactly relates to the near optimal bound of δK+1 < 1 √ K+1 for OMP (N = 1) as shown in [2].

Theoretical results for sparse signal recovery with noises using generalized OMP algorithm

“…which is the exact result obtained in [12,16]. This proves that (5) is in line with the known best sufficient condition for the gOMP algorithm if at least one correct atom is selected in each iteration.…”

“…Þ, is the same as the sufficient condition for the gOMP algorithm in noiseless compressive sensing proposed in [12]. When N¼1, δ K þ 1 o1=ð ffiffiffi ffi K p þ1Þ is the sufficient condition in noiseless case for the OMP algorithm [13,14].…”

A Sharp Bound on RIC in Generalized Orthogonal Matching Pursuit

“…can ensure the reconstruction of any K-sparse signals [16]. Later, Satpathi et al improved [32]. They also refined the bound further to [24] and [25] for N = 1.…”

A New Generalized Orthogonal Matching Pursuit Method

“…The generalized OMP (GOMP) has received increasing attention in recent years, because the method can enhance the recovery performance of OMP. Several papers have been published on the analysis of the theoretical performance of GOMP [13][14][15][16][17].…”