A New Look at Generalized Orthogonal Matching Pursuit: Stable Signal Recovery under Measurement Noise

Generalized orthogonal matching pursuit (gOMP) is an extension of orthogonal matching pursuit (OMP) algorithm designed to improve the recovery performance of sparse signals. In this paper, we provide a new analysis for the gOMP algorithm for both noiseless and noisy scenarios. We show that if the measurement matrix Φ ∈ R satisfies the restricted isometry property (RIP) with δ7K+N−1 ≤ 0.0231, then gOMP can perfectly recover any K-sparse signal x ∈ R from the measurements y = Φx within ⌈ 6K N ⌉ iterations (N is the number of indices chosen in each iteration). We also show that if Φ satisfies the RIP with δ11K+N−1 < 0.0627, then gOMP can perform a stable recovery of K-sparse signal x from the noisy measurements y = Φx+v within ⌈ 10K N ⌉ iterations. For Gaussian random measurements, the results indicate that the required measurement size is m = O(K log( n K )), which is much smaller than the existing result m = O(K log( n K )).