Recovery of Sparse Signals via Generalized Orthogonal Matching Pursuit: A New Analysis

Abstract: As an extension of orthogonal matching pursuit (OMP) improving the recovery performance of sparse signals, generalized OMP (gOMP) has recently been studied in the literature. In this paper, we present a new analysis of the gOMP algorithm using restricted isometry property (RIP). We show that if the measurement matrix $\mathbf{\Phi} \in \mathcal{R}^{m \times n}$ satisfies the RIP with $$\delta_{\max \left{9, S + 1 \right}K} \leq \frac{1}{8},$$ then gOMP performs stable reconstruction of all $K$-sparse signals $\mathbf{x} \in \mathcal{R}^n$ from the noisy measurements $\mathbf{y} = \mathbf{\Phi x} + \mathbf{v}$ within $\max \left{K, \left\lfloor \frac{8K}{S} \right\rfloor \right}$ iterations where $\mathbf{v}$ is the noise vector and $S$ is the number of indices chosen in each iteration of the gOMP algorithm. For Gaussian random measurements, our results indicate that the number of required measurements is essentially $m = \mathcal{O}(K \log \frac{n}{K})$, which is a significant improvement over the existing result $m = \mathcal{O}(K^2 \log \frac{n}{K})$, especially for large $K$.