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

TL;DR

This paper provides a new RIP-based analysis of generalized orthogonal matching pursuit (gOMP), showing improved measurement bounds for stable sparse signal recovery, especially for large sparsity levels.

Contribution

It introduces a novel RIP-based analysis of gOMP, demonstrating improved measurement bounds and stable recovery guarantees for sparse signals.

Findings

gOMP performs stable reconstruction under RIP with δ ≤ 1/8

Number of measurements needed is O(K log(n/K)) for Gaussian matrices

Significant improvement over previous O(K^2 log(n/K)) bounds

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 $K$.