Improved Bounds on RIP for Generalized Orthogonal Matching Pursuit

TL;DR

This paper improves the theoretical bounds on the restricted isometry property (RIP) needed for generalized orthogonal matching pursuit (gOMP) to successfully recover sparse signals, advancing understanding of its performance guarantees.

Contribution

The paper provides new, tighter RIP bounds for gOMP, extending the theoretical analysis and improving upon recent results in sparse signal recovery.

Findings

gOMP can recover signals under less restrictive RIP conditions

New bounds relate gOMP's performance to known bounds for OMP

Improved theoretical guarantees for sparse signal reconstruction

Abstract

Generalized Orthogonal Matching Pursuit (gOMP) is a natural extension of OMP algorithm where unlike OMP, it may select $N (\geq1)$ atoms in each iteration. In this paper, we demonstrate that gOMP can successfully reconstruct a $K$-sparse signal from a compressed measurement $ {\bf y}={\bf \Phi x}$ by $K^{th}$ iteration if the sensing matrix ${\bf \Phi}$ satisfies restricted isometry property (RIP) of order $NK$ where $\delta_{NK} < \frac {\sqrt{N}}{\sqrt{K}+2\sqrt{N}}$. Our bound offers an improvement over the very recent result shown in \cite{wang_2012b}. Moreover, we present another bound for gOMP of order $NK+1$ with $\delta_{NK+1} < \frac {\sqrt{N}}{\sqrt{K}+\sqrt{N}}$ which exactly relates to the near optimal bound of $\delta_{K+1} < \frac {1}{\sqrt{K}+1}$ for OMP (N=1) as shown in \cite{wang_2012a}.