A Sharp RIP Condition for Orthogonal Matching Pursuit

Abstract

A restricted isometry property (RIP) condition is known to be sufficient for orthogonal matching pursuit (OMP) to exactly recover every K-sparse signal x from measurements $y=\mathrm{\Phi }x$. This paper is devoted to demonstrate that this condition is sharp. We construct a specific matrix with ${\delta }_K+\sqrt{K}{\theta }_{K,1}=1$ such that OMP cannot exactly recover some K-sparse signals.