Near optimal bound of orthogonal matching pursuit using restricted isometric constant

As a paradigm for reconstructing sparse signals using a set of under sampled measurements, compressed sensing has received much attention in recent years. In identifying the sufficient condition under which the perfect recovery of sparse signals is ensured, a property of the sensing matrix referred to as the restricted isometry property (RIP) is popularly employed. In this article, we propose the RIP based bound of the orthogonal matching pursuit (OMP) algorithm guaranteeing the exact reconstruction of sparse signals. Our proof is built on an observation that the general step of the OMP process is in essence the same as the initial step in the sense that the residual is considered as a new measurement preserving the sparsity level of an input vector. Our main conclusion is that if the restricted isometry constant δ_K of the sensing matrix satisfies δ_K < √ K-1/√K-1+K then the OMP algorithm can perfectly recover K(> 1)-sparse signals from measurements. We show that our bound is sharp and indeed close to the limit conjectured by Dai and Milenkovic.