Improved bounds on restricted isometry constant for orthogonal multi matching pursuit

Abstract

Orthogonal Multi Matching Pursuit (OMMP) is a super greedy-type algorithm for sparse approximation. We an- alyze the convergence property of OMMP based on Restricted Isometry Property (RIP). Our main conclusion is that if the sampling matrix Φ satisfies the Restricted Isometry Property of order (sK) with isometry constant δ< 1+1 /2 K/s − 1/2 K/s +4 K/s, then OMMP (s) can exactly recover an arbitrary K-sparse signal x from y =Φ x in at most K steps. where y ∈ R M is an observation vector and Φ ∈ R M ×N is a given M × N matrix with M< N ,w e usually call it measurement matrix or sampling matrix. Our goal is to recover x based on Φ and y. In this paper, we study the efficiency of compressed sensing via Orthogonal Matching Multi Pursuit (OMMP). OMMP (s) is a natural generation of Orthogonal Matching Pursuit (OMP) in the sense that it selects s candidates at each step while OMP selects one. Let us begin with the demonstration of the use of super greedy algorithms in the compressed sensing problem. Now let us recall some necessary concepts of CS. Assume that 1 K M N and 0 <δ< 1 .L et||x||0 denotes the number of nonzero coordinates of a signal x =( xj) N=1 ∈ R N , we say that the signal x is K-sparse if ||x||0 K.