A Remark on the Restricted Isometry Property in Orthogonal Matching Pursuit

• Published in: IEEE transactions on information theory Vol. 58; no. 6; pp. 3654 - 3656
• Main Authors: Mo, Qun, Shen, Yi
• Format: Journal Article
• Language: English
• Published: New York IEEE 01-06-2012; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Compressed sensing; Construction; Educational institutions; Greedy algorithms; Indexes; Information theory; Linear algebra; Matching; Matching pursuit algorithms; Matrix; Minimization; orthogonal matching pursuit; restricted isometry property; Signal reconstruction; sparse signal reconstruction; Vectors
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2012.2185923

This paper demonstrates that if the restricted isometry constant δ K +1 of the measurement matrix A satisfies [δ K +1 <; 1 √K+1] then a greedy algorithm called Orthogonal Matching Pursuit (OMP) can recover every K-sparse signal x in K iterations from Ax. By contrast, a matrix is also constructed with the restricted isometry constant [δ K +1 = 1 √K] such that OMP can not recover some K-sparse signal x in K iterations. This result positively verifies the conjecture given by Dai and Milenkovic in 2009.