TG07-Signal-Recovery - IEEE TRANSACTIONS ON INFORMATION...

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Unformatted text preview: IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 53, NO. 12, DECEMBER 2007 4655 Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit Joel A. Tropp , Member, IEEE , and Anna C. Gilbert Abstract— This paper demonstrates theoretically and empiri- cally that a greedy algorithm called Orthogonal Matching Pursuit (OMP) can reliably recover a signal with m nonzero entries in dimension d given O( m ln d ) random linear measurements of that signal. This is a massive improvement over previous results, which require O( m P ) measurements. The new results for OMP are comparable with recent results for another approach called Basis Pursuit (BP). In some settings, the OMP algorithm is faster and easier to implement, so it is an attractive alternative to BP for signal recovery problems. Index Terms— Algorithms, approximation, basis pursuit, com- pressed sensing, group testing, orthogonal matching pursuit, signal recovery, sparse approximation. I. INTRODUCTION L ET be a-dimensional real signal with at most non- zero components. This type of signal is called-sparse. Let be a sequence of measurement vectors in that does not depend on the signal. We use these vectors to col- lect linear measurements of the signal where denotes the usual inner product. The problem of signal recovery asks two distinct questions. 1) How many measurements are necessary to reconstruct the signal? 2) Given these measurements, what algorithms can perform the reconstruction task? As we will see, signal recovery is dual to sparse approximation, a problem of significant interest [1]–[5]. To the first question, we can immediately respond that no fewer than measurements will do. Even if the measurements were adapted to the signal, it would still take pieces of in- formation to determine the nonzero components of an-sparse Manuscript received April 20, 2005; revised August 15, 2007. The work of J. A. Tropp was supported by the National Science Foundation under Grant DMS 0503299. The work of A. C. Gilbert was supported by the National Sci- ence Foundation under Grant DMS 0354600. J. A. Tropp was with the Department of Mathematics, The University of Michigan, Ann Arbor, MI 48109-1043 USA. He is now with Applied and Com- putational Mathematics, MC 217-50, The California Institute of Technology, Pasadena, CA 91125 USA (e-mail: jtropp@acm.caltech.edu). A. C. Gilbert is with the Department of Mathematics, The University of Michigan, Ann Arbor, MI 48109-1043 USA (e-mail: annacg@umich.edu). Communicated by A. Høst-Madsen, Associate Editor for Detection Estimation. Color versions of Figures 1–6 in this paper are available online at http://iee- explore.ieee.org. Digital Object Identifier 10.1109/TIT.2007.909108 signal. In the other direction, nonadaptive measurements al- ways suffice because we could simply list the components of the signal. Although it is not obvious, sparse signals can be re- constructed with far less information....
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TG07-Signal-Recovery - IEEE TRANSACTIONS ON INFORMATION...

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