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
nonzero entries in
dimension
given
random linear measurements of
that signal. This is a massive improvement over previous results,
which require
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. I
NTRODUCTION
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 481091043 USA. He is now with Applied and Com
putational Mathematics, MC 21750, The California Institute of Technology,
Pasadena, CA 91125 USA (email: [email protected]).
A. C. Gilbert is with the Department of Mathematics, The University of
Michigan, Ann Arbor, MI 481091043 USA (email: [email protected]).
Communicated
by
A.
HøstMadsen,
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.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 RunyiYu
 Probability theory, Greedy algorithm, OMP, signal recovery

Click to edit the document details