Small ensembles of sampling matrices constructed from coding theory

Alexander Barg, Arya Mazumdar

Research output: Chapter in Book/Report/Conference proceedingConference contribution

3 Scopus citations

Abstract

In the area of compressed sensing it is known that high-dimensional, approximately sparse signals x ∈ ℝN can be accurately recovered from a small number m of linear samples. Most known recovery procedures rely in some form on a special property of the sampling matrix, called the Restricted Isometry Property (RIP). It has been shown that random Gaussian matrices possess the RIP and allow an efficient reconstruction of the signal, for instance, by linear programming or by other methods, and similar results are known for the ensembles of random binary matrices. We pursue a link between coding theory and compressed sensing, discussing the construction problem of sampling matrices with short sketches. The best known constructions of "small-bias probability spaces" (aka codes with narrow distance distribution) account for deterministic sampling matrices with sketch length m = O(k3 log N/log k) while the best randomized constructions require only m = Ω(klog(N/k)) samples, where k is the number of essential entries of the signal. We address the construction problem of sampling matrices in the region between these two extremes, allowing the sketch length of order k 2 log N. We show that in this case it is possible to construct ensembles of sampling matrices of size much smaller than the ensembles known previously. For instance, the number of random bits required to construct a matrix in one of our ensembles equals O(k2 log k log N), which compares favorably with the previously known estimate of O(k N log(N/k)) random bits for ensembles of random binary matrices.

Original languageEnglish (US)
Title of host publication2010 IEEE International Symposium on Information Theory, ISIT 2010 - Proceedings
Pages1963-1967
Number of pages5
DOIs
StatePublished - Aug 23 2010
Event2010 IEEE International Symposium on Information Theory, ISIT 2010 - Austin, TX, United States
Duration: Jun 13 2010Jun 18 2010

Publication series

NameIEEE International Symposium on Information Theory - Proceedings
ISSN (Print)2157-8103

Other

Other2010 IEEE International Symposium on Information Theory, ISIT 2010
Country/TerritoryUnited States
CityAustin, TX
Period6/13/106/18/10

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