Constructive Approximation

, Volume 28, Issue 3, pp 253–263

A Simple Proof of the Restricted Isometry Property for Random Matrices


  • Richard Baraniuk
    • Department of Electrical and Computer EngineeringRice University
  • Mark Davenport
    • Department of Electrical and Computer EngineeringRice University
    • Industrial Mathematics Institute, Department of Mathematics and StatisticsUniversity of South Carolina
  • Michael Wakin
    • Department of Electrical Engineering and Computer ScienceUniversity of Michigan

DOI: 10.1007/s00365-007-9003-x

Cite this article as:
Baraniuk, R., Davenport, M., DeVore, R. et al. Constr Approx (2008) 28: 253. doi:10.1007/s00365-007-9003-x


We give a simple technique for verifying the Restricted Isometry Property (as introduced by Candès and Tao) for random matrices that underlies Compressed Sensing. Our approach has two main ingredients: (i) concentration inequalities for random inner products that have recently provided algorithmically simple proofs of the Johnson–Lindenstrauss lemma; and (ii) covering numbers for finite-dimensional balls in Euclidean space. This leads to an elementary proof of the Restricted Isometry Property and brings out connections between Compressed Sensing and the Johnson–Lindenstrauss lemma. As a result, we obtain simple and direct proofs of Kashin’s theorems on widths of finite balls in Euclidean space (and their improvements due to Gluskin) and proofs of the existence of optimal Compressed Sensing measurement matrices. In the process, we also prove that these measurements have a certain universality with respect to the sparsity-inducing basis.


Compressed sensingSamplingRandom matricesConcentration inequalities

Mathematics Subject Classification (2000)

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© Springer Science+Business Media, LLC 2008