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A Simple Proof of the Restricted Isometry Property for Random Matrices

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Abstract

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.

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Authors and Affiliations

  1. Department of Electrical and Computer Engineering, Rice University, MS-380, 6100 Main Street, Houston, TX, 77005, USA

    Richard Baraniuk & Mark Davenport

  2. Industrial Mathematics Institute, Department of Mathematics and Statistics, University of South Carolina, Columbia, SC, 29208, USA

    Ronald DeVore

  3. Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, 48109, USA

    Michael Wakin

Authors
  1. Richard Baraniuk
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  2. Mark Davenport
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  3. Ronald DeVore
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  4. Michael Wakin
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Corresponding author

Correspondence to Ronald DeVore.

Additional information

Communicated by Emmanuel J. Candès.

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Baraniuk, R., Davenport, M., DeVore, R. et al. A Simple Proof of the Restricted Isometry Property for Random Matrices. Constr Approx 28, 253–263 (2008). https://doi.org/10.1007/s00365-007-9003-x

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  • DOI: https://doi.org/10.1007/s00365-007-9003-x

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