A Simple Proof of the Restricted Isometry Property for Random Matrices

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A Simple Proof of the Restricted Isometry Property for Random Matrices (2008)

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by Richard Baraniuk , Mark Davenport , Ronald DeVore , Michael Wakin

Venue:	CONSTR APPROX
Citations:	625 - 64 self

BibTeX

@MISC{Baraniuk08asimple,
    author = {Richard Baraniuk and Mark Davenport and Ronald DeVore and Michael Wakin},
    title = { A Simple Proof of the Restricted Isometry Property for Random Matrices},
    year = {2008}
}

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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.

Keyphrases

restricted isometry property simple proof random matrix johnson lindenstrauss lemma euclidean space random inner product certain universality direct proof main ingredient simple technique finite-dimensional ball concentration inequality sparsity-inducing basis measurement matrix finite ball elementary proof

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