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

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Venue: | CONSTR APPROX |

Citations: | 303 - 56 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.

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Citation Context ...cing basis. 1 Introduction It has recently become clear that matrices and projections generated by certain random processes provide solutions to a number of fundamental questions in signal processing =-=[2, 8, 10]-=-. In Compressed Sensing (CS) [2, 8], for example, a random projection of a high-dimensional but sparse or compressible signal vector onto a lower-dimensional space has been shown, with high probabilit... |

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Citation Context ...e replaced by other ℓN p spaces. One of the main problems in CS is to understand which encoder-decoder pairs (Φ, ∆) provide estimates like (3.6). Independently Donoho [8] and Candès, Romberg, and Tao =-=[3]-=- have given sufficient conditions on the matrix Φ for (3.6) to hold. Both approaches show that certain random constructions generate matrices Φ with these properties. Moreover, in both of these settin... |

714 | Approximate nearest neighbors: towards removing the curse of dimensionality
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(Show Context)
Citation Context ...and proof of the JL lemma, as well as simpler and more efficient algorithms for constructing such embeddings, have been developed using elementary concentration inequalities for random inner products =-=[1, 7, 9, 12]-=-. The aims of this paper are twofold. First, we show an intimate linkage between the CS theory, classic results on n-widths, and the JL lemma. Second, we exploit this linkage to provide simple proofs ... |

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Citation Context ...ices Φ with these properties. Moreover, in both of these settings, they show that the decoding can be accomplished by the linear program n (3.6) ∆(y) := argmin �x�ℓN . (3.7) 1 x : Φx=y Candès and Tao =-=[4]-=- introduced the following isometry condition on matrices Φ and established its important role in CS. Given a matrix Φ and any set T of column indices, we denote by ΦT the n × #(T ) matrix composed of ... |

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Citation Context ...avily on random matrix constructions. These same constructions were later applied in the study of point clouds in highdimensional spaces. Specifically, the well-known Johnson-Lindenstrauss (JL) lemma =-=[13]-=- states ∗ This research was supported by Office of Naval Research grants ONR N00014-03-1-0051, ONR/DEPSCoR N00014-03-1-0675, ONR/DEPSCoR N00014-00-1-0470, and ONR N00014-02-1-0353; Army Research Offic... |

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Citation Context ... of points in Q. The study of concentration of measure inequalities like (4.3) is an important subject in probability and analysis. There is now a vast literature on this subject that is described in =-=[15]-=-. The proof of the RIP given in the next section can begin with any random matrices that satisfy (4.3). However, in the case of CS, it is of interest to have specific examples for which one can easily... |

169 |
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Citation Context ...ions have also played a central role in a number of related fields. In particular, random projections have been used as a fundamental tool in the asymptotic theory of finite-dimensional normed spaces =-=[18]-=- and approximation theory [16] since the 1970s. For example, the results of Kashin and Gluskin on n-widths [11, 14] relied heavily on random matrix constructions. These same constructions were later a... |

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Citation Context ...and proof of the JL lemma, as well as simpler and more efficient algorithms for constructing such embeddings, have been developed using elementary concentration inequalities for random inner products =-=[1, 7, 9, 12]-=-. The aims of this paper are twofold. First, we show an intimate linkage between the CS theory, classic results on n-widths, and the JL lemma. Second, we exploit this linkage to provide simple proofs ... |

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Citation Context ...he orthogonal complement of Y and use these basis vectors as the rows of a CS matrix Φ for estimating En,N(K)X. This correspondence between Y and Φ leads easily to the proof of (3.5) (see for example =-=[5]-=- for the simple details). As an example, consider the unit ball U(ℓN 1 ) in ℓN2 . Since U(ℓN1 ) + U(ℓN1 ) ⊂ 2U(ℓN1 ), we have from (2.3) and (3.5) that for all 0 < n < N � � log(N/n) + 1 log(N/n) + 1 ... |

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Citation Context ...and proof of the JL lemma, as well as simpler and more efficient algorithms for constructing such embeddings, have been developed using elementary concentration inequalities for random inner products =-=[1, 7, 9, 12]-=-. The aims of this paper are twofold. First, we show an intimate linkage between the CS theory, classic results on n-widths, and the JL lemma. Second, we exploit this linkage to provide simple proofs ... |

98 |
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Citation Context |

80 | Nearoptimal sparse Fourier representations via sampling
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Citation Context ...cing basis. 1 Introduction It has recently become clear that matrices and projections generated by certain random processes provide solutions to a number of fundamental questions in signal processing =-=[2, 8, 10]-=-. In Compressed Sensing (CS) [2, 8], for example, a random projection of a high-dimensional but sparse or compressible signal vector onto a lower-dimensional space has been shown, with high probabilit... |

66 |
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(Show Context)
Citation Context ... used as a fundamental tool in the asymptotic theory of finite-dimensional normed spaces [18] and approximation theory [16] since the 1970s. For example, the results of Kashin and Gluskin on n-widths =-=[11, 14]-=- relied heavily on random matrix constructions. These same constructions were later applied in the study of point clouds in highdimensional spaces. Specifically, the well-known Johnson-Lindenstrauss (... |

25 |
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(Show Context)
Citation Context ... used as a fundamental tool in the asymptotic theory of finite-dimensional normed spaces [18] and approximation theory [16] since the 1970s. For example, the results of Kashin and Gluskin on n-widths =-=[11, 14]-=- relied heavily on random matrix constructions. These same constructions were later applied in the study of point clouds in highdimensional spaces. Specifically, the well-known Johnson-Lindenstrauss (... |

10 |
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(Show Context)
Citation Context ...process. Gluskin and Garnaev’s improvement used matrices with Gaussian entries. It was not until quite recently that it was understood that the Bernoulli processes also yield the upper bound in (2.3) =-=[17, 19]-=-. The paper [17] also generalizes (2.3) to an arbitrary compact, convex body K ⊂ R N . 3 Compressed Sensing (CS) Similar to n-widths, CS exploits the fact that many signal classes have a low-dimension... |

6 |
Constructive Approximation: Advanced problems, Grundlehren vol. 304
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(Show Context)
Citation Context ...l role in a number of related fields. In particular, random projections have been used as a fundamental tool in the asymptotic theory of finite-dimensional normed spaces [18] and approximation theory =-=[16]-=- since the 1970s. For example, the results of Kashin and Gluskin on n-widths [11, 14] relied heavily on random matrix constructions. These same constructions were later applied in the study of point c... |

3 |
Near optimal approximation of arbitrary signals from highly incomplete measurements
- Cohen, Dahmen, et al.
(Show Context)
Citation Context ...probability 2 3 , � 3 − n with probability 1 6 . Again these matrices satisfy (4.3) with c0(ɛ) = ɛ 2 /4 − ɛ 3 /6 which is proved in [1] and can also be simply derived from Hoeffding’s inequality (see =-=[6]-=-). 5 Verifying the RIP from concentration inequalities We shall now show how the concentration of measure inequality (4.3) can be used together with covering arguments to prove the RIP for random matr... |

1 |
Reconstruction and subgaussian operators in Aymptotic Geometric Analysis
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- 2006
(Show Context)
Citation Context ...process. Gluskin and Garnaev’s improvement used matrices with Gaussian entries. It was not until quite recently that it was understood that the Bernoulli processes also yield the upper bound in (2.3) =-=[17, 19]-=-. The paper [17] also generalizes (2.3) to an arbitrary compact, convex body K ⊂ R N . 3 Compressed Sensing (CS) Similar to n-widths, CS exploits the fact that many signal classes have a low-dimension... |