### Abstract

A matrix A ∈ C^{q×N} satisfies the restricted isometry property of order k with constant ∈ if it preserves the l_{2} norm of all k-sparse vectors up to a factor of 1 ± ϵ. We prove that a matrix A obtained by randomly sampling q = O(k log^{2} k log N) rows from an N × N Fourier matrix satisfies the restricted isometry property of order k with a fixed ∈ with high probability. This improves on Rudelson and Vershynin (Comm Pure Appl Math, 2008), its subsequent improvements, and Bourgain (GAFA Seminar Notes, 2014).

Original language | English (US) |
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Title of host publication | Lecture Notes in Mathematics |

Publisher | Springer Verlag |

Pages | 163-179 |

Number of pages | 17 |

Volume | 2169 |

DOIs | |

State | Published - 2017 |

### Publication series

Name | Lecture Notes in Mathematics |
---|---|

Volume | 2169 |

ISSN (Print) | 00758434 |

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### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Lecture Notes in Mathematics*(Vol. 2169, pp. 163-179). (Lecture Notes in Mathematics; Vol. 2169). Springer Verlag. https://doi.org/10.1007/978-3-319-45282-1_11

**The restricted isometry property of subsampled fourier matrices.** / Haviv, Ishay; Regev, Oded.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*Lecture Notes in Mathematics.*vol. 2169, Lecture Notes in Mathematics, vol. 2169, Springer Verlag, pp. 163-179. https://doi.org/10.1007/978-3-319-45282-1_11

}

TY - CHAP

T1 - The restricted isometry property of subsampled fourier matrices

AU - Haviv, Ishay

AU - Regev, Oded

PY - 2017

Y1 - 2017

N2 - A matrix A ∈ Cq×N satisfies the restricted isometry property of order k with constant ∈ if it preserves the l2 norm of all k-sparse vectors up to a factor of 1 ± ϵ. We prove that a matrix A obtained by randomly sampling q = O(k log2 k log N) rows from an N × N Fourier matrix satisfies the restricted isometry property of order k with a fixed ∈ with high probability. This improves on Rudelson and Vershynin (Comm Pure Appl Math, 2008), its subsequent improvements, and Bourgain (GAFA Seminar Notes, 2014).

AB - A matrix A ∈ Cq×N satisfies the restricted isometry property of order k with constant ∈ if it preserves the l2 norm of all k-sparse vectors up to a factor of 1 ± ϵ. We prove that a matrix A obtained by randomly sampling q = O(k log2 k log N) rows from an N × N Fourier matrix satisfies the restricted isometry property of order k with a fixed ∈ with high probability. This improves on Rudelson and Vershynin (Comm Pure Appl Math, 2008), its subsequent improvements, and Bourgain (GAFA Seminar Notes, 2014).

UR - http://www.scopus.com/inward/record.url?scp=85018503011&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85018503011&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-45282-1_11

DO - 10.1007/978-3-319-45282-1_11

M3 - Chapter

AN - SCOPUS:85018503011

VL - 2169

T3 - Lecture Notes in Mathematics

SP - 163

EP - 179

BT - Lecture Notes in Mathematics

PB - Springer Verlag

ER -