### Abstract

A matrix A ∈ C^{q×N} satisfies the restricted isometry property of order k with constant e if it preserves the l_{2} norm of all κ-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 TV) rows from an N × N Fourier matrix satisfies the restricted isometry property of order k with a fixed e 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 | 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016 |

Editors | Robert Krauthgamer |

Publisher | Association for Computing Machinery |

Pages | 288-297 |

Number of pages | 10 |

ISBN (Electronic) | 9781510819672 |

DOIs | |

State | Published - 2016 |

Event | 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016 - Arlington, United States Duration: Jan 10 2016 → Jan 12 2016 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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Volume | 1 |

### Other

Other | 27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016 |
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Country | United States |

City | Arlington |

Period | 1/10/16 → 1/12/16 |

### ASJC Scopus subject areas

- Software
- Mathematics(all)

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## Cite this

Haviv, I., & Regev, O. (2016). The restricted isometry property of subsampled Fourier matrices. In R. Krauthgamer (Ed.),

*27th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016*(pp. 288-297). (Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms; Vol. 1). Association for Computing Machinery. https://doi.org/10.1137/1.9781611974331.ch22