The restricted isometry property of subsampled fourier matrices

Ishay Haviv, Oded Regev

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

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).

Original languageEnglish (US)
Title of host publicationLecture Notes in Mathematics
PublisherSpringer Verlag
Pages163-179
Number of pages17
Volume2169
DOIs
StatePublished - 2017

Publication series

NameLecture Notes in Mathematics
Volume2169
ISSN (Print)00758434

Fingerprint

Isometry
Norm

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

Haviv, I., & Regev, O. (2017). The restricted isometry property of subsampled fourier matrices. In 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.

Lecture Notes in Mathematics. Vol. 2169 Springer Verlag, 2017. p. 163-179 (Lecture Notes in Mathematics; Vol. 2169).

Research output: Chapter in Book/Report/Conference proceedingChapter

Haviv, I & Regev, O 2017, The restricted isometry property of subsampled fourier matrices. in 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
Haviv I, Regev O. The restricted isometry property of subsampled fourier matrices. In Lecture Notes in Mathematics. Vol. 2169. Springer Verlag. 2017. p. 163-179. (Lecture Notes in Mathematics). https://doi.org/10.1007/978-3-319-45282-1_11
Haviv, Ishay ; Regev, Oded. / The restricted isometry property of subsampled fourier matrices. Lecture Notes in Mathematics. Vol. 2169 Springer Verlag, 2017. pp. 163-179 (Lecture Notes in Mathematics).
@inbook{c2f1a4a03d704f309d38019bd85c91d2,
title = "The restricted isometry property of subsampled fourier matrices",
abstract = "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).",
author = "Ishay Haviv and Oded Regev",
year = "2017",
doi = "10.1007/978-3-319-45282-1_11",
language = "English (US)",
volume = "2169",
series = "Lecture Notes in Mathematics",
publisher = "Springer Verlag",
pages = "163--179",
booktitle = "Lecture Notes in Mathematics",
address = "Germany",

}

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 -