The Johnson-Lindenstrauss lemma almost characterizes Hilbert space, but not quite

William B. Johnson, Assaf Naor

Research output: Contribution to journalArticle

Abstract

Let X be a normed space that satisfies the Johnson-Lindenstrauss lemma (J-L lemma, in short) in the sense that for any integer n and any x1,...,xn∈X, there exists a linear mapping L:X→F, where F⊆X is a linear subspace of dimension O(log n), such that {double pipe}xi-xj{double pipe}≤{double pipe}L(xi)-L(xj){double pipe}≤O(1){dot operator}{double pipe}xi-xj{double pipe} for all i,j∈{1,...,n}. We show that this implies that X is almost Euclidean in the following sense: Every n-dimensional subspace of X embeds into Hilbert space with distortion,. On the other hand, we show that there exists a normed space Y which satisfies the J-L lemma, but for every n, there exists an n-dimensional subspace En⊆Y whose Euclidean distortion is at least 2Ω(α(n)), where α is the inverse Ackermann function.

Original languageEnglish (US)
Pages (from-to)542-553
Number of pages12
JournalDiscrete and Computational Geometry
Volume43
Issue number3
DOIs
StatePublished - Apr 2010

Fingerprint

Hilbert spaces
Lemma
Hilbert space
Pipe
Subspace
Normed Space
n-dimensional
Euclidean
Inverse function
Imply
Integer
Operator

Keywords

  • Dimension reduction
  • Johnson-Lindenstrauss lemma

ASJC Scopus subject areas

  • Computational Theory and Mathematics
  • Discrete Mathematics and Combinatorics
  • Geometry and Topology
  • Theoretical Computer Science

Cite this

The Johnson-Lindenstrauss lemma almost characterizes Hilbert space, but not quite. / Johnson, William B.; Naor, Assaf.

In: Discrete and Computational Geometry, Vol. 43, No. 3, 04.2010, p. 542-553.

Research output: Contribution to journalArticle

Johnson, William B. ; Naor, Assaf. / The Johnson-Lindenstrauss lemma almost characterizes Hilbert space, but not quite. In: Discrete and Computational Geometry. 2010 ; Vol. 43, No. 3. pp. 542-553.
@article{45dc5b1fbce24d09aafa7ade5c9ba454,
title = "The Johnson-Lindenstrauss lemma almost characterizes Hilbert space, but not quite",
abstract = "Let X be a normed space that satisfies the Johnson-Lindenstrauss lemma (J-L lemma, in short) in the sense that for any integer n and any x1,...,xn∈X, there exists a linear mapping L:X→F, where F⊆X is a linear subspace of dimension O(log n), such that {double pipe}xi-xj{double pipe}≤{double pipe}L(xi)-L(xj){double pipe}≤O(1){dot operator}{double pipe}xi-xj{double pipe} for all i,j∈{1,...,n}. We show that this implies that X is almost Euclidean in the following sense: Every n-dimensional subspace of X embeds into Hilbert space with distortion,. On the other hand, we show that there exists a normed space Y which satisfies the J-L lemma, but for every n, there exists an n-dimensional subspace En⊆Y whose Euclidean distortion is at least 2Ω(α(n)), where α is the inverse Ackermann function.",
keywords = "Dimension reduction, Johnson-Lindenstrauss lemma",
author = "Johnson, {William B.} and Assaf Naor",
year = "2010",
month = "4",
doi = "10.1007/s00454-009-9193-z",
language = "English (US)",
volume = "43",
pages = "542--553",
journal = "Discrete and Computational Geometry",
issn = "0179-5376",
publisher = "Springer New York",
number = "3",

}

TY - JOUR

T1 - The Johnson-Lindenstrauss lemma almost characterizes Hilbert space, but not quite

AU - Johnson, William B.

AU - Naor, Assaf

PY - 2010/4

Y1 - 2010/4

N2 - Let X be a normed space that satisfies the Johnson-Lindenstrauss lemma (J-L lemma, in short) in the sense that for any integer n and any x1,...,xn∈X, there exists a linear mapping L:X→F, where F⊆X is a linear subspace of dimension O(log n), such that {double pipe}xi-xj{double pipe}≤{double pipe}L(xi)-L(xj){double pipe}≤O(1){dot operator}{double pipe}xi-xj{double pipe} for all i,j∈{1,...,n}. We show that this implies that X is almost Euclidean in the following sense: Every n-dimensional subspace of X embeds into Hilbert space with distortion,. On the other hand, we show that there exists a normed space Y which satisfies the J-L lemma, but for every n, there exists an n-dimensional subspace En⊆Y whose Euclidean distortion is at least 2Ω(α(n)), where α is the inverse Ackermann function.

AB - Let X be a normed space that satisfies the Johnson-Lindenstrauss lemma (J-L lemma, in short) in the sense that for any integer n and any x1,...,xn∈X, there exists a linear mapping L:X→F, where F⊆X is a linear subspace of dimension O(log n), such that {double pipe}xi-xj{double pipe}≤{double pipe}L(xi)-L(xj){double pipe}≤O(1){dot operator}{double pipe}xi-xj{double pipe} for all i,j∈{1,...,n}. We show that this implies that X is almost Euclidean in the following sense: Every n-dimensional subspace of X embeds into Hilbert space with distortion,. On the other hand, we show that there exists a normed space Y which satisfies the J-L lemma, but for every n, there exists an n-dimensional subspace En⊆Y whose Euclidean distortion is at least 2Ω(α(n)), where α is the inverse Ackermann function.

KW - Dimension reduction

KW - Johnson-Lindenstrauss lemma

UR - http://www.scopus.com/inward/record.url?scp=77950521029&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77950521029&partnerID=8YFLogxK

U2 - 10.1007/s00454-009-9193-z

DO - 10.1007/s00454-009-9193-z

M3 - Article

VL - 43

SP - 542

EP - 553

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 3

ER -