Embedding the diamond graph in Lp and dimension reduction in L1

James R. Lee, Assaf Naor

Research output: Contribution to journalArticle

Abstract

We show that any embedding of the level k diamond graph of Newman and Rabinovich [NR] into Lp, 1 < p ≤ 2, requires distortion at least √k(p - 1) + 1. An immediate corollary is that there exist arbitrarily large n-point sets X ⊆ L1 such that any D-embedding of X into ℓ1d requires d ≥ n Ω(1/D2). This gives a simple proof of a recent result of Brinkman and Charikar [BrC] which settles the long standing question of whether there is an L1 analogue of the Johnson-Lindenstrauss dimension reduction lemma [JL].

Original languageEnglish (US)
Pages (from-to)745-747
Number of pages3
JournalGeometric and Functional Analysis
Volume14
Issue number4
DOIs
StatePublished - 2004

Fingerprint

Dimension Reduction
Strombus or kite or diamond
Graph in graph theory
Point Sets
Lemma
Corollary
Analogue

ASJC Scopus subject areas

  • Mathematics(all)
  • Analysis

Cite this

Embedding the diamond graph in Lp and dimension reduction in L1. / Lee, James R.; Naor, Assaf.

In: Geometric and Functional Analysis, Vol. 14, No. 4, 2004, p. 745-747.

Research output: Contribution to journalArticle

Lee, James R. ; Naor, Assaf. / Embedding the diamond graph in Lp and dimension reduction in L1. In: Geometric and Functional Analysis. 2004 ; Vol. 14, No. 4. pp. 745-747.
@article{f8b2876bce5240d1aef551a2c0a73ab0,
title = "Embedding the diamond graph in Lp and dimension reduction in L1",
abstract = "We show that any embedding of the level k diamond graph of Newman and Rabinovich [NR] into Lp, 1 < p ≤ 2, requires distortion at least √k(p - 1) + 1. An immediate corollary is that there exist arbitrarily large n-point sets X ⊆ L1 such that any D-embedding of X into ℓ1d requires d ≥ n Ω(1/D2). This gives a simple proof of a recent result of Brinkman and Charikar [BrC] which settles the long standing question of whether there is an L1 analogue of the Johnson-Lindenstrauss dimension reduction lemma [JL].",
author = "Lee, {James R.} and Assaf Naor",
year = "2004",
doi = "10.1007/s00039-004-0473-8",
language = "English (US)",
volume = "14",
pages = "745--747",
journal = "Geometric and Functional Analysis",
issn = "1016-443X",
publisher = "Birkhauser Verlag Basel",
number = "4",

}

TY - JOUR

T1 - Embedding the diamond graph in Lp and dimension reduction in L1

AU - Lee, James R.

AU - Naor, Assaf

PY - 2004

Y1 - 2004

N2 - We show that any embedding of the level k diamond graph of Newman and Rabinovich [NR] into Lp, 1 < p ≤ 2, requires distortion at least √k(p - 1) + 1. An immediate corollary is that there exist arbitrarily large n-point sets X ⊆ L1 such that any D-embedding of X into ℓ1d requires d ≥ n Ω(1/D2). This gives a simple proof of a recent result of Brinkman and Charikar [BrC] which settles the long standing question of whether there is an L1 analogue of the Johnson-Lindenstrauss dimension reduction lemma [JL].

AB - We show that any embedding of the level k diamond graph of Newman and Rabinovich [NR] into Lp, 1 < p ≤ 2, requires distortion at least √k(p - 1) + 1. An immediate corollary is that there exist arbitrarily large n-point sets X ⊆ L1 such that any D-embedding of X into ℓ1d requires d ≥ n Ω(1/D2). This gives a simple proof of a recent result of Brinkman and Charikar [BrC] which settles the long standing question of whether there is an L1 analogue of the Johnson-Lindenstrauss dimension reduction lemma [JL].

UR - http://www.scopus.com/inward/record.url?scp=4744370894&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4744370894&partnerID=8YFLogxK

U2 - 10.1007/s00039-004-0473-8

DO - 10.1007/s00039-004-0473-8

M3 - Article

VL - 14

SP - 745

EP - 747

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

SN - 1016-443X

IS - 4

ER -