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 1 2010

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Keywords

  • Dimension reduction
  • Johnson-Lindenstrauss lemma

ASJC Scopus subject areas

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

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