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Discrete & Computational Geometry

, Volume 43, Issue 3, pp 542–553 | Cite as

The Johnson–Lindenstrauss Lemma Almost Characterizes Hilbert Space, But Not Quite

  • William B. Johnson
  • Assaf Naor
Article

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 x 1,…,x n X, there exists a linear mapping L:XF, where FX is a linear subspace of dimension O(log n), such that ‖x i x j ‖≤‖L(x i )−L(x j )‖≤O(1)⋅‖x i x j ‖ 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 \(2^{2^{O(\log^{*}n)}}\) . 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 E n Y whose Euclidean distortion is at least 2Ω(α(n)), where α is the inverse Ackermann function.

Keywords

Dimension reduction Johnson–Lindenstrauss lemma 

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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Texas A&M UniversityCollege StationUSA
  2. 2.Courant InstituteNew YorkUSA

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