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 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.

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Correspondence to Assaf Naor.

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Research of W.B. Johnson supported in part by NSF grants DMS-0503688 and DMS-0528358.

Research of A. Naor supported in part by NSF grants DMS-0528387, CCF-0635078, and CCF-0832795, BSF grant 2006009, and the Packard Foundation.

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Johnson, W.B., Naor, A. The Johnson–Lindenstrauss Lemma Almost Characterizes Hilbert Space, But Not Quite. Discrete Comput Geom 43, 542–553 (2010). https://doi.org/10.1007/s00454-009-9193-z

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Keywords

  • Dimension reduction
  • Johnson–Lindenstrauss lemma