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

William B. Johnson, Assaf Naor

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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:X → F, where F ⊆ X 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 2O(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 2 Ω(α(n)), where α is the inverse Ackermann function.

Original languageEnglish (US)
Title of host publicationProceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms
Pages885-891
Number of pages7
StatePublished - 2009
Event20th Annual ACM-SIAM Symposium on Discrete Algorithms - New York, NY, United States
Duration: Jan 4 2009Jan 6 2009

Other

Other20th Annual ACM-SIAM Symposium on Discrete Algorithms
CountryUnited States
CityNew York, NY
Period1/4/091/6/09

Fingerprint

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

ASJC Scopus subject areas

  • Software
  • Mathematics(all)

Cite this

Johnson, W. B., & Naor, A. (2009). The Johnson-Lindenstrauss lemma almost characterizes Hilbert space, but not quite. In Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms (pp. 885-891)

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

Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms. 2009. p. 885-891.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Johnson, WB & Naor, A 2009, The Johnson-Lindenstrauss lemma almost characterizes Hilbert space, but not quite. in Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms. pp. 885-891, 20th Annual ACM-SIAM Symposium on Discrete Algorithms, New York, NY, United States, 1/4/09.
Johnson WB, Naor A. The Johnson-Lindenstrauss lemma almost characterizes Hilbert space, but not quite. In Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms. 2009. p. 885-891
Johnson, William B. ; Naor, Assaf. / The Johnson-Lindenstrauss lemma almost characterizes Hilbert space, but not quite. Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms. 2009. pp. 885-891
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