Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH scholarly article en Larsen, Kasper Green; Nelson, Jelani https://www.dagstuhl.de/lipics License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
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URN: urn:nbn:de:0030-drops-62032
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The Johnson-Lindenstrauss Lemma Is Optimal for Linear Dimensionality Reduction

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Abstract

For any n > 1, 0 < epsilon < 1/2, and N > n^C for some constant C > 0, we show the existence of an N-point subset X of l_2^n such that any linear map from X to l_2^m with distortion at most 1 + epsilon must have m = Omega(min{n, epsilon^{-2}*lg(N)). This improves a lower bound of Alon [Alon, Discre. Mathem., 1999], in the linear setting, by a lg(1/epsilon) factor. Our lower bound matches the upper bounds provided by the identity matrix and the Johnson-Lindenstrauss lemma [Johnson and Lindenstrauss, Contem. Mathem., 1984].

BibTeX - Entry

@InProceedings{larsen_et_al:LIPIcs:2016:6203,
  author =	{Kasper Green Larsen and Jelani Nelson},
  title =	{{The Johnson-Lindenstrauss Lemma Is Optimal for Linear Dimensionality Reduction}},
  booktitle =	{43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)},
  pages =	{82:1--82:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-013-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{55},
  editor =	{Ioannis Chatzigiannakis and Michael Mitzenmacher and Yuval Rabani and Davide Sangiorgi},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6203},
  URN =		{urn:nbn:de:0030-drops-62032},
  doi =		{10.4230/LIPIcs.ICALP.2016.82},
  annote =	{Keywords: dimensionality reduction, lower bounds, Johnson-Lindenstrauss}
}

Keywords: dimensionality reduction, lower bounds, Johnson-Lindenstrauss
Seminar: 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)
Issue date: 2016
Date of publication: 23.08.2016


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