License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2016.16
URN: urn:nbn:de:0030-drops-59081
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/5908/
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Bartal, Yair ; Gottlieb, Lee-Ad

Dimension Reduction Techniques for l_p (1<p<2), with Applications

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LIPIcs-SoCG-2016-16.pdf (0.6 MB)

Abstract

For Euclidean space (l_2), there exists the powerful dimension reduction transform of Johnson and Lindenstrauss [Conf. in modern analysis and probability, AMS 1984], with a host of known applications. Here, we consider the problem of dimension reduction for all l_p spaces 1<p<2. Although strong lower bounds are known for dimension reduction in l_1, Ostrovsky and Rabani [JACM 2002] successfully circumvented these by presenting an l_1 embedding that maintains fidelity in only a bounded distance range, with applications to clustering and nearest neighbor search. However, their embedding techniques are specific to l_1 and do not naturally extend to other norms.

In this paper, we apply a range of advanced techniques and produce bounded range dimension reduction embeddings for all of 1<p<2, thereby demonstrating that the approach initiated by Ostrovsky and Rabani for l_1 can be extended to a much more general framework. We also obtain improved bounds in terms of the intrinsic dimensionality. As a result we achieve improved bounds for proximity problems including snowflake embeddings and clustering.

BibTeX - Entry

@InProceedings{bartal_et_al:LIPIcs:2016:5908,
  author =	{Yair Bartal and Lee-Ad Gottlieb},
  title =	{{Dimension Reduction Techniques for l_p (1<p<2), with Applications}},
  booktitle =	{32nd International Symposium on Computational Geometry (SoCG 2016)},
  pages =	{16:1--16:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-009-5},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{51},
  editor =	{S{\'a}ndor Fekete and Anna Lubiw},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/5908},
  URN =		{urn:nbn:de:0030-drops-59081},
  doi =		{10.4230/LIPIcs.SoCG.2016.16},
  annote =	{Keywords: Dimension reduction, embeddings}
}

Keywords: Dimension reduction, embeddings
Collection: 32nd International Symposium on Computational Geometry (SoCG 2016)
Issue Date: 2016
Date of publication: 10.06.2016

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