Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH scholarly article en Deshpande, Amit; Harsha, Prahladh; Venkat, Rakesh http://www.dagstuhl.de/lipics License
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Embedding Approximately Low-Dimensional l_2^2 Metrics into l_1

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Abstract

Goemans showed that any n points x_1,..., x_n in d-dimensions satisfying l_2^2 triangle inequalities can be embedded into l_{1}, with worst-case distortion at most sqrt{d}. We consider an extension of this theorem to the case when the points are approximately low-dimensional as opposed to exactly low-dimensional, and prove the following analogous theorem, albeit with average distortion guarantees: There exists an l_{2}^{2}-to-l_{1} embedding with average distortion at most the stable rank, sr(M), of the matrix M consisting of columns {x_i-x_j}_{i<j}. Average distortion embedding suffices for applications such as the SPARSEST CUT problem. Our embedding gives an approximation algorithm for the SPARSEST CUT problem on low threshold-rank graphs, where earlier work was inspired by Lasserre SDP hierarchy, and improves on a previous result of the first and third author [Deshpande and Venkat, in Proc. 17th APPROX, 2014]. Our ideas give a new perspective on l_{2}^{2} metric, an alternate proof of Goemans' theorem, and a simpler proof for average distortion sqrt{d}.

BibTeX - Entry

@InProceedings{deshpande_et_al:LIPIcs:2016:6845,
  author =	{Amit Deshpande and Prahladh Harsha and Rakesh Venkat},
  title =	{{Embedding Approximately Low-Dimensional l_2^2 Metrics into l_1}},
  booktitle =	{36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)},
  pages =	{10:1--10:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-027-9},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{65},
  editor =	{Akash Lal and S. Akshay and Saket Saurabh and Sandeep Sen},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6845},
  URN =		{urn:nbn:de:0030-drops-68456},
  doi =		{10.4230/LIPIcs.FSTTCS.2016.10},
  annote =	{Keywords: Metric Embeddings, Sparsest Cut, Negative type metrics, Approximation Algorithms}
}

Keywords: Metric Embeddings, Sparsest Cut, Negative type metrics, Approximation Algorithms
Seminar: 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016)
Issue date: 2016
Date of publication: 2016


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