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Embedding Approximately Low-Dimensional l_2^2 Metrics into l_1

Authors Amit Deshpande, Prahladh Harsha, Rakesh Venkat



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Amit Deshpande
Prahladh Harsha
Rakesh Venkat

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Amit Deshpande, Prahladh Harsha, and Rakesh Venkat. Embedding Approximately Low-Dimensional l_2^2 Metrics into l_1. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 65, pp. 10:1-10:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.FSTTCS.2016.10

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}.
Keywords
  • Metric Embeddings
  • Sparsest Cut
  • Negative type metrics
  • Approximation Algorithms

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