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APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2020.45
Learning Lines with Ordinal Constraints

Authors: Bohan Fan, Diego Ihara, Neshat Mohammadi, Francesco Sgherzi, Anastasios Sidiropoulos, and Mina Valizadeh

Published in: LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

Abstract
We study the problem of finding a mapping f from a set of points into the real line, under ordinal triple constraints. An ordinal constraint for a triple of points (u,v,w) asserts that |f(u)-f(v)| < |f(u)-f(w)|. We present an approximation algorithm for the dense case of this problem. Given an instance that admits a solution that satisfies (1-ε)-fraction of all constraints, our algorithm computes a solution that satisfies (1-O(ε^{1/8}))-fraction of all constraints, in time O(n⁷) + (1/ε)^{O(1/ε^{1/8})} n.
Cite as

Bohan Fan, Diego Ihara, Neshat Mohammadi, Francesco Sgherzi, Anastasios Sidiropoulos, and Mina Valizadeh. Learning Lines with Ordinal Constraints. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 45:1-45:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{fan_et_al:LIPIcs.APPROX/RANDOM.2020.45,
  author =	{Fan, Bohan and Ihara, Diego and Mohammadi, Neshat and Sgherzi, Francesco and Sidiropoulos, Anastasios and Valizadeh, Mina},
  title =	{{Learning Lines with Ordinal Constraints}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
  pages =	{45:1--45:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-164-1},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{176},
  editor =	{Byrka, Jaros{\l}aw and Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.45},
  URN =		{urn:nbn:de:0030-drops-126486},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2020.45},
  annote =	{Keywords: metric learning, embedding into the line, ordinal constraints, approximation algorithms}
}
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