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.ICALP.2020.64
URN: urn:nbn:de:0030-drops-124711
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12471/
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Har-Peled, Sariel ; Jones, Mitchell ; Rahul, Saladi

Active Learning a Convex Body in Low Dimensions

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LIPIcs-ICALP-2020-64.pdf (0.7 MB)

Abstract

Consider a set P ⊆ ℝ^d of n points, and a convex body C provided via a separation oracle. The task at hand is to decide for each point of P if it is in C using the fewest number of oracle queries. We show that one can solve this problem in two and three dimensions using O(⬡_P log n) queries, where ⬡_P is the largest subset of points of P in convex position. In 2D, we provide an algorithm which efficiently generates these adaptive queries.
Furthermore, we show that in two dimensions one can solve this problem using O(⊚(P,C) log² n) oracle queries, where ⊚(P,C) is a lower bound on the minimum number of queries that any algorithm for this specific instance requires. Finally, we consider other variations on the problem, such as using the fewest number of queries to decide if C contains all points of P.
As an application of the above, we show that the discrete geometric median of a point set P in ℝ² can be computed in O(n log² n (log n log log n + ⬡(P))) expected time.

BibTeX - Entry

@InProceedings{harpeled_et_al:LIPIcs:2020:12471,
  author =	{Sariel Har-Peled and Mitchell Jones and Saladi Rahul},
  title =	{{Active Learning a Convex Body in Low Dimensions}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{64:1--64:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Artur Czumaj and Anuj Dawar and Emanuela Merelli},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12471},
  URN =		{urn:nbn:de:0030-drops-124711},
  doi =		{10.4230/LIPIcs.ICALP.2020.64},
  annote =	{Keywords: Approximation algorithms, computational geometry, separation oracles, active learning}
}

Keywords: Approximation algorithms, computational geometry, separation oracles, active learning
Collection: 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)
Issue Date: 2020
Date of publication: 29.06.2020

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