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Active Learning a Convex Body in Low Dimensions

Authors Sariel Har-Peled, Mitchell Jones, Saladi Rahul

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LIPIcs.ICALP.2020.64.pdf
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Author Details

Sariel Har-Peled
  • Department of Computer Science, University of Illinois at Urbana-Champaign, IL, USA
Mitchell Jones
  • Department of Computer Science, University of Illinois at Urbana-Champaign, IL, USA
Saladi Rahul
  • Dept. of Computer Science and Automation, Indian Institute of Science, Bangalore, India

Funding

  • Har-Peled, Sariel: Supported in part by NSF AF award CCF-1907400.
  • Jones, Mitchell: Supported in part by NSF AF award CCF-1907400.

Cite AsGet BibTex

Sariel Har-Peled, Mitchell Jones, and Saladi Rahul. Active Learning a Convex Body in Low Dimensions. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 64:1-64:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ICALP.2020.64

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Approximation algorithms
  • computational geometry
  • separation oracles
  • active learning

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