Active learning a convex body in low dimensions

Har-Peled, S and Jones, M and Rahul, S (2020) Active learning a convex body in low dimensions. In: 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020, 8-11, July 2020, Germany.

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Abstract

Consider a set P � Rd 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(9P log n) queries, where 9P 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 + 9P )) expected time.

Item Type:	Conference Paper
Publication:	Leibniz International Proceedings in Informatics, LIPIcs
Publisher:	Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Additional Information:	The copyright of this article belongs to Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
Keywords:	Robots, Active Learning; Convex body; Expected time; Lower bounds; Point set; Three dimensions; Two-dimension, Automata theory
Department/Centre:	Division of Electrical Sciences > Computer Science & Automation
Date Deposited:	25 Aug 2020 11:59
Last Modified:	25 Aug 2020 11:59
URI:	http://eprints.iisc.ac.in/id/eprint/66369

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