文章基本信息

标题：Active Learning a Convex Body in Low Dimensions
本地全文：下载
作者：Sariel Har-Peled ; Mitchell Jones ; Saladi Rahul 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2020
卷号：168
页码：64:1-64:17
DOI：10.4230/LIPIcs.ICALP.2020.64
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：Consider a set P âS â"^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(â¬¡_P log n) queries, where â¬¡_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(âSS(P,C) logÂ² n) oracle queries, where âSS(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 â"Â² can be computed in O(n logÂ² n (log n log log n + â¬¡(P))) expected time.
关键词：Approximation algorithms; computational geometry; separation oracles; active learning