{"title":"Some new bounds for Epsilon-nets","authors":"J. Pach, G. Woeginger","doi":"10.1145/98524.98529","DOIUrl":null,"url":null,"abstract":"Given any natural number <italic>d</italic>, 0 < <italic>ε</italic> < 1, let ƒ<italic><subscrpt>d</subscrpt></italic>(<italic>ε</italic>) denote the smallest integer ƒ such that every range space of Vapnik-Chervonenkis dimension <italic>d</italic> has an <italic>ε</italic>-net of size at most ƒ We solve a problem of Haussler and Welzl by showing that if <italic>d</italic> ≥ 2, then ƒ<italic><subscrpt>d</subscrpt></italic>(<italic>ε</italic>) > 1/48 <italic>d</italic>/<italic>ε</italic> log 1/ <italic>ε</italic> which is not far from being optimal, if <italic>d</italic> is fixed and <italic>ε</italic> → 0. Further, we prove that ƒ<subscrpt>1</subscrpt>(<italic>ε</italic>) = max(2,⌈1/<italic>ε</italic>⌉ - 1), and similar bounds are established for some special classes of range spaces of Vapnik-Chervonenkis dimension three.","PeriodicalId":113850,"journal":{"name":"SCG '90","volume":"16 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"1990-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"44","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"SCG '90","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/98524.98529","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 44
Abstract
Given any natural number d, 0 < ε < 1, let ƒd(ε) denote the smallest integer ƒ such that every range space of Vapnik-Chervonenkis dimension d has an ε-net of size at most ƒ We solve a problem of Haussler and Welzl by showing that if d ≥ 2, then ƒd(ε) > 1/48 d/ε log 1/ ε which is not far from being optimal, if d is fixed and ε → 0. Further, we prove that ƒ1(ε) = max(2,⌈1/ε⌉ - 1), and similar bounds are established for some special classes of range spaces of Vapnik-Chervonenkis dimension three.
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