{"title":"关于Epsilon-nets的一些新边界","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":"{\"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}","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}
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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