{"title":"关于实值布尔函数的最大 L1 影响","authors":"Andrew J. Young, Henry D. Pfister","doi":"arxiv-2406.10772","DOIUrl":null,"url":null,"abstract":"We show that any sequence of well-behaved (e.g. bounded and non-constant)\nreal-valued functions of $n$ boolean variables $\\{f_n\\}$ admits a sequence of\ncoordinates whose $L^1$ influence under the $p$-biased distribution, for any\n$p\\in(0,1)$, is $\\Omega(\\text{var}(f_n) \\frac{\\ln n}{n})$.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"346 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the maximal L1 influence of real-valued boolean functions\",\"authors\":\"Andrew J. Young, Henry D. Pfister\",\"doi\":\"arxiv-2406.10772\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We show that any sequence of well-behaved (e.g. bounded and non-constant)\\nreal-valued functions of $n$ boolean variables $\\\\{f_n\\\\}$ admits a sequence of\\ncoordinates whose $L^1$ influence under the $p$-biased distribution, for any\\n$p\\\\in(0,1)$, is $\\\\Omega(\\\\text{var}(f_n) \\\\frac{\\\\ln n}{n})$.\",\"PeriodicalId\":501216,\"journal\":{\"name\":\"arXiv - CS - Discrete Mathematics\",\"volume\":\"346 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-06-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - CS - Discrete Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2406.10772\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - CS - Discrete Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2406.10772","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
On the maximal L1 influence of real-valued boolean functions
We show that any sequence of well-behaved (e.g. bounded and non-constant)
real-valued functions of $n$ boolean variables $\{f_n\}$ admits a sequence of
coordinates whose $L^1$ influence under the $p$-biased distribution, for any
$p\in(0,1)$, is $\Omega(\text{var}(f_n) \frac{\ln n}{n})$.
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