Sofya Raskhodnikova, D. Ron, Amir Shpilka, Adam D. Smith
{"title":"近似分布支撑尺寸的强下界及不同元问题","authors":"Sofya Raskhodnikova, D. Ron, Amir Shpilka, Adam D. Smith","doi":"10.1109/FOCS.2007.67","DOIUrl":null,"url":null,"abstract":"We consider the problem of approximating the support size of a distribution from a small number of samples, when each element in the distribution appears with probability at least 1/n. This problem is closely related to the problem of approximating the number of distinct elements in a sequence of length n. For both problems, we prove a nearly linear in n lower bound on the query complexity, applicable even for approximation with additive error. At the heart of the lower bound is a construction of two positive integer random variables. X<sub>1</sub> and X<sub>2</sub>, with very different expectations and the following condition on the first k moments: E[X<sub>1</sub>]/E[X<sub>2</sub>] = E[X<sub>1</sub> <sup>2</sup>]/E[X<sub>2</sub> <sup>2</sup>] = ... = E[X<sub>1</sub> <sup>k</sup>]/E[X<sub>2</sub> <sup>k</sup>]. Our lower bound method is also applicable to other problems. In particular, it gives new lower bounds for the sample complexity of (1) approximating the entropy of a distribution and (2) approximating how well a given string is compressed by the Lempel-Ziv scheme.","PeriodicalId":197431,"journal":{"name":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2007-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"136","resultStr":"{\"title\":\"Strong Lower Bounds for Approximating Distribution Support Size and the Distinct Elements Problem\",\"authors\":\"Sofya Raskhodnikova, D. Ron, Amir Shpilka, Adam D. Smith\",\"doi\":\"10.1109/FOCS.2007.67\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We consider the problem of approximating the support size of a distribution from a small number of samples, when each element in the distribution appears with probability at least 1/n. This problem is closely related to the problem of approximating the number of distinct elements in a sequence of length n. For both problems, we prove a nearly linear in n lower bound on the query complexity, applicable even for approximation with additive error. At the heart of the lower bound is a construction of two positive integer random variables. X<sub>1</sub> and X<sub>2</sub>, with very different expectations and the following condition on the first k moments: E[X<sub>1</sub>]/E[X<sub>2</sub>] = E[X<sub>1</sub> <sup>2</sup>]/E[X<sub>2</sub> <sup>2</sup>] = ... = E[X<sub>1</sub> <sup>k</sup>]/E[X<sub>2</sub> <sup>k</sup>]. Our lower bound method is also applicable to other problems. In particular, it gives new lower bounds for the sample complexity of (1) approximating the entropy of a distribution and (2) approximating how well a given string is compressed by the Lempel-Ziv scheme.\",\"PeriodicalId\":197431,\"journal\":{\"name\":\"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2007-10-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"136\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1109/FOCS.2007.67\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1109/FOCS.2007.67","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Strong Lower Bounds for Approximating Distribution Support Size and the Distinct Elements Problem
We consider the problem of approximating the support size of a distribution from a small number of samples, when each element in the distribution appears with probability at least 1/n. This problem is closely related to the problem of approximating the number of distinct elements in a sequence of length n. For both problems, we prove a nearly linear in n lower bound on the query complexity, applicable even for approximation with additive error. At the heart of the lower bound is a construction of two positive integer random variables. X1 and X2, with very different expectations and the following condition on the first k moments: E[X1]/E[X2] = E[X12]/E[X22] = ... = E[X1k]/E[X2k]. Our lower bound method is also applicable to other problems. In particular, it gives new lower bounds for the sample complexity of (1) approximating the entropy of a distribution and (2) approximating how well a given string is compressed by the Lempel-Ziv scheme.