{"title":"用Lewis权值进行Lp行抽样","authors":"Michael B. Cohen, Richard Peng","doi":"10.1145/2746539.2746567","DOIUrl":null,"url":null,"abstract":"We give a simple algorithm to efficiently sample the rows of a matrix while preserving the p-norms of its product with vectors. Given an n * d matrix A, we find with high probability and in input sparsity time an A' consisting of about d log d rescaled rows of A such that |Ax|1 is close to |A'x|1 for all vectors x. We also show similar results for all Lp that give nearly optimal sample bounds in input sparsity time. Our results are based on sampling by \"Lewis weights\", which can be viewed as statistical leverage scores of a reweighted matrix. We also give an elementary proof of the guarantees of this sampling process for L1.","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":"21 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2015-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"93","resultStr":"{\"title\":\"Lp Row Sampling by Lewis Weights\",\"authors\":\"Michael B. Cohen, Richard Peng\",\"doi\":\"10.1145/2746539.2746567\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We give a simple algorithm to efficiently sample the rows of a matrix while preserving the p-norms of its product with vectors. Given an n * d matrix A, we find with high probability and in input sparsity time an A' consisting of about d log d rescaled rows of A such that |Ax|1 is close to |A'x|1 for all vectors x. We also show similar results for all Lp that give nearly optimal sample bounds in input sparsity time. Our results are based on sampling by \\\"Lewis weights\\\", which can be viewed as statistical leverage scores of a reweighted matrix. We also give an elementary proof of the guarantees of this sampling process for L1.\",\"PeriodicalId\":20566,\"journal\":{\"name\":\"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing\",\"volume\":\"21 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2015-06-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"93\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1145/2746539.2746567\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/2746539.2746567","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
We give a simple algorithm to efficiently sample the rows of a matrix while preserving the p-norms of its product with vectors. Given an n * d matrix A, we find with high probability and in input sparsity time an A' consisting of about d log d rescaled rows of A such that |Ax|1 is close to |A'x|1 for all vectors x. We also show similar results for all Lp that give nearly optimal sample bounds in input sparsity time. Our results are based on sampling by "Lewis weights", which can be viewed as statistical leverage scores of a reweighted matrix. We also give an elementary proof of the guarantees of this sampling process for L1.
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