{"title":"稀疏无关子空间嵌入的下界","authors":"Yi Li, Mingmou Liu","doi":"10.1145/3517804.3526224","DOIUrl":null,"url":null,"abstract":"An oblivious subspace embedding (OSE), characterized by parameters m,n,d,ε,δ, is a random matrix Π ∈ Rm x n such that for any d-dimensional subspace T ⊆ Rn, PrΠ[◨x ∈ T, (1-ε)|x|2 ≤ |Π x|2 ≤ (1+ε)|x|2] ≥ 1-δ. For ε and δ at most a small constant, we show that any OSE with one nonzero entry in each column must satisfy that m = Ω(d2/(ε2δ)), establishing the optimality of the classical Count-Sketch matrix. When an OSE has 1/(9ε) nonzero entries in each column, we show it must hold that m = Ω(εO(δ) d2), improving on the previous Ω(ε2 d2) lower bound due to Nelson and Nguyen (ICALP 2014).","PeriodicalId":230606,"journal":{"name":"Proceedings of the 41st ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems","volume":"356 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2021-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"Lower Bounds for Sparse Oblivious Subspace Embeddings\",\"authors\":\"Yi Li, Mingmou Liu\",\"doi\":\"10.1145/3517804.3526224\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"An oblivious subspace embedding (OSE), characterized by parameters m,n,d,ε,δ, is a random matrix Π ∈ Rm x n such that for any d-dimensional subspace T ⊆ Rn, PrΠ[◨x ∈ T, (1-ε)|x|2 ≤ |Π x|2 ≤ (1+ε)|x|2] ≥ 1-δ. For ε and δ at most a small constant, we show that any OSE with one nonzero entry in each column must satisfy that m = Ω(d2/(ε2δ)), establishing the optimality of the classical Count-Sketch matrix. When an OSE has 1/(9ε) nonzero entries in each column, we show it must hold that m = Ω(εO(δ) d2), improving on the previous Ω(ε2 d2) lower bound due to Nelson and Nguyen (ICALP 2014).\",\"PeriodicalId\":230606,\"journal\":{\"name\":\"Proceedings of the 41st ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems\",\"volume\":\"356 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2021-12-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the 41st ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1145/3517804.3526224\",\"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 41st ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/3517804.3526224","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Lower Bounds for Sparse Oblivious Subspace Embeddings
An oblivious subspace embedding (OSE), characterized by parameters m,n,d,ε,δ, is a random matrix Π ∈ Rm x n such that for any d-dimensional subspace T ⊆ Rn, PrΠ[◨x ∈ T, (1-ε)|x|2 ≤ |Π x|2 ≤ (1+ε)|x|2] ≥ 1-δ. For ε and δ at most a small constant, we show that any OSE with one nonzero entry in each column must satisfy that m = Ω(d2/(ε2δ)), establishing the optimality of the classical Count-Sketch matrix. When an OSE has 1/(9ε) nonzero entries in each column, we show it must hold that m = Ω(εO(δ) d2), improving on the previous Ω(ε2 d2) lower bound due to Nelson and Nguyen (ICALP 2014).
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