{"title":"使用长度平方抽样的乘法秩-1近似","authors":"Ragesh Jaiswal, Amit Kumar","doi":"10.1137/1.9781611976014.4","DOIUrl":null,"url":null,"abstract":"We show that the span of $\\Omega(\\frac{1}{\\varepsilon^4})$ rows of any matrix $A \\subset \\mathbb{R}^{n \\times d}$ sampled according to the length-squared distribution contains a rank-$1$ matrix $\\tilde{A}$ such that $||A - \\tilde{A}||_F^2 \\leq (1 + \\varepsilon) \\cdot ||A - \\pi_1(A)||_F^2$, where $\\pi_1(A)$ denotes the best rank-$1$ approximation of $A$ under the Frobenius norm. Length-squared sampling has previously been used in the context of rank-$k$ approximation. However, the approximation obtained was additive in nature. We obtain a multiplicative approximation albeit only for rank-$1$ approximation.","PeriodicalId":93491,"journal":{"name":"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)","volume":"24 1","pages":"18-23"},"PeriodicalIF":0.0000,"publicationDate":"2019-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Multiplicative Rank-1 Approximation using Length-Squared Sampling\",\"authors\":\"Ragesh Jaiswal, Amit Kumar\",\"doi\":\"10.1137/1.9781611976014.4\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We show that the span of $\\\\Omega(\\\\frac{1}{\\\\varepsilon^4})$ rows of any matrix $A \\\\subset \\\\mathbb{R}^{n \\\\times d}$ sampled according to the length-squared distribution contains a rank-$1$ matrix $\\\\tilde{A}$ such that $||A - \\\\tilde{A}||_F^2 \\\\leq (1 + \\\\varepsilon) \\\\cdot ||A - \\\\pi_1(A)||_F^2$, where $\\\\pi_1(A)$ denotes the best rank-$1$ approximation of $A$ under the Frobenius norm. Length-squared sampling has previously been used in the context of rank-$k$ approximation. However, the approximation obtained was additive in nature. We obtain a multiplicative approximation albeit only for rank-$1$ approximation.\",\"PeriodicalId\":93491,\"journal\":{\"name\":\"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)\",\"volume\":\"24 1\",\"pages\":\"18-23\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2019-09-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1137/1.9781611976014.4\",\"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 SIAM Symposium on Simplicity in Algorithms (SOSA)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1137/1.9781611976014.4","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Multiplicative Rank-1 Approximation using Length-Squared Sampling
We show that the span of $\Omega(\frac{1}{\varepsilon^4})$ rows of any matrix $A \subset \mathbb{R}^{n \times d}$ sampled according to the length-squared distribution contains a rank-$1$ matrix $\tilde{A}$ such that $||A - \tilde{A}||_F^2 \leq (1 + \varepsilon) \cdot ||A - \pi_1(A)||_F^2$, where $\pi_1(A)$ denotes the best rank-$1$ approximation of $A$ under the Frobenius norm. Length-squared sampling has previously been used in the context of rank-$k$ approximation. However, the approximation obtained was additive in nature. We obtain a multiplicative approximation albeit only for rank-$1$ approximation.