Randomized Algorithms for Low-Rank Matrix Factorizations: Sharp Performance Bounds
The development of randomized algorithms for numerical linear algebra, e.g. for computing approximate QR and SVD factorizations, has recently become an intense area of research. This paper studies one of the most frequently discussed algorithms in the literature for dimensionality reduction—specific...
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| Vydáno v: | Algorithmica Ročník 72; číslo 1; s. 264 - 281 |
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| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
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Springer US
01.05.2015
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| ISSN: | 0178-4617, 1432-0541 |
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| Abstract | The development of randomized algorithms for numerical linear algebra, e.g. for computing approximate QR and SVD factorizations, has recently become an intense area of research. This paper studies one of the most frequently discussed algorithms in the literature for dimensionality reduction—specifically for approximating an input matrix with a low-rank element. We introduce a novel and rather intuitive analysis of the algorithm in [
6
], which allows us to derive sharp estimates and give new insights about its performance. This analysis yields theoretical guarantees about the approximation error and at the same time, ultimate limits of performance (lower bounds) showing that our upper bounds are tight. Numerical experiments complement our study and show the tightness of our predictions compared with empirical observations. |
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| AbstractList | The development of randomized algorithms for numerical linear algebra, e.g. for computing approximate QR and SVD factorizations, has recently become an intense area of research. This paper studies one of the most frequently discussed algorithms in the literature for dimensionality reduction—specifically for approximating an input matrix with a low-rank element. We introduce a novel and rather intuitive analysis of the algorithm in [
6
], which allows us to derive sharp estimates and give new insights about its performance. This analysis yields theoretical guarantees about the approximation error and at the same time, ultimate limits of performance (lower bounds) showing that our upper bounds are tight. Numerical experiments complement our study and show the tightness of our predictions compared with empirical observations. |
| Author | Witten, Rafi Candès, Emmanuel |
| Author_xml | – sequence: 1 givenname: Rafi surname: Witten fullname: Witten, Rafi organization: Bit Body, Inc – sequence: 2 givenname: Emmanuel surname: Candès fullname: Candès, Emmanuel email: candes@stanford.edu organization: Departments of Mathematics and of Statistics, Stanford University |
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| Cites_doi | 10.1137/080736417 10.1016/j.acha.2007.12.002 10.1016/j.acha.2010.02.003 10.1214/aop/1176994775 10.1137/090771806 10.1017/CBO9780511810817 10.1137/040616413 10.1214/aop/1176992819 10.1142/9789814324359_0111 |
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| Keywords | Matrix approximation Dimension reduction Random matrix Randomized linear algebra Pass efficient algorithm |
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| References | Rokhlin, Szlam, Tygert (CR7) 2009; 31 Silverstein (CR10) 1985; 13 Martinsson, Rokhlin, Tygert (CR6) 2011; 30 CR8 Halko, Martinsson, Tropp (CR3) 2011; 53 Geman (CR2) 1980; 8 CR9 Chen, Dongarrar (CR1) 2005; 27 Woolfe, Liberty, Rokhlin, Tygert (CR11) 2008; 25 Horn, Johnson (CR4) 1985 Kent, Mardia, Bibby (CR5) 1976 9891_CR9 F Woolfe (9891_CR11) 2008; 25 JW Silverstein (9891_CR10) 1985; 13 JT Kent (9891_CR5) 1976 9891_CR8 Z Chen (9891_CR1) 2005; 27 RA Horn (9891_CR4) 1985 V Rokhlin (9891_CR7) 2009; 31 N Halko (9891_CR3) 2011; 53 S Geman (9891_CR2) 1980; 8 P-G Martinsson (9891_CR6) 2011; 30 |
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| SubjectTerms | Algorithm Analysis and Problem Complexity Algorithms Computer Science Computer Systems Organization and Communication Networks Data Structures and Information Theory Mathematics of Computing Theory of Computation |
| Title | Randomized Algorithms for Low-Rank Matrix Factorizations: Sharp Performance Bounds |
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