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Author Rudelson, Mark ♦ Vershynin, Roman
Source ACM Digital Library
Content type Text
Publisher Association for Computing Machinery (ACM)
File Format PDF
Copyright Year ©2007
Language English
Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Subject Keyword Monte-Carlo methods ♦ Randomized algorithms ♦ Low-rank approximations ♦ Massive data sets ♦ Singular-value decompositions
Abstract We study random submatrices of a large matrix $\textit{A}.$ We show how to approximately compute $\textit{A}$ from its random submatrix of the smallest possible size $\textit{O}(\textit{r}log$ $\textit{r})$ with a small error in the spectral norm, where $\textit{r}$ = $‖A‖^{2}_{F}/‖A‖^{2}_{2}$ is the numerical rank of $\textit{A}.$ The numerical rank is always bounded by, and is a stable relaxation of, the rank of $\textit{A}.$ This yields an asymptotically optimal guarantee in an algorithm for computing low-rank approximations of $\textit{A}.$ We also prove asymptotically optimal estimates on the spectral norm and the $\textit{cut-norm}$ of random submatrices of $\textit{A}.$ The result for the cut-norm yields a slight improvement on the best-known sample complexity for an approximation algorithm for MAX-2CSP problems. We use methods of Probability in Banach spaces, in particular the law of large numbers for operator-valued random variables.
ISSN 00045411
Age Range 18 to 22 years ♦ above 22 year
Educational Use Research
Education Level UG and PG
Learning Resource Type Article
Publisher Date 2007-07-01
Publisher Place New York
e-ISSN 1557735X
Journal Journal of the ACM (JACM)
Volume Number 54
Issue Number 4

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Source: ACM Digital Library