### Sampling from large matrices: An approach through geometric functional analysisSampling from large matrices: An approach through geometric functional analysis

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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