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Fast monte-carlo algorithms for finding low-rank approximations
Abstract
We consider the problem of approximating a given m × n matrix A by another matrix of specified rank k , which is smaller than m and n . The Singular Value Decomposition (SVD) can be used to find the "best" such approximation. However, it takes time polynomial in m, n which is prohibitive for some modern applications. In this article, we develop an algorithm that is qualitatively faster, provided we may sample the entries of the matrix in accordance with a natural probability distribution. In many applications, such sampling can be done efficiently. Our main result is a randomized algorithm to find the description of a matrix D * of rank at most k so that holds with probability at least 1 − δ (where |·| F is the Frobenius norm). The algorithm takes time polynomial in k ,1/ϵ, log(1/δ) only and is independent of m and n . In particular, this implies that in constant time, it can be determined if a given matrix of arbitrary size has a good low-rank approximation.
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Cites
Golub, G. (2012) · doi:10.56021/9781421407944 · Cited by 34,505
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Metadata
- Authors
- Frieze, A., Kannan, R., & Vempala, S.
- Venue
- Journal of the ACM
- Published
- 2004-11
- Volume
- 51
- Issue
- 6
- Pages
- 1025-1041
- ISSN
- 0004-5411 (ISSN-L)
- 1557-735X
- OpenAlex
- W1979750072
- Cited by
- 500
- References
- 1
- FWCI
- 12.24
- Type
- journal article
- Language
- en
- Open access
- closed
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