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Efficient Algorithms for Computing a Strong Rank-Revealing QR Factorization
Abstract
Given an $m \times n$ matrix M with $m \geqslant n$, it is shown that there exists a permutation $\Pi $ and an integer k such that the QR factorization \[ M\Pi = Q\left( {\begin{array}{*{20}c} {A_k } & {B_k } \\ {} & {C_k } \\ \end{array} } \right) \] reveals the numerical rank of M: the $k \times k$ upper-triangular matrix $A_k $ is well conditioned, $\|C_k \|_2 $ is small, and $B_k $is linearly dependent on $A_k $ with coefficients bounded by a low-degree polynomial in n. Existing rank-revealing QR (RRQR) algorithms are related to such factorizations and two algorithms are presented for computing them. The new algorithms are nearly as efficient as QR with column pivoting for most problems and take $O(mn^2 )$ floating-point operations in the worst case.
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Randomized algorithms for the low-rank approximation of matrices
Liberty, E., Woolfe, F., Martinsson, P.-G., Rokhlin, V., & Tygert, M. (2007) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.0709640104 · Cited by 559
A Randomized Algorithm for Principal Component Analysis
Rokhlin, V., Szlam, A., & Tygert, M. (2010) · SIAM Journal on Matrix Analysis and Applications · doi:10.1137/080736417 · Cited by 422
Structured Pruning for Deep Convolutional Neural Networks: A Survey
He, Y., & Xiao, L. (2024) · IEEE Transactions on Pattern Analysis and Machine Intelligence · doi:10.1109/tpami.2023.3334614 · Cited by 363
On the Compression of Low Rank Matrices
Cheng, H., Gimbutas, Z., Martinsson, P. G., & Rokhlin, V. (2005) · SIAM Journal on Scientific Computing · doi:10.1137/030602678 · Cited by 301
A fast randomized algorithm for the approximation of matrices
Woolfe, F., Liberty, E., Rokhlin, V., & Tygert, M. (2008) · Applied and Computational Harmonic Analysis · doi:10.1016/j.acha.2007.12.002 · Cited by 280
Metadata
- Authors
- Gu, M., & Eisenstat, S. C.
- Venue
- SIAM Journal on Scientific Computing
- Published
- 1996-07
- Volume
- 17
- Issue
- 4
- Pages
- 848-869
- ISSN
- 1064-8275 (ISSN-L)
- 1095-7197
- OpenAlex
- W2119233169
- Cited by
- 677
- References
- 24
- FWCI
- 4.12
- Type
- journal article
- Language
- en
- Open access
- closed
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