The twin to random matrices, and elder sibling of vector random projections; doing linear algebra operations using randomised matrix projections. Useful for, e.g. randomised regression
Quite an old method, but extra hot recently.
Obligatory Igor Carron mention:
IBM has a research group in this although they seem to have gone silent since a good year in 2014.
Martinsson (2016) seems to be a fresh review of the action.
🏗 make coherent.
To learn: Are people doing this with quasi monte carlo? Surely.
Implementations
- R: rSVD, Randomised SVD in R (Erichson et al. 2016)
- Python/C++: IBM’s libskylark
- c: RSVDPACK implements low rank SVD, ID, and CUR factorizations of matrices, also does GPU calculations. (Martinsson and Voronin 2015; Voronin and Martinsson 2014)
- MATLAB: RandMatrixMatlab (Wang 2015)
Random regression
Achlioptas, Dimitris. 2003. “Database-Friendly Random Projections: Johnson-Lindenstrauss with Binary Coins.” Journal of Computer and System Sciences, Special Issue on PODS 2001, 66 (4): 671–87. https://doi.org/10.1016/S0022-0000(03)00025-4.
Achlioptas, Dimitris, and Frank Mcsherry. 2007. “Fast Computation of Low-Rank Matrix Approximations.” J. ACM 54 (2). https://doi.org/10.1145/1219092.1219097.
Alaoui, Ahmed El, and Michael W. Mahoney. 2014. “Fast Randomized Kernel Methods with Statistical Guarantees,” November. http://arxiv.org/abs/1411.0306.
Bingham, Ella, and Heikki Mannila. 2001. “Random Projection in Dimensionality Reduction: Applications to Image and Text Data.” In Proceedings of the Seventh ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 245–50. KDD ’01. New York, NY, USA: ACM. https://doi.org/10.1145/502512.502546.
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Erichson, N. Benjamin, Sergey Voronin, Steven L. Brunton, and J. Nathan Kutz. 2016. “Randomized Matrix Decompositions Using R,” August. http://arxiv.org/abs/1608.02148.
Gottwald, Georg A., and Sebastian Reich. 2020. “Supervised Learning from Noisy Observations: Combining Machine-Learning Techniques with Data Assimilation,” July. http://arxiv.org/abs/2007.07383.
Gower, Robert M. 2016. “Sketch and Project: Randomized Iterative Methods for Linear Systems and Inverting Matrices,” December. http://arxiv.org/abs/1612.06013.
Gower, Robert M., and Peter Richtárik. 2016. “Randomized Quasi-Newton Updates Are Linearly Convergent Matrix Inversion Algorithms,” February. http://arxiv.org/abs/1602.01768.
Halko, Nathan, Per-Gunnar Martinsson, and Joel A. Tropp. 2009. “Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions,” September. http://arxiv.org/abs/0909.4061.
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Kumar, N. Kishore, and Jan Shneider. 2016. “Literature Survey on Low Rank Approximation of Matrices,” June. http://arxiv.org/abs/1606.06511.
Lahiri, Subhaneil, Peiran Gao, and Surya Ganguli. 2016. “Random Projections of Random Manifolds,” July. http://arxiv.org/abs/1607.04331.
Lee, Daniel D., and H. Sebastian Seung. 2001. “Algorithms for Non-Negative Matrix Factorization.” In Advances in Neural Information Processing Systems 13, edited by T. K. Leen, T. G. Dietterich, and V. Tresp, 556–62. MIT Press. http://papers.nips.cc/paper/1861-algorithms-for-non-negative-matrix-factorization.pdf.
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Liberty, Edo, Franco Woolfe, Per-Gunnar Martinsson, Vladimir Rokhlin, and Mark Tygert. 2007. “Randomized Algorithms for the Low-Rank Approximation of Matrices.” Proceedings of the National Academy of Sciences 104 (51): 20167–72. https://doi.org/10.1073/pnas.0709640104.
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Martinsson, Per-Gunnar. 2016. “Randomized Methods for Matrix Computations and Analysis of High Dimensional Data,” July. http://arxiv.org/abs/1607.01649.
Martinsson, Per-Gunnar, Vladimir Rockhlin, and Mark Tygert. 2006. “A Randomized Algorithm for the Approximation of Matrices.” DTIC Document. http://oai.dtic.mil/oai/oai?verb=getRecord&metadataPrefix=html&identifier=ADA458932.
Martinsson, Per-Gunnar, and Sergey Voronin. 2015. “A Randomized Blocked Algorithm for Efficiently Computing Rank-Revealing Factorizations of Matrices,” March. http://arxiv.org/abs/1503.07157.
Pourkamali-Anaraki, Farhad, and Stephen Becker. 2016. “A Randomized Approach to Efficient Kernel Clustering,” August. http://arxiv.org/abs/1608.07597.
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