Randomised linear algebra
2016-08-16 — 2022-10-22
See random matrices, vector random projections and many other related tricks Notes on doing linear algebra operations using randomised matrix projections. Useful for, e.g. randomised regression.
1 Context
Obligatory Igor Carron mention: Random matrices are too damn large. Martinsson (2016) seems to be a fresh review of the action.
2 Log det and trace estimation
Machine Learning Trick of the Day (3): Hutchinson’s Trick — Shakir Mohammed
- RaphaelArkadyMeyerNYU/HutchPlusPlus: Code for Hutch++: Optimal Stochastic Trace Estimation Saibaba, Alexanderian, and Ipsen (2017)
3 Random Fourier Features
4 Randomisation in matrix factorization
See various matrix factorisation methods.
5 Random regression
6 Hutchinson trace estimator
Shakir Mohamed mentions Hutchinson’s Trick, and was introduced to it, as I was, by Dr Maurizio Filippone. This trick also works efficiently with the ensemble Kalman filter, where the randomised products are cheap.
7 Stochastic Lanczos Quadrature
8 Tools
8.1 imate
The main purpose of ımate is to estimate the algebraic quantity
where is a square matrix, is a matrix function, and trace is the trace operator. Imate can also compute variants of , such as and where and are matrices. Other variations include the cases where is replaced by in the above expressions.
8.2 Misc
- 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)
9 References
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