Randomised linear algebra

Context
Log det and trace estimation
Random Fourier Features
Randomisation in matrix factorization
Random regression
Hutchinson trace estimator
Stochastic Lanczos Quadrature
Tools
- imate
- Misc
References

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.

Context

Obligatory Igor Carron mention: Random matrices are too damn large. Martinsson (2016) seems to be a fresh review of the action.

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)

Random Fourier Features

See Random Fourier Features.

Randomisation in matrix factorization

See various matrix factorisation methods.

Random regression

See randomised regression

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 with efficiently with the ensemble Kalman filter, where the randomised products are cheap.

Stochastic Lanczos Quadrature

Overview — imate Manual

Tools

imate

Overview — imate

The main purpose of ımate is to estimate the algebraic quantity \[ \operatorname{trace}(f(\mathbf{A})) \] where \(\mathbf{A}\) is a square matrix, \(f\) is a matrix function, and trace \((\cdot)\) is the trace operator. Imate can also compute variants of \((1)\), such as \[ \operatorname{trace}(\mathbf{B} f(\mathbf{A})) \] and \[ \operatorname{trace}(\mathbf{B} f(\mathbf{A}) \mathbf{C} f(\mathbf{A})) \] where \(\mathbf{B}\) and \(\mathbf{C}\) are matrices. Other variations include the cases where \(\mathbf{A}\) is replaced by \(\mathbf{A}^{\top} \mathbf{A}\) in the above expressions.

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)

References

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.

Achlioptas, Dimitris, and Frank Mcsherry. 2007. “Fast Computation of Low-Rank Matrix Approximations.” J. ACM 54 (2).

Alaoui, Ahmed El, and Michael W. Mahoney. 2014. “Fast Randomized Kernel Methods With Statistical Guarantees.” arXiv:1411.0306 [Cs, Stat], November.

Bardenet, Rémi, and Odalric-Ambrym Maillard. 2015. “A Note on Replacing Uniform Subsampling by Random Projections in MCMC for Linear Regression of Tall Datasets.”

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.

Boutsidis, Christos, Petros Drineas, Prabhanjan Kambadur, Eugenia-Maria Kontopoulou, and Anastasios Zouzias. 2017. “A Randomized Algorithm for Approximating the Log Determinant of a Symmetric Positive Definite Matrix.” Linear Algebra and Its Applications 533 (November): 95–117.

Choromanski, Krzysztof, Mark Rowland, and Adrian Weller. 2017. “The Unreasonable Effectiveness of Random Orthogonal Embeddings.” arXiv:1703.00864 [Stat], March.

Dasgupta, Sanjoy. 2000. “Experiments with Random Projection.” In Proceedings of the Sixteenth Conference on Uncertainty in Artificial Intelligence, 143–51. UAI’00. San Francisco, CA, USA: Morgan Kaufmann Publishers Inc.

Dasgupta, Sanjoy, and Anupam Gupta. 2003. “An Elementary Proof of a Theorem of Johnson and Lindenstrauss.” Random Structures & Algorithms 22 (1): 60–65.

Dezfouli, Amir, and Edwin V. Bonilla. 2015. “Scalable Inference for Gaussian Process Models with Black-Box Likelihoods.” In Advances in Neural Information Processing Systems 28, 1414–22. NIPS’15. Cambridge, MA, USA: MIT Press.

Drineas, Petros, Ravi Kannan, and Michael W. Mahoney. 2006a. “Fast Monte Carlo Algorithms for Matrices II: Computing a Low-Rank Approximation to a Matrix.” SIAM Journal on Computing 36 (1): 158–83.

———. 2006b. “Fast Monte Carlo Algorithms for Matrices III: Computing a Compressed Approximate Matrix Decomposition.” SIAM Journal on Computing 36 (1): 184–206.

Drineas, Petros, and Michael W. Mahoney. 2005. “On the Nyström Method for Approximating a Gram Matrix for Improved Kernel-Based Learning.” Journal of Machine Learning Research 6 (December): 2153–75.

Edelman, Alan. 1989. “Eigenvalues and Condition Numbers of Random Matrices.”

Erichson, N. Benjamin, Sergey Voronin, Steven L. Brunton, and J. Nathan Kutz. 2016. “Randomized Matrix Decompositions Using R.” arXiv:1608.02148 [Cs, Stat], August.

Golub, Gene H., and Gérard Meurant. 2010. Matrices, Moments and Quadrature with Applications. USA: Princeton University Press.

Gottwald, Georg A., and Sebastian Reich. 2020. “Supervised Learning from Noisy Observations: Combining Machine-Learning Techniques with Data Assimilation.” arXiv:2007.07383 [Physics, Stat], July.

Gower, Robert M. 2016. “Sketch and Project: Randomized Iterative Methods for Linear Systems and Inverting Matrices.” arXiv:1612.06013 [Math], December.

Gower, Robert M., and Peter Richtárik. 2016. “Randomized Quasi-Newton Updates Are Linearly Convergent Matrix Inversion Algorithms.” arXiv:1602.01768 [Cs, Math], February.

Halko, Nathan, Per-Gunnar Martinsson, Yoel Shkolnisky, and Mark Tygert. 2011. “An Algorithm for the Principal Component Analysis of Large Data Sets.” SIAM Journal on Scientific Computing 33 (5): 2580–94.

Halko, Nathan, Per-Gunnar Martinsson, and Joel A. Tropp. 2010. “Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions.” arXiv.

Hesslow, Daniel, Alessandro Cappelli, Igor Carron, Laurent Daudet, Raphaël Lafargue, Kilian Müller, Ruben Ohana, Gustave Pariente, and Iacopo Poli. 2021. “Photonic Co-Processors in HPC: Using LightOn OPUs for Randomized Numerical Linear Algebra.” arXiv:2104.14429 [Cs, Stat], May.

Hutchinson, M.F. 1990. “A Stochastic Estimator of the Trace of the Influence Matrix for Laplacian Smoothing Splines.” Communications in Statistics - Simulation and Computation 19 (2): 433–50.

Kapralov, Michael, Vamsi K. Potluru, and David P. Woodruff. 2016. “How to Fake Multiply by a Gaussian Matrix.” arXiv:1606.05732 [Cs], June.

Krummenacher, Gabriel, Brian McWilliams, Yannic Kilcher, Joachim M Buhmann, and Nicolai Meinshausen. 2016. “Scalable Adaptive Stochastic Optimization Using Random Projections.” In Advances in Neural Information Processing Systems 29, edited by D. D. Lee, M. Sugiyama, U. V. Luxburg, I. Guyon, and R. Garnett, 1750–58. Curran Associates, Inc.

Kumar, N. Kishore, and Jan Shneider. 2016. “Literature Survey on Low Rank Approximation of Matrices.” arXiv:1606.06511 [Cs, Math], June.

Lahiri, Subhaneil, Peiran Gao, and Surya Ganguli. 2016. “Random Projections of Random Manifolds.” arXiv:1607.04331 [Cs, q-Bio, Stat], July.

Launay, Julien, Iacopo Poli, François Boniface, and Florent Krzakala. 2020. “Direct Feedback Alignment Scales to Modern Deep Learning Tasks and Architectures.” In Advances in Neural Information Processing Systems, 33:15.

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.

Li, Ping, Trevor J. Hastie, and Kenneth W. Church. 2006. “Very Sparse Random Projections.” In Proceedings of the 12th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 287–96. KDD ’06. New York, NY, USA: ACM.

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.

Mahoney, Michael W. 2010. Randomized Algorithms for Matrices and Data. Vol. 3.

Martinsson, Per-Gunnar. 2016. “Randomized Methods for Matrix Computations and Analysis of High Dimensional Data.” arXiv:1607.01649 [Math], July.

Martinsson, Per-Gunnar, Vladimir Rockhlin, and Mark Tygert. 2006. “A Randomized Algorithm for the Approximation of Matrices.” DTIC Document.

Martinsson, Per-Gunnar, and Sergey Voronin. 2015. “A Randomized Blocked Algorithm for Efficiently Computing Rank-Revealing Factorizations of Matrices.” arXiv:1503.07157 [Math], March.

Meyer, Raphael A., Cameron Musco, Christopher Musco, and David P. Woodruff. 2021. “Hutch++: Optimal Stochastic Trace Estimation.” arXiv.

Nakatsukasa, Yuji. 2017. “Accuracy of Singular Vectors Obtained by Projection-Based SVD Methods.” BIT Numerical Mathematics 57 (4): 1137–52.

Pourkamali-Anaraki, Farhad, and Stephen Becker. 2016. “A Randomized Approach to Efficient Kernel Clustering.” arXiv:1608.07597 [Stat], August.

Rabani, Eran, and Sivan Toledo. 2001. “Out-of-Core SVD and QR Decompositions.” In PPSC.

Rahimi, Ali, and Benjamin Recht. 2007. “Random Features for Large-Scale Kernel Machines.” In Advances in Neural Information Processing Systems, 1177–84. Curran Associates, Inc.

———. 2009. “Weighted Sums of Random Kitchen Sinks: Replacing Minimization with Randomization in Learning.” In Advances in Neural Information Processing Systems, 1313–20. Curran Associates, Inc.

Refinetti, Maria, Stéphane d’Ascoli, Ruben Ohana, and Sebastian Goldt. 2021. “Align, Then Memorise: The Dynamics of Learning with Feedback Alignment.” arXiv:2011.12428 [Cond-Mat, Stat], June.

Rokhlin, Vladimir, Arthur Szlam, and Mark Tygert. 2009. “A Randomized Algorithm for Principal Component Analysis.” SIAM J. Matrix Anal. Appl. 31 (3): 1100–1124.

Rokhlin, Vladimir, and Mark Tygert. 2008. “A Fast Randomized Algorithm for Overdetermined Linear Least-Squares Regression.” Proceedings of the National Academy of Sciences 105 (36): 13212–17.

Saibaba, Arvind K., Alen Alexanderian, and Ilse C. F. Ipsen. 2017. “Randomized Matrix-Free Trace and Log-Determinant Estimators.” arXiv.

Shi, Jiaxin, Shengyang Sun, and Jun Zhu. 2018. “A Spectral Approach to Gradient Estimation for Implicit Distributions.” In. arXiv.

Skorski, Maciej. 2020. “A Modern Analysis of Hutchinson’s Trace Estimator.” arXiv.

Sterge, Nicholas, and Bharath Sriperumbudur. 2021. “Statistical Optimality and Computational Efficiency of Nyström Kernel PCA.” arXiv:2105.08875 [Cs, Math, Stat], May.

Tropp, Joel A. 2015. An Introduction to Matrix Concentration Inequalities.

Tropp, Joel A., Alp Yurtsever, Madeleine Udell, and Volkan Cevher. 2016. “Randomized Single-View Algorithms for Low-Rank Matrix Approximation.” arXiv:1609.00048 [Cs, Math, Stat], August.

Voronin, Sergey, and Per-Gunnar Martinsson. 2014. “Efficient Algorithms for CUR and Interpolative Matrix Decompositions.” arXiv:1412.8447 [Cs, Math], December.

Wang, Shusen. 2015. “A Practical Guide to Randomized Matrix Computations with MATLAB Implementations.” arXiv:1505.07570 [Cs], May.

Woolfe, Franco, Edo Liberty, Vladimir Rokhlin, and Mark Tygert. 2008. “A Fast Randomized Algorithm for the Approximation of Matrices.” Applied and Computational Harmonic Analysis 25 (3): 335–66.

Yang, Jiyan, Xiangrui Meng, and Michael W. Mahoney. 2015. “Implementing Randomized Matrix Algorithms in Parallel and Distributed Environments.” arXiv:1502.03032 [Cs, Math, Stat], February.

Yang, Jiyan, Vikas Sindhwani, Haim Avron, and Michael Mahoney. 2014. “Quasi-Monte Carlo Feature Maps for Shift-Invariant Kernels.” arXiv:1412.8293 [Cs, Math, Stat], December.

Yang, Tianbao, Yu-Feng Li, Mehrdad Mahdavi, Rong Jin, and Zhi-Hua Zhou. 2012. “Nyström Method Vs Random Fourier Features: A Theoretical and Empirical Comparison.” In Advances in Neural Information Processing Systems, 476–84.

Yu, Hsiang-Fu, Cho-Jui Hsieh, Si Si, and Inderjit S. Dhillon. 2012. “Scalable Coordinate Descent Approaches to Parallel Matrix Factorization for Recommender Systems.” In IEEE International Conference of Data Mining, 765–74.

Zhang, Kai, Chuanren Liu, Jie Zhang, Hui Xiong, Eric Xing, and Jieping Ye. 2017. “Randomization or Condensation?: Linear-Cost Matrix Sketching Via Cascaded Compression Sampling.” In Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 615–23. KDD ’17. New York, NY, USA: ACM.

Zhao, Liang. 2017. “Fast Algorithms on Random Matrices and Structured Matrices.”

Zhou, Tianyi, and Dacheng Tao. 2011. “GoDec: Randomized Low-Rank & Sparse Matrix Decomposition in Noisy Case.” In Proceedings of the 28th International Conference on International Conference on Machine Learning, 33–40. ICML’11. Madison, WI, USA: Omnipress.