2011 FindingStructurewithRandomnessP

(Halko et al., 2011) ⇒ Nathan Halko, Per-Gunnar Martinsson, and Joel A. Tropp. (2011). “Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions.” In: SIAM Review Journal, 53(2). doi:10.1137/090771806

Subject Headings: Low-Rank Matrix Approximation, Randomized Matrix Decomposition, Truncated SVD, sklearn.decomposition.PCA.

Notes

Cited By

Quotes

Abstract

Low-rank matrix approximations, such as the truncated singular value decomposition and the rank-revealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys and extends recent research which demonstrates that randomization offers a powerful tool for performing low-rank matrix approximation. These techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets. This paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions. These methods use random sampling to identify a subspace that captures most of the action of a matrix. The input matrix is then compressed --- either explicitly or implicitly --- to this subspace, and the reduced matrix is manipulated deterministically to obtain the desired low-rank factorization. In many cases, this approach beats its classical competitors in terms of accuracy, robustness, and/or speed. These claims are supported by extensive numerical experiments and a detailed error analysis. The specific benefits of randomized techniques depend on the computational environment. Consider the model problem of finding the k dominant components of the singular value decomposition of an [math]\displaystyle{ m \times n }[/math] matrix. (i) For a dense input matrix, randomized algorithms require [math]\displaystyle{ \mathcal{O}(mn \log (k)) }[/math] floating-point operations (flops) in contrast to [math]\displaystyle{ \mathcal{O}(mnk) }[/math] for classical algorithms. (ii) For a sparse input matrix, the flop count matches classical Krylov subspace methods, but the randomized approach is more robust and can easily be reorganized to exploit multiprocessor architectures. (iii) For a matrix that is too large to fit in fast memory, the randomized techniques require only a constant number of passes over the data, as opposed to [math]\displaystyle{ \mathcal{O}(k) }[/math] passes for classical algorithms. In fact, it is sometimes possible to perform matrix approximation with a single pass over the data.

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164. Md. Mehrab Tanjim, Muhammad Abdullah Adnan, SSketch: A Scalable Sketching Technique for PCA in the Cloud, Proceedings of the Eleventh ACM International Conference on Web Search and Data Mining, February 05-09, 2018, Marina Del Rey, CA, USA
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198. Jacob Steinhardt, Percy Liang, Unsupervised Risk Estimation Using Only Conditional Independence Structure, Proceedings of the 30th International Conference on Neural Information Processing Systems, p.3664-3672, December 05-10, 2016, Barcelona, Spain
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230. Yin Yang, Dingzeyu Li, Weiwei Xu, Yuan Tian, Changxi Zheng, Expediting Precomputation for Reduced Deformable Simulation, ACM Transactions on Graphics (TOG), v.34 n.6, November 2015
231. Jinfeng Yi, Lijun Zhang, Jun Wang, Rong Jin, Anil K. Jain, A Single-pass Algorithm for Efficiently Recovering Sparse Cluster Centers of High-dimensional Data, Proceedings of the 31st International Conference on International Conference on Machine Learning, June 21-26, 2014, Beijing, China
232. Borja Balle, William L. Hamilton, Joelle Pineau, Methods of Moments for Learning Stochastic Languages: Unified Presentation and Empirical Comparison, Proceedings of the 31st International Conference on International Conference on Machine Learning, June 21-26, 2014, Beijing, China
233. Pablo Arias, Jean-Michel Morel, Video Denoising via Empirical Bayesian Estimation of Space-Time Patches, Journal of Mathematical Imaging and Vision, v.60 n.1, p.70-93, January 2018
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235. Evgeny Frolov, Ivan Oseledets, HybridSVD: When Collaborative Information is Not Enough, Proceedings of the 13th ACM Conference on Recommender Systems, September 16-20, 2019, Copenhagen, Denmark
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239. Xixian Chen, Michael R. Lyu, Irwin King, Toward Efficient and Accurate Covariance Matrix Estimation on Compressed Data, Proceedings of the 34th International Conference on Machine Learning, p.767-776, August 06-11, 2017, Sydney, NSW, Australia
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242. Jalaj Upadhyay, The Price of Privacy for Low-rank Factorization, Proceedings of the 32nd International Conference on Neural Information Processing Systems, p.4180-4191, December 03-08, 2018, Montréal, Canada
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246. Alex Gittens, Michael W. Mahoney, Revisiting the Nyström Method for Improved Large-scale Machine Learning, The Journal of Machine Learning Research, v.17 n.1, p.3977-4041, January 2016
247. Lin Lin, Randomized Estimation of Spectral Densities of Large Matrices Made Accurate, Numerische Mathematik, v.136 n.1, p.183-213, May 2017
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	Author	volume	Date Value	title	type	journal	titleUrl	doi	note	year
2011 FindingStructurewithRandomnessP	Nathan Halko Per-Gunnar Martinsson Joel A. Tropp			Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions				10.1137/090771806		2011

2011 FindingStructurewithRandomnessP

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