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On randomized sketching algorithms and the Tracy-Widom law.

Published version
Peer-reviewed

Type

Article

Change log

Authors

Astle, William J 
Richardson, Sylvia 

Abstract

UNLABELLED: There is an increasing body of work exploring the integration of random projection into algorithms for numerical linear algebra. The primary motivation is to reduce the overall computational cost of processing large datasets. A suitably chosen random projection can be used to embed the original dataset in a lower-dimensional space such that key properties of the original dataset are retained. These algorithms are often referred to as sketching algorithms, as the projected dataset can be used as a compressed representation of the full dataset. We show that random matrix theory, in particular the Tracy-Widom law, is useful for describing the operating characteristics of sketching algorithms in the tall-data regime when the sample size n is much greater than the number of variables d. Asymptotic large sample results are of particular interest as this is the regime where sketching is most useful for data compression. In particular, we develop asymptotic approximations for the success rate in generating random subspace embeddings and the convergence probability of iterative sketching algorithms. We test a number of sketching algorithms on real large high-dimensional datasets and find that the asymptotic expressions give accurate predictions of the empirical performance. SUPPLEMENTARY INFORMATION: The online version contains supplementary material available at 10.1007/s11222-022-10148-5.

Description

Keywords

Random matrix theory, Random projection, Sketching

Journal Title

Stat Comput

Conference Name

Journal ISSN

0960-3174
1573-1375

Volume Title

33

Publisher

Springer Science and Business Media LLC
Sponsorship
National Institute for Health Research (NIHR) (BTRU-2014-10024)
Medical Research Council (MC_UU_00002/10)
Alan Turing Institute (TU/B/00092)