A randomized generalized low rank approximations of matrices algorithm for high dimensionality reduction and image compression

Summary High‐dimensionality reduction techniques are very important tools in machine learning and data mining. The method of generalized low rank approximations of matrices (GLRAM) is a popular technique for dimensionality reduction and image compression. However, it suffers from heavily computation...

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Bibliographic Details
Published in:Numerical linear algebra with applications Vol. 28; no. 1
Main Authors: Li, Ke, Wu, Gang
Format: Journal Article
Language:English
Published: Oxford Wiley Subscription Services, Inc 01.01.2021
Subjects:
Algorithms
Data mining
generalized low rank approximations of matrices
high dimensionality reduction
Image compression
Machine learning
randomized singular value decomposition
Reconstruction
Singular value decomposition
ISSN:1070-5325, 1099-1506
Online Access:Get full text
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Summary:Summary High‐dimensionality reduction techniques are very important tools in machine learning and data mining. The method of generalized low rank approximations of matrices (GLRAM) is a popular technique for dimensionality reduction and image compression. However, it suffers from heavily computational overhead in practice, especially for data with high dimension. In order to reduce the cost of this algorithm, we propose a randomized GLRAM algorithm based on randomized singular value decomposition (RSVD). The theoretical contribution of our work is threefold. First, we discuss the decaying property of singular values of the matrices during iterations of the GLRAM algorithm, and provide a target rank required in the RSVD process from a theoretical point of view. Second, we establish the relationship between the reconstruction errors generated by the standard GLRAM algorithm and the randomized GLRAM algorithm. It is shown that the reconstruction errors generated by the former and the latter are comparable, even if the solutions are computed inaccurately during iterations. Third, the convergence of the randomized GLRAM algorithm is investigated. Numerical experiments on some real‐world data sets illustrate the superiority of our proposed algorithm over its original counterpart and some state‐of‐the‐art GLRAM‐type algorithms.
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ISSN:1070-5325
1099-1506
DOI:10.1002/nla.2338