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A randomized tensor singular value decomposition based on the t‐product
Author(s) -
Zhang Jiani,
Saibaba Arvind K.,
Kilmer Misha E.,
Aeron Shuchin
Publication year - 2018
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2179
Subject(s) - singular value decomposition , mathematics , tensor (intrinsic definition) , matrix decomposition , convergence (economics) , matrix (chemical analysis) , singular value , algorithm , pure mathematics , eigenvalues and eigenvectors , physics , materials science , quantum mechanics , economics , composite material , economic growth
Summary The tensor SVD (t‐SVD) for third‐order tensors, previously proposed in the literature, has been applied successfully in many fields, such as computed tomography, facial recognition, and video completion. In this paper, we propose a method that extends a well‐known randomized matrix method to the t‐SVD. This method can produce a factorization with similar properties to the t‐SVD, but it is more computationally efficient on very large data sets. We present details of the algorithms and theoretical results and provide numerical results that show the promise of our approach for compressing and analyzing image‐based data sets. We also present an improved analysis of the randomized and simultaneous iteration for matrices, which may be of independent interest to the scientific community. We also use these new results to address the convergence properties of the new and randomized tensor method as well.