Randomized algorithms for the approximations of Tucker and the tensor train decompositions

Randomized algorithms provide a powerful tool for scientific computing. Compared with standard deterministic algorithms, randomized algorithms are often faster and robust. The main purpose of this paper is to design adaptive randomized algorithms for computing the approximate tensor decompositions....

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Bibliographic Details
Published in:Advances in computational mathematics Vol. 45; no. 1; pp. 395 - 428
Main Authors: Che, Maolin, Wei, Yimin
Format: Journal Article
Language:English
Published: New York Springer US 05.02.2019
Springer Nature B.V
Subjects:
Adaptive algorithms
Algorithms
Computation
Computational mathematics
Computational Mathematics and Numerical Analysis
Computational Science and Engineering
Decomposition
Error analysis
Mathematical analysis
Mathematical and Computational Biology
Mathematical Modeling and Industrial Mathematics
Mathematics
Mathematics and Statistics
Randomization
Robustness (mathematics)
Software
Tensors
Visualization
Kronecker structures
Multilinear rank
TT-approximation
TT-rank
15A69
Adaptive randomized algorithms
Tensor train decomposition
15A18
65F10
Low multilinear rank approximation
Randomized algorithms
65F15
Tucker decomposition
ISSN:1019-7168, 1572-9044
Online Access:Get full text
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Summary:Randomized algorithms provide a powerful tool for scientific computing. Compared with standard deterministic algorithms, randomized algorithms are often faster and robust. The main purpose of this paper is to design adaptive randomized algorithms for computing the approximate tensor decompositions. We give an adaptive randomized algorithm for the computation of a low multilinear rank approximation of the tensors with unknown multilinear rank and analyze its probabilistic error bound under certain assumptions. Finally, we design an adaptive randomized algorithm for computing the tensor train approximations of the tensors. Based on the bounds about the singular values of sub-Gaussian matrices with independent columns or independent rows, we analyze these randomized algorithms. We illustrate our adaptive randomized algorithms via several numerical examples.
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ISSN:1019-7168
1572-9044
DOI:10.1007/s10444-018-9622-8