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

• 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
• DOI: 10.1007/s10444-018-9622-8

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.