SVD-based algorithms for tensor wheel decomposition

• Published in: Advances in computational mathematics Vol. 50; no. 5
• Main Authors: Wang, Mengyu, Cui, Honghua, Li, Hanyu
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.10.2024; Springer Nature B.V
• Subjects: Algorithms; Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Controllability; Decomposition; Mathematical analysis; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Robustness; Tensors; Visualization; SVD; 68W20; 65F55; Tensor wheel decomposition; Alternating least squares; 49M27; Deterministic algorithm; 15A69; Randomized algorithm; Sketching
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-024-10194-9

Tensor wheel (TW) decomposition combines the popular tensor ring and fully connected tensor network decompositions and has achieved excellent performance in tensor completion problem. A standard method to compute this decomposition is the alternating least squares (ALS). However, it usually suffers from slow convergence and numerical instability. In this work, the fast and robust SVD-based algorithms are investigated. Based on a result on TW-ranks, we first propose a deterministic algorithm that can estimate the TW decomposition of the target tensor under a controllable accuracy. Then, the randomized versions of this algorithm are presented, which can be divided into two categories and allow various types of sketching. Numerical results on synthetic and real data show that our algorithms have much better performance than the ALS-based method and are also quite robust. In addition, with one SVD-based algorithm, we also numerically explore the variability of TW decomposition with respect to TW-ranks and the comparisons between TW decomposition and other famous formats in terms of the performance on approximation and compression.