Numerical CP Decomposition of Some Difficult Tensors

0468385 - ÚTIA 2018 RIV NL eng J - Journal Article
Tichavský, Petr - Phan, A. H. - Cichocki, A.
Numerical CP Decomposition of Some Difficult Tensors.
Journal of Computational and Applied Mathematics. Roč. 317, č. 1 (2017), s. 362-370. ISSN 0377-0427. E-ISSN 1879-1778
R&D Projects: GA ČR(CZ) GA14-13713S
Institutional support: RVO:67985556
Keywords : Small matrix multiplication * Canonical polyadic tensor decomposition * Levenberg-Marquardt method
OECD category: Applied mathematics
Impact factor: 1.632, year: 2017
http://library.utia.cas.cz/separaty/2017/SI/tichavsky-0468385.pdf

In this paper, a numerical method is proposed for canonical polyadic (CP) decomposition of small size tensors. The focus is primarily on decomposition of tensors that correspond to small matrix multiplications. Here, rank of the tensors is equal to the smallest number of scalar multiplications that are necessary to accomplish the matrix multiplication. The proposed method is based on a constrained Levenberg-Marquardt optimization. Numerical results indicate the rank and border ranks of tensors that correspond to multiplication of matrices of the size 2x3 and 3x2, 3x3 and 3x2,
3x3 and 3x3, and 3x4 and 4x3. The ranks are 11, 15, 23 and 29, respectively. In particular, a novel algorithm for computing product of matrices of the sizes 3x4 and 4x3 using 29 multiplications is presented.
Permanent Link: http://hdl.handle.net/11104/0270594