Numerical CP decomposition of some difficult tensors

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:317

Article

Tichavsky, Petr¹  Anh-Huy Phan²  Cichocki, Andrzej² 

[1] Czech Acad Sci, Inst Informat Theory & Automat, Pod Vodarenskou Vezi 4, Prague 18208 8, Czech Republic

[2] RIKEN, Brain Sci Inst, Wako, Saitama, Japan

关键词: Small matrix multiplication;    Canonical polyadic tensor decomposition;    Levenberg-Marquardt method;   

DOI  :  10.1016/j.cam.2016.12.007

来源: Elsevier

【 摘 要 】

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 2 x 3 and 3 x 2, 3 x 3 and 3 x 2, 3 x 3 and 3 x 3, and 3 x 4 and 4 x 3. The ranks are 11, 15, 23 and 29, respectively. In particular, a novel algorithm for computing product of matrices of the sizes 3 x 4 and 4 x 3 using 29 multiplications is presented. (C) 2016 Elsevier B.V. All rights reserved.

【 授权许可】

Free