Computational Mathematics and Computer Modeling with Applications (CMCMA)

On rank decomposition and semi-symmetric rank decomposition of semi-symmetric tensors

https://doi.org/10.52547/CMCMA.1.1.37
Volume 1, Issue 1 - Serial Number 1
June 2022
Pages 37-47
Computational Mathematics and Computer Modeling with Applications (CMCMA)

Document Type : Regular paper

Authors

Hassan Bozorgmanesh 1
Anthony Theodore Chronopoulos 2

1 Department of Applied Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran

2 Department of Computer Science, University of Texas, San Antonio, Texas 78249, USA. (Visiting Faculty) Department of Computer Engineering & Informatics, University of Patras, Rio 26500, Greece.

Abstract
A  tensor is called  semi-symmetric  if all modes but one, are symmetric. In this paper, we study the CP decomposition of semi-symmetric tensors or higher-order individual difference scaling (INDSCAL). Comon's conjecture states that for any symmetric tensor, the CP rank and symmetric CP rank are equal, while it is known that Comon's conjecture is not true in the general case but it is proved under several assumptions in the literature. In the paper, Comon's conjecture is extended for semi-symmetric CP decomposition and CP decomposition of semi-symmetric tensors under suitable assumptions. Specially, we show that if a semi-symmetric tensor has a CP rank  smaller or equal to its order, or when the semi-symmetric CP rank is less than/or equal to the dimension, then the semi-symmetric CP rank is equal to the CP rank.

Keywords

INDSCAL
semi-symmetric tensor
CP decomposition
CP rank
semi-symmetric decomposition
  • Receive Date 24 November 2021
  • Revise Date 07 January 2022
  • Accept Date 07 January 2022

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