Operations Research Transactions ›› 2025, Vol. 29 ›› Issue (3): 34-60.doi: 10.15960/j.cnki.issn.1007-6093.2025.03.002

Special Issue: 第九届中国运筹学会科学技术奖获奖者专辑

• Research Article • Previous Articles     Next Articles

Uniqueness of tensor canonical polyadic decomposition

Shenglong HU1,*()   

  1. 1. College of Sciences, National University of Defense Technology, Changsha 410072, Hunan, China
  • Received:2025-04-10 Online:2025-09-15 Published:2025-09-09
  • Contact: Shenglong HU E-mail:hushenglong@nudt.edu.cn

Abstract:

Uniqueness of tensor decomposition is a cornerstone for modeling optimization problems of low-rank tensor decomposition and low-rank tensor approximation in diverse areas of applications, and it is a powerful theory for system parameter identification. This paper briefly summarizes the basic concepts of unique decomposition theory, classical conclusions such as the Kruskal theorem, necessary conditions for uniqueness, the Jennrich-Harshman theory and its extensions, partial uniqueness theory of decomposition, uniqueness of block decomposition, and uniqueness in the statistical sense, etc. Understanding these fundamental properties provides a theoretical basis for further research on the modeling, analysis, solution, and verification of corresponding low-rank tensor decomposition and low-rank tensor approximation optimization models.

Key words: tensor decomposition, uniqueness, sufficient condition, necessary condition, tensor low-rank approximation, properties of solutions, Kruskal's theorem

CLC Number: