张量分解的唯一性

doi:10.15960/j.cnki.issn.1007-6093.2025.03.002

摘要/Abstract

摘要：

张量分解的唯一性是多个应用问题中张量低秩分解和张量低秩逼近优化问题建模的关键基础, 是进行系统参数识别的强有力理论。本文简要归纳唯一分解理论的基本概念、Kruskal定理等经典结论、唯一性成立的必要条件、Jennrich-Harshman理论及其延伸、分解的部分唯一性理论、块分解唯一性以及统计意义下唯一性等。通过对这些基本性质的了解, 为相应张量低秩分解和张量低秩逼近优化模型的建模、分析、求解和验证等理论和方法的进一步研究提供理论基础。

关键词: 张量分解, 唯一性, 充分条件, 必要条件, 张量低秩逼近, 解的性质, Kruskal定理

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

中图分类号:

O221.2

胡胜龙. 张量分解的唯一性[J]. 运筹学学报（中英文）, 2025, 29(3): 34-60.

Shenglong HU. Uniqueness of tensor canonical polyadic decomposition[J]. Operations Research Transactions, 2025, 29(3): 34-60.

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