运筹学学报 ›› 2019, Vol. 23 ›› Issue (1): 53-60.doi: 10.15960/j.cnki.issn.1007-6093.2019.01.006

• 运筹学 • 上一篇    下一篇

基于块循环矩阵的对称张量的最佳秩-1逼近

徐娇娇1, 杨志霞1,*, 蒋耀林1,2   

  1. 1. 新疆大学 数学与系统科学学院, 乌鲁木齐 830046;
    2. 西安交通大学 数学与统计学院, 西安 710049
  • 收稿日期:2017-04-26 出版日期:2019-03-15 发布日期:2019-03-15
  • 通讯作者: 杨志霞 E-mail:xjyangzhx@sina.com
  • 基金资助:

    国家自然科学基金(No.11561066)

The best rank-one approximation of the symmetric tensor based on the block circulant matrix

XU Jiaojiao1, YANG Zhixia1,*, JIANG Yaolin1,2   

  1. 1. College of Mathematics and System Science, Xinjiang University, Urumqi, 830046, China;
    2. College of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China
  • Received:2017-04-26 Online:2019-03-15 Published:2019-03-15

摘要:

对称张量的最佳秩-1问题是张量研究中非常重要的部分.首先,基于三阶张量的块循环矩阵,提出了求解对称张量最佳秩-1逼近问题的一个新方法.其次,针对求解对称张量的最佳秩-1逼近方法,给出了对称张量的最佳秩-1逼近不变性的一个充要条件,以及逼近误差上界的估计.最后,数值算例表明了上述方法的可行性和误差上界的正确性.

关键词: 对称张量, 秩-1张量, 最佳秩-1逼近

Abstract:

In this paper we mainly study the best rank-one approximation problem of a symmetric tensor. This problem plays an important role in our investigation of the tensor. Firstly, we propose a new method to solve the best rank-one approximation problem of a symmetric tensor, which is based on the block circulant matrix of a third-order tensor. Secondly, sufficient and necessary conditions and an estimation of error upper bound are provided for the best rank-one approximation method. Finally, the numerical example is presented to illustrate the feasibility of our approach and the correctness of the error upper bound.

Key words: symmetric tensor, rank-one tensor, the best rank-one approximation

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