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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2017, Vol. 34 ›› Issue (1): 87-99.doi: 10.3969/j.issn.1005-3085.2017.01.009

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广义特征值问题与两线性流形之间夹角的计算（英）

张圣贵   

  1. 福建师范大学数学与计算机科学学院，福州  350007
  • 收稿日期:2014-09-19 接受日期:2016-07-06 出版日期:2017-02-15 发布日期:2017-04-15
  • 基金资助:
    国家自然科学基金(11301080)；福建省自然科学基金(2013J05002).

Generalized Eigenvalue Problems and Computation of Angles between Two Linear Manifolds

ZHANG Sheng-gui   

  1. School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350007
  • Received:2014-09-19 Accepted:2016-07-06 Online:2017-02-15 Published:2017-04-15
  • Supported by:
    The National Natural Science Foundation of China (11301080); the Natural Science Foundation of Fujian Province (2013J05002).

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340

摘要：

高维欧氏空间中的两线性流形的夹角可用带二次等式约束的二次规划（QP-QEC）刻画．这样的夹角计算在统计学和数据分析中有许多重要应用，比如，两组随机变量的典型相关分析和核典型相关分析．本文用KKT条件探讨了更一般的QP-QEC与其对应的一般特征值问题之间的关系．在此基础上，借助一般特征值问题的解法，给出了这种夹角的算法．

关键词: 广义特征值问题, 线性流形, 二次规划, 夹角

Abstract:

The angle between two nontrivial linear manifolds in the high dimensional Euclidean space can be characterized as a quadratic programming with quadratic equation constraints (QP-QEC). The computing of such angles has many important applications in statistics and data analysis, such as the canonical correlation analysis and the kernel correlation analysis between two multivariate random vectors. This paper explores the relationship between a more general QP-QEC and its corresponding generalized eigenvalue problem in terms of the KKT conditions. On this basis, we design an algorithm for computing such an angle by means of the solution method of the generalized eigenvalue problem.

Key words: generalized eigenvalue problem, linear manifold, quadratic programming, angle

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