求解不同阶对称张量组特征值的带位移高阶幂法  

Shifted Symmetric Higher-Order Power Method for Computing Eigenvalues of Multiple Order Symmetric Tensors

作　　者：张小双 陈震 刘奇龙 ZHANG Xiao-shuang;CHEN Zhen;LIU Qi-long(School of Mathematical Sciences,Guizhou Normal University,Guiyang 550025,China)

机构地区：[1]贵州师范大学数学科学学院,贵阳550025

出　　处：《西南大学学报（自然科学版）》2020年第8期81-87,共7页Journal of Southwest University(Natural Science Edition)

基　　金：国家自然科学基金项目(11671105);贵州省科学技术基金重点项目(1Z002);贵州师范大学2017年博士科研启动项目(GZNUD26号)。

摘　　要：求解不同阶对称张量组的特征值和特征向量问题在超图匹配中具有重要的作用.首先,基于求解对称张量Z-特征值的带位移高阶幂法(SS-HOPM),利用系数张量组构造一个带位移因子的辅助函数,将求解不同阶对称张量组的特征值问题转化为求解辅助函数的极值点问题,提出了求解不同阶对称张量组特征值和特征向量的带位移高阶幂法.其次,利用凸函数的性质和单调有界原理,讨论了辅助函数的性质,确定了位移因子的取值范围,使得所给算法是收敛的.最后,通过数值算例对理论结果进行了验证,数值结果表明所提出的算法是有效的,并且该算法也能有效求出不同阶对称非半正定张量组的特征值和特征向量.The problem of computing the eigenvalues and eigenvectors of multiple order symmetric tensors plays an important role in hyper-graph matching.Firstly,based on the shifted symmetric higher-order power method(SS-HOPM),we use the coefficient tensors to construct an auxiliary function with a shift factor,and transform the problem of solving the eigenvalues of multiple order symmetric tensors into one of solving extreme points of the auxiliary function.Thus we propose herein a shifted symmetric higher-order power method for computing eigenvalues and eigenvectors of multiple order symmetric tensors.Secondly,we utilize the properties of convex functions and the monotonic bounded principle to discuss the properties of the auxiliary function and determine the range of the shifted factor,which makes the given algorithm convergent.Finally,numerical examples are given to verify the theoretical results and show the effectiveness of the proposed algorithm,and the algorithm can also effectively compute the eigenvalues and eigenvectors of multiple order symmetric tensors that are not semi-positive definite.

关 键 词：不同阶对称张量组 特征值 特征向量 带位移高阶幂法 

分 类 号：O151.23[理学—数学]