张量奇异值及高阶奇异值分解具有的若干性质

发布时间：2018-04-14 16:26

本文选题：张量奇异值 + 高阶奇异值分解　；参考：《哈尔滨工业大学》2017年硕士论文

【摘要】：张量的奇异值及高阶奇异值分解(HOSVD)被广泛地应用于多个学科之中。近年来,该理论引起了诸多学者的普遍关注,是国内外专家和学者研究的热门课题。本文主要研究张量奇异值及HOSVD具有的性质。首先,将正交变换不改变矩阵的奇异值,合同变换保持实二次型正定性的结论推广到张量上。其次,研究矩形张量的奇异值在Kronecker积下具有的性质,定义张量的Kronecker和,并研究其H-特征值具有的性质,随后探究张量的HOSVD在Kronecker积下具有的性质。最后,研究具有特殊结构的张量HOSVD具有的性质,给出并证明3阶2维的Hankel张量通过HOSVD得到的核心张量是对角张量的充分必要条件,以及若干循环张量具有保结构的HOSVD.
[Abstract]:Zhang Liang's singular value and higher order singular value decomposition (HOSVD) are widely used in many disciplines.In recent years, the theory has attracted the widespread attention of many scholars and is a hot topic for experts and scholars at home and abroad.In this paper, we study the singular value of Zhang Liang and the properties of HOSVD.Firstly, the conclusion that the orthogonal transformation does not change the singular value of the matrix and the contract transformation keeps the positive definiteness of the real quadratic form is extended to Zhang Liang.Secondly, we study the properties of the singular value of rectangle Zhang Liang under Kronecker product, define the Kronecker sum of Zhang Liang, and study the properties of its H-eigenvalue, and then explore the properties of the HOSVD of under the Kronecker product.Finally, the properties of Zhang Liang HOSVD with special structure are studied, and the sufficient and necessary conditions that the core Zhang Liang obtained by HOSVD obtained by HOSVD is the necessary and sufficient condition of the diagonal Zhang Liang, as well as the existence of some conserved Hos SVDs, are given and proved.
【学位授予单位】：哈尔滨工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O183.2

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