Max半环上张量的分解

发布时间：2018-10-07 20:53

【摘要】：张量的分解在信号处理、数据挖掘、数值分析和神经系统科学等领域有着重要的应用。本文主要考虑Max半环上张量的分解及其相关问题。首先,给出例子说明在Max半环上满秩的矩阵并不能在矩阵与张量的乘法中保持张量的秩不变。利用数学归纳法证明能够在矩阵和张量的乘法中保持张量秩不变的矩阵为广义置换阵。其次,给出例子指出Max半环上的对称矩阵并不一定存在对称分解。进而详细地讨论Max半环上对称张量的分解,并且给出对称张量存在对称分解的充要条件。当对称张量存在对称分解时,通过构造的方法给出一种分解。最后,讨论Hankel张量的Vandermonde分解及Vandermonde秩。由于Max半环上的Hankel张量并不总是存在Vandermonde分解,所以给出Hankel张量存在Vandermonde分解的充要条件。当Hankel张量存在Vandermonde分解时,构造出一种分解。进一步,当Hankel张量存在Vandermonde分解时,给出Hankel张量Vandermonde秩的刻画。
[Abstract]:Zhang Liang's decomposition plays an important role in signal processing, data mining, numerical analysis and neuroscience. In this paper, we consider the decomposition of Zhang Liang over Max semirings and its related problems. Firstly, an example is given to show that a matrix with full rank over a Max semiring can not keep the rank of Zhang Liang unchanged in the multiplication of a matrix with Zhang Liang. By means of mathematical induction, it is proved that the matrix which can keep the rank invariant of Zhang Liang in the matrix and Zhang Liang's multiplication is a generalized permutation matrix. Secondly, an example is given to show that symmetric matrices over Max semirings do not necessarily have symmetric decomposition. Furthermore, the decomposition of symmetric Zhang Liang over Max semirings is discussed in detail, and the necessary and sufficient conditions for the existence of symmetric decompositions are given. When symmetric Zhang Liang has symmetric decomposition, a kind of decomposition is given by the method of construction. Finally, the Vandermonde decomposition and Vandermonde rank of Hankel Zhang Liang are discussed. Because Hankel Zhang Liang on Max semirings does not always have Vandermonde decomposition, the necessary and sufficient condition for the existence of Vandermonde decomposition for Hankel Zhang Liang is given. When Hankel Zhang Liang has Vandermonde decomposition, a decomposition is constructed. Furthermore, when Hankel Zhang Liang has Vandermonde decomposition, the rank of Hankel Zhang Liang Vandermonde is described.
【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】：O153.3;O183.2

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