An Adaptive Shifted Power Method for Computing Generalized Tensor Eigenpairs
An Adaptive Shifted Power Method for Computing Generalized Tensor Eigenpairs
复制标题
DOI：
10.1137/140951758
复制
发表时间：
2014-01
期刊：
影响因子：
--
通讯作者：
T. Kolda;J. Mayo
中科院分区：
文献类型：
--
作者：
T. Kolda;J. Mayo
在过去的十年中，已经提出了几种张量特征对的定义，但这些定义都可以统一在广义张量特征对框架下，由Chang， Pearson， and Zhang [J]引入。数学。分析的。苹果。， 350 (2009), pp. 416—422]。给定m阶，n维实值对称张量${\mathscr{A}}$和$\boldsymbol{\mathscr{B}}$，目标是找到$\lambda \in \mathbb{R}$和$\mathbf{x} \in \mathbb{R}^{n}, \mathbf{x} \neq 0$，使得${\mathscr{A}}\mathbf{x}^{m-1} = \lambda {\mathscr{B}}\mathbf{x}^{m-1}$。${\mathscr{B}}$的不同选择产生不同版本的张量特征值问题。本文提出了求解z -特征对的广义特征问题自适应幂方法（GEAP），它是求解z -特征对的位移对称高阶幂方法（SS-HOPM）的扩展。SS-HOPM的一个主要缺点是它的性能取决于选择合适的移位，但是我们的GEAP方法还包括一个自动选择移位的自适应方法。
Several tensor eigenpair definitions have been put forth in the past decade, but these can all be unified under generalized tensor eigenpair framework, introduced by Chang, Pearson, and Zhang [J. Math. Anal. Appl., 350 (2009), pp. 416--422]. Given mth-order, n-dimensional real-valued symmetric tensors ${\mathscr{A}}$ and $\boldsymbol{\mathscr{B}}$, the goal is to find $\lambda \in \mathbb{R}$ and $\mathbf{x} \in \mathbb{R}^{n}, \mathbf{x} \neq 0$ such that ${\mathscr{A}}\mathbf{x}^{m-1} = \lambda {\mathscr{B}}\mathbf{x}^{m-1}$. Different choices for ${\mathscr{B}}$ yield different versions of the tensor eigenvalue problem. We present our generalized eigenproblem adaptive power (GEAP) method for solving the problem, which is an extension of the shifted symmetric higher-order power method (SS-HOPM) for finding Z-eigenpairs. A major drawback of SS-HOPM is that its performance depended on choosing an appropriate shift, but our GEAP method also includes an adaptive method for choosing the shift automatically.