Efficient solvers for tensor eigenvalue problems are important tools for the analysis of higher-order data sets. Here we introduce, analyze and demonstrate an extrapolation method to accelerate the widely used shifted symmetric higher order power method for tensor $Z$-eigenvalue problems. We analyze the asymptotic convergence of the method, determining the range of extrapolation parameters that induce acceleration, as well as the parameter that gives the optimal convergence rate. We then introduce an automated method to dynamically approximate the optimal parameter, and demonstrate it's efficiency when the base iteration is run with either static or adaptively set shifts. Our numerical results on both even and odd order tensors demonstrate the theory and show we achieve our theoretically predicted acceleration.

翻译：张量特征值问题的高效求解器是分析高阶数据集的重要工具。本文提出、分析并展示了一种外推方法，用于加速广泛使用的移位对称高阶幂方法求解张量 $Z$-特征值问题。我们分析了该方法的渐近收敛性，确定了可引发加速的外推参数范围，以及能使收敛速率最优的参数。随后，我们引入了一种自动动态逼近最优参数的方法，并证明了当基迭代采用静态或自适应设置移位时，该方法的有效性。我们对偶数阶与奇数阶张量的数值实验验证了理论分析，并表明我们实现了理论预测的加速效果。