An Adaptive Shifted Power Method for Computing Generalized Tensor Eigenpairs

Abstract

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 (2009). Given mth-order, n-dimensional real-valued symmetric tensors A and B, the goal is to find $λ\in R$ and $x \in R^n$, $x \neq 0$, such that $Ax^{m-1} = λBx^{m-1}$. Different choices for B yield different versions of the tensor eigenvalue problem. We present our generalized eigenproblem adaptive power method (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 was that its performance depended in choosing an appropriate shift, but our GEAP method also includes an adaptive method for choosing the shift automatically.