An efficient randomized homotopy method to approximate eigenpairs of tensors

Abstract

Let $A \in (\mathbb{C}^{n})^{\otimes p}$ be a complex tensor of order $p$. The pair $(v,η)\in\mathbb{C}^n\times \mathbb{C}$ is called an h-eigenpair of $A$, if $v\neq0$ and it satisfies $Av^{p-1}=η^{p-2} v$, where $Av^{p-1}$ is the contraction of $A$ by $v$ in all but the first modes. We describe a randomized algorithm to compute approximations of h-eigenpairs of complex tensors. Assuming random input, the average number of arithmetic operations it performs is polynomially bounded in the input size.