An efficient randomized homotopy method to approximate eigenpairs of tensors

TL;DR

This paper introduces a randomized homotopy method for efficiently approximating eigenpairs of complex tensors, with polynomial average complexity under random input assumptions.

Contribution

It presents a novel randomized algorithm for computing approximate eigenpairs of complex tensors with polynomial average complexity.

Findings

Algorithm performs polynomially bounded average operations

Effective for complex tensors of arbitrary order

Provides probabilistic guarantees under random input

Abstract

Let $A \in (\mathbb{C}^{n})^{\otimes p}$ be a complex tensor of order $p$. The pair $(v,\eta)\in\mathbb{C}^n\times \mathbb{C}$ is called an h-eigenpair of $A$, if $v\neq0$ and it satisfies $Av^{p-1}=\eta^{p-2} v$, 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.