Computing Eigenvalues of Large Scale Hankel Tensors

TL;DR

This paper introduces an efficient inexact curvilinear search method leveraging FFT to compute eigenvalues of large-scale Hankel tensors, capable of handling tensors with dimensions up to one million.

Contribution

It proposes a novel optimization algorithm for eigenvalue computation of large Hankel tensors, with proven convergence properties and high efficiency demonstrated through extensive numerical experiments.

Findings

Method achieves computational cost of O(mn log(mn)) per iteration.

Algorithm converges to eigen-pairs without second-order conditions.

Numerical tests show effectiveness on tensors up to one million in dimension.

Abstract

Large scale tensors, including large scale Hankel tensors, have many applications in science and engineering. In this paper, we propose an inexact curvilinear search optimization method to compute Z- and H-eigenvalues of $m$th order $n$ dimensional Hankel tensors, where $n$ is large. Owing to the fast Fourier transform, the computational cost of each iteration of the new method is about $\mathcal{O}(mn\log(mn))$. Using the Cayley transform, we obtain an effective curvilinear search scheme. Then, we show that every limiting point of iterates generated by the new algorithm is an eigen-pair of Hankel tensors. Without the assumption of a second-order sufficient condition, we analyze the linear convergence rate of iterate sequence by the Kurdyka-{\L}ojasiewicz property. Finally, numerical experiments for Hankel tensors, whose dimension may up to one million, are reported to show the efficiency of the proposed curvilinear search method.