Computing Extreme Eigenvalues of Large Scale Hankel Tensors

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 mth 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 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-Lojasiewicz 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.

Computing the p-Spectral Radii of Uniform Hypergraphs with Applications

“…That is to say the orthogonal transform is in fact the Cayley transform. The proof is then similar to Lemma 3.2 in [8,12].…”

A semidefinite program approach for computing the maximum eigenvalue of a class of structured tensors and its applications in hypergraphs and copositivity test

“…However, for tensors with specific structures, the extreme eigenvalues of large‐scale tensors can be computed in reasonable time by exploiting the underlying structure. For instance, an inexact curvilinear search optimization method was established to compute the extreme eigenvalues of Hankel tensors, whose dimension may reach up to one million …”

“…For instance, an inexact curvilinear search optimization method was established to compute the extreme eigenvalues of Hankel tensors, whose dimension may reach up to one million. 20 Nonnegative tensors are an important class of structured tensors, which arise from the study of image science, statistics, and hypergraph theory. 21,22 Ng et al 3 proposed an iterative method for finding the maximum H-eigenvalue of an irreducible nonnegative tensor.…”

An adaptive gradient method for computing generalized tensor eigenpairs

“…In [49], Ni and Qi employed Newton method for the KKT system of optimization problem, and obtained a quadratically convergent algorithm for finding the largest eigenvalue of a nonnegative homogeneous polynomial map. In [10], an inexact steepest descent method was proposed for computing eigenvalues of large scale Hankel tensors. Since nonlinear optimization methods may stop at a local optimum, a sequential semi-definite programming method was proposed by Hu et al [19] for finding the extremal Z-eigenvalues of tensors.…”

“…Where S n−1 denote the unit sphere, i.e., S n−1 = {x ∈ R n | x 2 = 1}, · denotes the Euclidean norm. By some simple calculations, we can get its gradient and Hessian, as follows [21,10]:…”