A locally and cubically convergent algorithm for computing 𝒵‐eigenpairs of symmetric tensors

Abstract

This paper is concerned with computing -eigenpairs of symmetric tensors.We first show that computing -eigenpairs of a symmetric tensor is equivalent to finding the nonzero solutions of a nonlinear system of equations, and then propose a modified normalized Newton method (MNNM) for it. Our proposed MNNM method is proved to be locally and cubically convergent under some suitable conditions, which greatly improves the Newton correction method and the orthogonal Newton correction method recently provided by Jaffe, Weiss and Nadler since these two methods only enjoy a quadratic rate of convergence. As an application, the unitary symmetric eigenpairs of a complex-valued symmetric tensor arising from the computation of quantum entanglement in quantum physics are calculated by the MNNM method. Some numerical results are presented to illustrate the efficiency and effectiveness of our method. K E Y W O R D S-eigenpairs, US-eigenpairs, cubical convergence, modified normalized Newton method, Newton correction method, symmetric tensors INTRODUCTIONThe eigenvalue problems of tensors have attracted much attention in recent years because of their applications in high-order Markov chains, 1 magnetic resonance imaging, 2 quantum entanglement, 3,4 automatical control, 5 the best rank-one approximations in statistical data analysis, 6-10 learning binary latent variable models, 11 and so on. Unlike matrices, there are various definitions of eigenvalues for tensors, including -eigenvalues, -eigenvalues, and -eigenvalues; see, for example, other works 12-18 and the references therein. In this paper, we are mainly concerned with the calculation of -eigenvalues and corresponding -eigenvectors. This kind of eigenvalues and eigenvectors has practical applications, for example, the smallest -eigenvalue can reflect the stability of a nonlinear autonomous system in automatic control, and the principal -eigenvector can depict the orientations of nerve fibers in the voxel of white matter of human brain. 5,19 Computing -eigenpairs of a third or higher order tensor is equivalent to finding nontrivial solutions of a system of inhomogeneous polynomial equations in several variables. This implies that the task in the computation of -eigenpairs of a tensor will be much more difficult when its order and dimension are very large. For this reason, some iterative methods have been developed to compute one or more -eigenpairs of the tensors with special structure, such as nonnegative tensors and symmetric tensors.Recently, for nonnegative tensors, Guo et al. 20 proposed a modified Newton iteration to compute some nonnegative -eigenpairs and showed that this method has local quadratic convergence under appropriate assumptions. Kuo et al. 21 presented a homotopy continuation method for the same purpose. For symmetric tensors, Cui, Dai and Numer Linear Algebra Appl. 2020;27:e2284.wileyonlinelibrary.com/journal/nla

A Rayleigh quotient‐gradient neural network method for computing 𝒵‐eigenpairs of general tensors

“…For RGNN model, computing -eigenpairs of a tensor is equivalent to finding the nonzero solutions of a nonlinear system of equations. 1 Finally, numerical experiments are given to show that RGNN has efficient computation for some large-scale tensors besides computing all the -eigenpairs of small-scale general tensors. Whereas, how to choose a suitable parameter still remains an open problem.…”

“…In recent years, there has been tremendous interest in computing ‐eigenpairs of tensors defined as follows: 1 Let be an m th‐order n ‐dimensional tensor. The goal is to find a real vector‐scalar pair with satisfying where is a vector in with .…”

“…For instance, Zhao et al 1 presented a modified normalized Newton method (MNNM) for getting ‐eigenpairs of symmetric tensors, which locally and cubically converges to a ‐eigenpair under the ‐Newton‐stable. Moreover, it also overcomes the drawback of the Newton correction method (NCM) 9 which may fail to converge to a ‐eigenpair with ‐eigenvalue zero.…”

“…In this article, we wish to investigate the computation of ‐eigenpairs for general tensors (not limited to symmetric or nonnegative tensors). Although MNI 18 and MNNM 1 can capture most or all of the eigenvalues, the computations are very expensive since these methods need to solve the system of different equations at each step. On the other hand, the computation of ‐eigenpairs will become a very challenging problem when the order and dimension of tensors are very large.…”

The Projected Newton Iteration Approach for Computing the Nonnegative Z-Eigenpairs of Nonnegative Tensors

“…To address this issue, Kolda and Mayo in [11] further added an adaptive procedure for choosing the shift. Most recently, Zhao et al [20] proposed a modified normalized Newton method (MNNM) for computing Z-eigenpairs of symmetric tensors which can be convergent cubically. For the nonnegative tensors, Guo et al [7] proposed a modified Newton iteration (MNI) to find some positive Z-eigenpairs and showed that their method has a local quadratic convergence under appropriate assumptions.…”