A modified Newton iteration for finding nonnegative Z-eigenpairs of a nonnegative tensor

Guo, Chun Hua; Lin, Wen‐Wei; Liu, Ching Sung

doi:10.1007/s11075-018-0498-y

Cited by 15 publications

(27 citation statements)

References 19 publications

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“…Here we derive a Newton-based method for computing the eigenpairs of symmetric tensors. Recently, [12] considered a similar approach for finding some nonnegative eigenpairs of a nonnegative tensor.…”

Section: Newton Correction Methodsmentioning

confidence: 99%

“…However, solving (12) exactly for y * is as difficult as finding the eigenpair (x * , λ * ) of the tensor T we started from. Instead, we devise an iterative Newton correction method (NCM) that solves (12) only approximately. Given the approximation x (k) of x * at the k th iteration, NCM computes a new approximation x (k+1) by neglecting the high order terms ∆(x, y * ) in (12).…”

Section: Newton Correction Methodsmentioning

confidence: 99%

“…Instead, we devise an iterative Newton correction method (NCM) that solves (12) only approximately. Given the approximation x (k) of x * at the k th iteration, NCM computes a new approximation x (k+1) by neglecting the high order terms ∆(x, y * ) in (12). This amounts to solving the system of n linear equations…”

Section: Newton Correction Methodsmentioning

confidence: 99%

“…Hence, it suffices to bound y * − y (0) . To this end, we view the exact system of non-linear equations (12), whose solution is y * , as a perturbation of the approximate system of linear equations (14), whose solution is y (0) . Consider the matrix A of Eq.…”

Section: Lemma 2 Consider One Update Step Of Algorithm 1 As In Equationmentioning

confidence: 99%

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Newton Correction Methods for Computing Real Eigenpairs of Symmetric Tensors

Jaffe

Weiss

Nadler

2018

SIAM J. Matrix Anal. & Appl.

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Real eigenpairs of symmetric tensors play an important role in multiple applications. In this paper we propose and analyze a fast iterative Newton-based method to compute real eigenpairs of symmetric tensors. We derive sufficient conditions for a real eigenpair to be a stable fixed point for our method, and prove that given a sufficiently close initial guess, the convergence rate is quadratic. Empirically, our method converges to a significantly larger number of eigenpairs compared to previously proposed iterative methods, and with enough random initializations typically finds all real eigenpairs. In particular, for a generic symmetric tensor, the sufficient conditions for local convergence of our Newton-based method hold simultaneously for all its real eigenpairs.

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Section: Newton Correction Methodsmentioning

confidence: 99%

Section: Newton Correction Methodsmentioning

confidence: 99%

Section: Newton Correction Methodsmentioning

confidence: 99%

Section: Lemma 2 Consider One Update Step Of Algorithm 1 As In Equationmentioning

confidence: 99%

See 2 more Smart Citations

Newton Correction Methods for Computing Real Eigenpairs of Symmetric Tensors

Jaffe

Weiss

Nadler

2018

SIAM J. Matrix Anal. & Appl.

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“…Observe that x * =S(x * ) and x (k+1) =S(x (k) ) for the MNNM method, therefore, using Taylor's expansion and Equation (20) we have…”

Section: F I G U R E 1 the Value Of H(x) On The Unit Sphere For Exampmentioning

confidence: 99%

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

Zhao

Zheng

Liang

et al. 2020

Numerical Linear Algebra App

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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

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Continuation methods for nonnegative rank‐1 approximation of nonnegative tensors

Hsu

Kuo

Liu

2021

Numerical Linear Algebra App

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In this article, the rank-1 approximation of a nonnegative tensor  ∈ Ris considered. Mathematically, the approximation problem can be formulated as an optimization problem. The Karush-Kuhn-Tucker (KKT) point of the optimization problem can be obtained by computing the nonnegative Z-eigenvector y of enlarged tensor . Therefore, we propose an iterative method with prediction and correction steps for computing nonnegative Z-eigenvector y of enlarged tensor , called the continuation method. In the theoretical part, we show that the computation requires only O( ∏ m i=1 n i ) flops for each iteration and the computed Z-eigenvector y has nonzero component block, and hence, the KKT point can be obtained. In addition, we show that the KKT point is a local optimizer of the optimization problem. Numerical experiments are provided to support the theoretical results.

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A modified Newton iteration for finding nonnegative Z-eigenpairs of a nonnegative tensor

Cited by 15 publications

References 19 publications

Newton Correction Methods for Computing Real Eigenpairs of Symmetric Tensors

Newton Correction Methods for Computing Real Eigenpairs of Symmetric Tensors

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

Continuation methods for nonnegative rank‐1 approximation of nonnegative tensors

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