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Computing Extreme Eigenvalues of Large Scale Hankel Tensors

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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 \(\mathcal {O}(mn\log (mn))\). 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–Łojasiewicz 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.

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Notes

  1. See “http://en.wikipedia.org/wiki/Cayley_transform”.

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Acknowledgments

We thank Mr. Weiyang Ding and Dr. Ziyan Luo for the discussion on numerical experiments, and two referees for their valuable comments.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, China

    Yannan Chen

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

    Liqun Qi & Qun Wang

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  1. Yannan Chen
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  2. Liqun Qi
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  3. Qun Wang
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Corresponding author

Correspondence to Liqun Qi.

Additional information

Yannan Chen: This author’s work was supported by the National Natural Science Foundation of China (Grant No. 11401539) and the Development Foundation for Excellent Youth Scholars of Zhengzhou University (Grant No. 1421315070). Liqun Qi: This author’s work was partially supported by the Hong Kong Research Grant Council (Grant No. PolyU 502111, 501212, 501913 and 15302114).

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Chen, Y., Qi, L. & Wang, Q. Computing Extreme Eigenvalues of Large Scale Hankel Tensors. J Sci Comput 68, 716–738 (2016). https://doi.org/10.1007/s10915-015-0155-8

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