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Computing the generalized eigenvalues of weakly symmetric tensors

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Abstract

Tensor is a hot topic in the past decade and eigenvalue problems of higher order tensors become more and more important in the numerical multilinear algebra. Several methods for finding the Z-eigenvalues and generalized eigenvalues of symmetric tensors have been given. However, the convergence of these methods when the tensor is not symmetric but weakly symmetric is not assured. In this paper, we give two convergent gradient projection methods for computing some generalized eigenvalues of weakly symmetric tensors. The gradient projection method with Armijo step-size rule (AGP) can be viewed as a modification of the GEAP method. The spectral gradient projection method which is born from the combination of the BB method with the gradient projection method is superior to the GEAP, AG and AGP methods. We also make comparisons among the four methods. Some competitive numerical results are reported at the end of this paper.

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References

  1. Bader, B.W., Kolda, T.G., et al.: Matlab tensor toolbox version 2.6, (Feb 2015). http://www.sandia.gov/~tgkolda/TensorToolbox/

  2. Barzilai, J., Borwein, J.M.: Two-point step size gradient methods. IMA J. Numer. Anal. 8, 141–148 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bertsekas, D.P.: On the Goldstein-Levitin-Polyak gradient projection method. IEEE Trans. Autom. Control 21, 174–184 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bertsekas, D.P., Tsitsiklis, J.N.: Parallel and Distributed Computation: Numerical Methods. Prentice-Hall, Englewood Cliffs, NJ (1989)

    MATH  Google Scholar 

  5. Bertsekas, D.P.: Nonlinear Programming, 2nd edn. Athena Scientific, Belmont (1999)

    MATH  Google Scholar 

  6. Bi, Y., Liu, X.: A new nonmonotone spectral projected gradient method for bound constrained optimization. Math. Numer. Sin. 35, 419–430 (2013)

    MathSciNet  MATH  Google Scholar 

  7. Birgin, E.G., Martinez, J.M., Raydan, M.: Nonmonotone spectral projected gradient methods on convex sets. SIAM J. Optim. 10, 1196–1211 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  8. Birgin, E.G., Martínez, J. M., Raydan, M.: Spectral projected gradient methods: review and perspectives, J. Stat. Softw., 60 (2014). http://www.jstatsoft.org/

  9. Cartwright, D., Sturmfels, B.: The number of eigenvalues of a tensor. Linear Algebra Appl. 438, 942–952 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chang, K.C., Pearson, K., Zhang, T.: Perron-Frobenius theorem for nonnegative tensors. Commun. Math. Sci. 6, 507–520 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chang, K.C., Pearson, K., Zhang, T.: On eigenvalue problems of real symmetric tensors. J. Math. Anal. Appl. 350, 416–422 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chang, K.C., Qi, L., Zhou, G.: Singular values of a real rectangular tensor. J. Math. Anal. Appl. 370, 284–294 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chang, K.C., Pearson, K., Zhang, T.: Some variational principles for \(Z\)-eigenvalues of nonnegative tensors. Linear Algebra Appl. 438, 4166–4182 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen, Y., Qi, L., Wang, Q.: Computing extreme eigenvalues of large scale Hankel tensors. J. Sci. Comput. 68, 716–738 (2016)

    Article  MathSciNet  Google Scholar 

  15. Combettes, P.L., Pesquet, J.C.: Proximal splitting methods in signal processing. In: Bauschke, H.H., Burachik, S.R., Combettes, P.L., Elser, V., Luke, R.D., Wolkowicz, H. (eds.) Fixed-Point Algorithms for Inverse Problems in Science and Engineering, vol. 49, pp. 185–212. Springer, New York (2011). doi:10.1007/978-1-4419-9569-8_10

  16. Cui, C., Dai, Y., Nie, J.: All real eigenvalues of symmetric tensors. SIAM J. Matrix Anal. Appl. 35, 1582–1601 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Han, L.: An unconstrained optimization approach for finding real eigenvalues of even order symmetric tensors. Numer. Algebra Control Optim. 3, 583–599 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hao, C., Cui, C., Dai, Y.: A sequential subspace projection method for extreme \(Z\)-eigenvalues of supersymmetric tensors. Numer. Linear Algebra Appl. 22, 283–298 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hu, S., Li, G., Qi, L., Song, Y.: Finding the maximum eigenvalue of essentially nonnegative symmetric tensors via sum of squares programming. J. Optim. Theory Appl. 158, 717–738 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kofidis, E., Regalia, P.A.: On the best rank-1 approximation of higher-order symmetric tensors. SIAM J. Matrix Anal. Appl. 23, 863–884 (2002)

    Article  MATH  Google Scholar 

  21. Kolda, T.G., Mayo, J.R.: Shifted power method for computing tensor eigenpairs. SIAM J. Matrix Anal. Appl. 32, 1095–1124 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kolda, T.G., Mayo, J.R.: An adaptive shifted power method for computing generalized tensor eigenpairs. SIAM J. Matrix Anal. Appl. 35, 1563–1581 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Li, G., Qi, L., Yu, G.: The \(Z\)-eigenvalues of a symmetric tensor and its application to spectral hypergraph theory. Numer. Linear Algebra Appl. 20, 1001–1029 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lim, L. H.: Singular values and eigenvalues of tensors: a variational approach. In: Proceedings of the IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, pp. 129–132 (2005)

  25. Nocebal, J., Wright, S.J.: Numerical Optimization. Springer, New York (1999)

    Book  Google Scholar 

  26. Qi, L.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40, 1302–1324 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Qi, L., Sun, W., Wang, Y.: Numerical multilinear algebra and its applications. Front. Math. China 2, 501–526 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. Qi, L.: Eigenvalues and invariants of tensors. J. Math. Anal. Appl. 325, 1363–1377 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Qi, L., Wang, Y., Wu, E.X.: \(D\)-eigenvalues of diffusion kurtosis tensors. J. Comput. Appl. Math. 221, 150–157 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  30. Qi, L., Wang, F., Wang, Y.: \(Z\)-eigenvalue methods for a global polynomial optimization problem. Math. Program. 118, 301–316 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. Qi, L.: The best rank-one approximation ratio of a tensor space. SIAM J. Matrix Anal. Appl. 32, 430–442 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  32. Schatz, M.D., Low, T.M., Van De Geijn, R.A., Kolda, T.G.: Exploiting symmetry in tensors for high performance: multiplication with symmetric tensors. SIAM J. Sci. Comput. 36, 453–479 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  33. Wang, Y., Xiu, N.: Nonlinear Optimization Theory and Methods. Science Press, Beijing (2012)

    Google Scholar 

  34. Wolfram Research, INC., Mathematica, Version 7.0, (2008)

  35. Yang, Y., Yang, Q.: Further results for Perron–Frobenius theorem for nonnegative tensors. SIAM J. Matrix Anal. Appl. 31, 2517–2530 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  36. Yu, G., Yu, Z., Xu, Y., Song, Y., Zhou, Y.: An adaptive gradient method for computing generalized tensor eigenpairs. Comput. Optim. Appl. (2016). doi:10.1007/s10589-016-9846-9

  37. Zhang, T., Golub, G.H.: Rank-1 approximation of higher-order tensors. SIAM J. Matrix Anal. Appl. 23, 534–550 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

We are grateful to Mr. Zhongming Chen for his helpful discussion and the anonymous referees for their constructive comments and valuable suggestions which have contributed to the revision of this paper. Q. Yang’s work is supported by the National Natural Science Foundation of China (Grant no. 11271206).

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Authors and Affiliations

  1. School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, People’s Republic of China

    Na Zhao, Qingzhi Yang & Yajun Liu

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  1. Na Zhao
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  2. Qingzhi Yang
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  3. Yajun Liu
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Correspondence to Qingzhi Yang.

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Zhao, N., Yang, Q. & Liu, Y. Computing the generalized eigenvalues of weakly symmetric tensors. Comput Optim Appl 66, 285–307 (2017). https://doi.org/10.1007/s10589-016-9865-6

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  • DOI: https://doi.org/10.1007/s10589-016-9865-6

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