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Further Results for \(Z\)-Eigenvalue Localization Theorem for Higher-Order Tensors and Their Applications

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Abstract

In this paper, we present some new \(Z\)-eigenvalue inclusion theorem for tensors by categorizing the entries of tensors, and prove that these sets are more precise than existing results. On this basis, some lower and upper bounds for the \(Z\)-spectral radius of weakly symmetric nonnegative tensors are proposed, which improves some of the existing results. As applications, we give some estimates of the best rank-one approximation rate in weakly symmetric nonnegative tensors and the maximal orthogonal rank of real orthogonal tensors, and our results are more precise than existing result in some situations. In particular, for a given symmetric multipartite pure state with nonnegative amplitudes in real field, some theoretical lower and upper bounds for the geometric measure of entanglement are also derived in terms of the bounds for \(Z\)-spectral radius. Numerical examples are given to illustrate validity and superiority of our results.

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References

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

    MathSciNet  MATH  Google Scholar 

  2. Chang, K., Pearson, K., Zhang, T.: Some variational principles for \(Z\)-eigenvalues of nonnegative tensors. Linear Algebra Appl. 438(11), 4166–4182 (2013)

    MathSciNet  MATH  Google Scholar 

  3. Lim, L.: Singular values and eigenvalues of tensors: a variational approach. In: CAMSAP’05: Proceeding of the IEEE International Workshop on Computational Advances in Multi-SensorAdaptive Processing, pp. 129–132 (2005)

    Google Scholar 

  4. Lathauwer, L., Moor, B., Vandewalle, J.: On the best rank-1 and rank-(\(R_{1},R_{2},\dots,R_{N}\)) approximation of higher-order tensors. SIAM J. Matrix Anal. Appl. 21, 1324–1342 (2000)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  6. Qi, L.: Rank and eigenvalues of a supersymmetric tensor, the multivariate homogeneous polynomial and the algebraic hypersurface it defines. J. Symb. Comput. 41(12), 1309–1327 (2006)

    MathSciNet  MATH  Google Scholar 

  7. Ng, M.K., Qi, L., Zhou, G.: Finding the largest eigenvalue of a nonnegative tensor. SIAM J. Matrix Anal. Appl. 31, 1090–1099 (2009)

    MathSciNet  MATH  Google Scholar 

  8. Li, W., Ng, M.K.: On the limiting probability distribution of a transition probability tensor. Linear Multilinear Algebra 62(3), 362–385 (2014)

    MathSciNet  MATH  Google Scholar 

  9. Peter, M.: Tensor Methods in Statistics. Chapman and Hall, London (1987)

    MATH  Google Scholar 

  10. Pierre, C.: Blind channel identification and extraction of more sources than sensors. Proc. SPIE Int. Soc. Opt. Eng. 3461, 2–13 (1998)

    Google Scholar 

  11. Goldberg Zimring, D., Mewes, A., Maddah, M., et al.: Diffusion tensor magnetic resonance imaging in multiple sclerosis. J. Neuroimaging 15(Supplement s4), 68–81 (2005)

    Google Scholar 

  12. André, L.d.A., Gérard, F., Mota, J.C.M.: Parafac-based unified tensor modeling for wireless communication systems with application to blind multiuser equalization. Signal Process. 87(2), 337–351 (2007)

    MATH  Google Scholar 

  13. da Costa, M.N., Favier, G., Romano, J.M.T.: Tensor modelling of MIMO communication systems with performance analysis and Kronecker receivers. Signal Process. 145, 304–316 (2018)

    Google Scholar 

  14. Cardoso, J.-F.: High-order contrasts for independent component analysis. Neural Comput. 11(1), 157 (1999)

    MathSciNet  Google Scholar 

  15. Che, M., Cichocki, A., Wei, Y.: Neural networks for computing best rank-one approximations of tensors and its applications. Neurocomputing 267, 114–133 (2017)

    Google Scholar 

  16. Qi, L.: The best rank-one approximation ratio of a tensor space. SIAM J. Matrix Anal. Appl. 32(2), 430–442 (2011)

    MathSciNet  MATH  Google Scholar 

  17. Ammar, A., Chinesta, F., Falcó, A.: On the convergence of a greedy rank-one update algorithm for a class of linear systems. Arch. Comput. Methods Eng. 17(4), 473–486 (2010)

    MathSciNet  MATH  Google Scholar 

  18. DeVore, R.A., Temlyakov, V.N.: Some remarks on greedy algorithm. Adv. Comput. Math. 5(1), 173–187 (1996)

    MathSciNet  MATH  Google Scholar 

  19. Falcó, A., Nouy, A.: A proper generalized decomposition for the solution of elliptic problems in abstract form by using a functional Eckart-Young approach. J. Math. Anal. Appl. 376(2), 469–480 (2011)

    MathSciNet  MATH  Google Scholar 

  20. Leibovici, D., ElMaache, H.: A singular value decomposition of an element belonging to a tensor product of k separable Hilbert spaces. C. R. Math. Acad. Sci. Paris 325, 779–782 (1997)

    MATH  Google Scholar 

  21. Wang, Y., Qi, L.: On the successive supersymmetric rank-1 decomposition of higher order supersymmetric tensors. Numer. Linear Algebra Appl. 14(6), 503–519 (2007)

    MathSciNet  MATH  Google Scholar 

  22. Uschmajew, A.: The best rank-1 approximation of a symmetric tensor and related spherical optimization problems. In: Proceedings of the International Conference on Sampling Theory and Applications, pp. 415–419 (2015)

    Google Scholar 

  23. Nielsen, M.A., Chuang, I.L.: Quantum Computing and Quantum Information. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  24. Wei, T.-C., Goldbart, P.M.: Geometric measure of entanglement and applications to bipartite and multipartite quantum states. Phys. Rev. A 68, 042307 (2003)

    Google Scholar 

  25. Hu, S., Qi, L., Zhang, G.: Computing the geometric measure of entanglement of multipartite pure states by means of non-negative tensors. Phys. Rev. A 93(1), 012304 (2016)

    MathSciNet  Google Scholar 

  26. Shimony, A.: Degree of entanglement. Ann. N.Y. Acad. Sci. 755, 675 (1995)

    MathSciNet  Google Scholar 

  27. Hübener, R., Kleinmann, M., Wei, T.-C., et al.: Geometric measure of entanglement for symmetric states. Phys. Rev. A 80(3), 032324 (2009)

    MathSciNet  Google Scholar 

  28. Qi, L.: Eigenvalues and invariants of tensors. J. Math. Anal. Appl. 325(2), 1363–1377 (2007)

    MathSciNet  MATH  Google Scholar 

  29. Qi, L., Zhang, G., Ni, G.: How entangled can a multi-party system possibly be? Phys. Lett. A 382(22), 1465–1471 (2018)

    MathSciNet  MATH  Google Scholar 

  30. Kolda, T.G., Mayo, J.R.: Shifted power method for computing tensor eigenpairs. SIAM J. Matrix Anal. Appl. 32(4), 1095–1124 (2011)

    MathSciNet  MATH  Google Scholar 

  31. Kolda, T.G., Mayo, J.R.: An adaptive shifted power method for computing generalized tensor eigenpairs. SIAM J. Matrix Anal. Appl. 35(4), 1563–1581 (2014)

    MathSciNet  MATH  Google Scholar 

  32. Cui, C.-F., Dai, Y.-H., Nie, J.: All real eigenvalues of symmetric tensors. SIAM J. Matrix Anal. Appl. 35(4), 1582–1601 (2014)

    MathSciNet  MATH  Google Scholar 

  33. Chen, L., Han, L., Zhou, L.: Computing tensor eigenvalues via homotopy methods. SIAM J. Matrix Anal. Appl. 37(1), 290–319 (2016)

    MathSciNet  MATH  Google Scholar 

  34. Chen, L., Han, L., Yin, H., et al.: A homotopy method for computing the largest eigenvalue of an irreducible nonnegative tensor. J. Comput. Appl. Math. 355(1), 174–181 (2019)

    MathSciNet  MATH  Google Scholar 

  35. Wang, G., Zhou, G., Caccetta, L.: \(Z\)-eigenvalue inclusion theorems for tensors. Discrete Contin. Dyn. Syst., Ser. B 22(1), 187–198 (2017)

    MathSciNet  MATH  Google Scholar 

  36. Zhao, J.: A new \(Z\)-eigenvalue localization set for tensors. J. Inequal. Appl. 2017, 85 (2017)

    MathSciNet  MATH  Google Scholar 

  37. Sang, C.: A new Brauer-type \(Z\)-eigenvalue inclusion set for tensors. Numer. Algorithms 80, 781–794 (2019)

    MathSciNet  MATH  Google Scholar 

  38. Varga, R.: Gersgorin and His Circles. Springer, Berlin (2004)

    MATH  Google Scholar 

  39. Li, C., Li, Y.: An eigenvalue localization set for tensors with applications to determine the positive (semi-)definiteness of tensors. Linear Multilinear Algebra 64(4), 587–601 (2016)

    MathSciNet  MATH  Google Scholar 

  40. Song, Y., Qi, L.: Spectral properties of positively homogeneous operators induced by higher order tensors. SIAM J. Matrix Anal. Appl. 34(4), 1581–1595 (2013)

    MathSciNet  MATH  Google Scholar 

  41. He, J., Huang, T.-Z.: Upper bound for the largest \(Z\)-eigenvalue of positive tensors. Appl. Math. Lett. 38, 110–114 (2014)

    MathSciNet  MATH  Google Scholar 

  42. Li, W., Liu, D., Vong, S.-W.: \(z\)-Eigenpair bounds for an irreducible nonnegative tensor. Linear Algebra Appl. 483, 182–199 (2015)

    MathSciNet  MATH  Google Scholar 

  43. Zhang, X., Ling, C., Qi, L.: The best rank-1 approximation of a symmetric tensor and related spherical optimization problems. SIAM J. Matrix Anal. Appl. 33(3), 806–821 (2012)

    MathSciNet  MATH  Google Scholar 

  44. Jiang, Y.-L., Kong, X.: On the uniqueness and perturbation to the best rank-one approximation of a tensor. SIAM J. Matrix Anal. Appl. 36(2), 775–792 (2015)

    MathSciNet  MATH  Google Scholar 

  45. Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51(3), 455–500 (2009)

    MathSciNet  MATH  Google Scholar 

  46. Uschmajew, A.: Local convergence of the alternating least squares algorithm for canonical tensor approximation. SIAM J. Matrix Anal. Appl. 33(2), 639–652 (2012)

    MathSciNet  MATH  Google Scholar 

  47. Zhang, T., Golub, G.H.: Rank-one approximation to high order tensors. SIAM J. Matrix Anal. Appl. 23(2), 534–550 (2001)

    MathSciNet  MATH  Google Scholar 

  48. Friedland, S., Mehrmann, V., Pajarola, R., et al.: On best rank one approximation of tensors. Numer. Linear Algebra Appl. 20(6), 942–955 (2013)

    MathSciNet  MATH  Google Scholar 

  49. Ishteva, M., Absil, P.-A., Huffel, S., Lathauwer, L.: Best low multilinear rank approximation of higher-order tensors, based on the Riemannian trust-region scheme. SIAM J. Matrix Anal. Appl. 32(1), 115–135 (2011)

    MathSciNet  MATH  Google Scholar 

  50. Savas, B., Lim, L.: Quasi-Newton methods on Grassmannians and multilinear approximations of tensors. SIAM J. Sci. Comput. 32(6), 3352–3393 (2009)

    MathSciNet  MATH  Google Scholar 

  51. Nie, J., Wang, L.: Semidefinite relaxations for best rank-1 tensor approximations. SIAM J. Matrix Anal. Appl. 35(3), 1155–1179 (2014)

    MathSciNet  MATH  Google Scholar 

  52. Hillar, C., Lim, L.-H.: Most tensor problems are NP-hard (2013). arXiv:0911.1393

  53. Li, Z., Nakatsukasa, Y., Soma, T., et al.: On orthogonal tensors and best rank-one approximation ratio. SIAM J. Matrix Anal. Appl. 39(1), 400–425 (2018)

    MathSciNet  MATH  Google Scholar 

  54. Cobos, F., Kühn, T., Peetre, J.: On \(\mathfrak{G}_{p}\)-classes of trilinear forms. J. Lond. Math. Soc. 59(1), 1003–1022 (1999)

    Google Scholar 

  55. Wei, T.-C., Das, D., Mukhopadyay, S., et al.: Global entanglement and quantum criticality in spin chains. Phys. Rev. A 71(6), 362 (2005)

    Google Scholar 

  56. Yoshifumi, N., Damian, M., Mio, M.: Thermal robustness of multipartite entanglement of the 1-D spin 1/2 XY model. Phys. Rev. A 79(79), 126 (2009)

    Google Scholar 

  57. Hayashi, M., Markham, D., Murao, M., et al.: The geometric measure of entanglement for a symmetric pure state with non-negative amplitudes. J. Math. Phys. 50, 122104 (2009)

    MathSciNet  MATH  Google Scholar 

  58. Orús, R., Dusuel, S., Vidal, J.: Equivalence of critical scaling laws for many-body entanglement in the Lipkin-Meshkov-Glick model. Phys. Rev. Lett. 101, 025701 (2008)

    Google Scholar 

  59. Qiang, W.-C., Zhang, L.: Geometric measure of quantum discord for entanglement of Dirac fields in noninertial frames. Phys. Lett. B 742, 383–389 (2015)

    MATH  Google Scholar 

  60. Chen, L., Xu, A., Zhu, H.: Computation of the geometric measure of entanglement for pure multiqubit states. Phys. Rev. A 82, 032301 (2010)

    MathSciNet  MATH  Google Scholar 

  61. Ragnarsson, S., Loan, C.F.: Block tensors and symmetric embeddings. Linear Algebra Appl. 438(2), 853–874 (2013)

    MathSciNet  MATH  Google Scholar 

  62. Wei, T.-C., Severini, S.: Matrix permanent and quantum entanglement of permutation invariant states. J. Math. Phys. 51(9), 092203 (2010)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors are grateful to the anonymous referees for their helpful suggestions and comments.

This work was supported by the National Natural Science Foundation of China (grant 11971413, 11571292).

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  1. School of Mathematics and Computational Science & Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, Hunan, 411105, China

    Liang Xiong & Jianzhou Liu

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  1. Liang Xiong
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  2. Jianzhou Liu
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Correspondence to Jianzhou Liu.

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Xiong, L., Liu, J. Further Results for \(Z\)-Eigenvalue Localization Theorem for Higher-Order Tensors and Their Applications. Acta Appl Math 170, 229–264 (2020). https://doi.org/10.1007/s10440-020-00332-y

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