Abstract
This paper considers the convergence analysis of the H-eigenvalues for a class of real symmetric and convergent tensor sequences. We first establish convergence results of some sequences of points. Then we study the behaviors of the H-eigenvalues and H-eigenvectors of the convergent tensor sequence. In particular, we obtain the convergence properties of the largest and smallest H-eigenvalues of the tensor sequence. Eventually, the corresponding numerical results are presented to verify our theoretical findings.
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References
Sidiropoulos, N.D., De Lathauwer, L., Fu, X., et al.: Tensor decomposition for signal processing and machine learning. IEEE J. Trans. Signal Process. 65(13), 3551–3582 (2017)
Che, M.L., Wei, Y.M.: Theory and Computation of Complex Tensors and its Applications. Springer, Singapore (2020)
Novikov, A., Trofimov, M., Oseledets, I.: Exponential machines, J. ArXiv:1605.03795 (2016)
Stoudenmire, E.M., Schwab, D.J.: Supervised learning with tensor networks. J. Adv. Neural Inf. Process. Syst. 29, 4799 (2016)
Sengupta, R., et al.: Tensor networks in machine learning. Eur. Math. Soc. Mag. 126, 4–12 (2022)
Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM. J. Rev. 51(3), 455–500 (2009)
Liu, Y.P., Liu, J.N., Long, Z., Zhu, C.: Tensor Computation for Data Analysis. Springer, Cham (2022)
Emil, F.: Tensor Calculus for Engineers and Physicists. Springer, Cham (2016)
Qi, L.Q.: Eigenvalues of a real supersymmetric tensor. J. Symbol Comput. 40(6), 1302–1324 (2005)
Qi, L.Q.: Eigenvalues of a Supersymmetric Tensor and Positive Definiteness of an Even Degree Multivariate Form. The Hong Kong Polytechnic University, Department of Applied Mathematics (2004)
Hu, S., Qi, L.: The Laplacian of a uniform hypergraph. J. Comb. Optim. 29(2), 331–366 (2015)
Bulò, M, S.R.: Pelillo, New bounds on the clique number of graphs based on spectral hypergraph theory. International Conference on Learning and Intelligent Optimization, Springer Verlag, Berlin. (2009) 45–58
Xie, J., Chang, A.: H-Eigenvalues of signless Laplacian tensor for an even uniform hypergraph. J. Front. Math. China. 8, 107–127 (2013)
Yuan, X., Qi, L.Q., Shao, J.: The proof of a conjecture on largest Laplacian and signless Laplacian H-eigenvalues of uniform hypergraphs. J. Linear Algebra Appl. 490, 18–30 (2016)
Sun, R., Wang, W.H.: On the spectral radius of uniform weighted hypergraph. J. Discret. Math. Algorithms Appl. 15(01), 2250067 (2023)
Qi, L.Q., Chen, H.B., Chen, Y.N.: Tensor Eigenvalues and Their Applications. Springer, Singapore (2018)
Li, R.C.: Relative perturbation theory: I. Eigenvalue and singular value variations. SIAM J. Matr. Anal. Appl. 19(4), 956–982 (1998)
Kofidis, E., Regalia, P.A.: On the best rank-1 approximation of higher-order supersymmetric tensors. SIAM J. Matrix Anal. Appl. 23(3), 863–884 (2002)
Wang, Y., Qi, L.Q.: On the successive supersymmetric rank-1 decomposition of higher-order supersymmetric tensors. J. Numer. Linear Algebra Appl. 14(6), 503–519 (2007)
Song, Y.S., Qi, L.Q.: Properties of some classes of structured tensors. J. Optim. Theory Appl. 165(3), 854–873 (2015)
Chen, H.B., Chen, Y.N., Li, G.Y., Qi, L.Q.: A semidefinite program approach for computing the maximum eigenvalue of a class of structured tensors and its applications in hypergraphs and copositivity test. J. Numer. Linear Algebra Appl. 25(1), e2125 (2018)
Shang, T.T., Tang, G.J.: Expected residual minimization method for stochastic tensor variational inequalities, J. Oper. Res. Soc. China. (2022) 1-24
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, CAMSAP 05,vol.1,IEEE Computer Society Press, Piscataway,NJ, (2005), pp. 129-132
Wang, Y., Huang, Z.H., Qi, L.Q.: Global uniqueness and solvability of tensor variational inequalities. J. Optim. Theory Appl. 177, 137–152 (2018)
Barnett, S.: Matrices, Methods and Applications. Clarenden Press, Oxford (1990)
Luo, M.J., Lin, G.H.: Expected residual minimization method for stochastic variational inequality problems. J. Optim. Theory Appl. 140(1), 103–116 (2009)
Acknowledgements
The authors thank anonymous referees for their professional and helpful comments. The research was partially supported by the Scientific Research Project of Guangxi Minzu University (2021KJQD05), the Guangxi Science and Technology Plan Project (guikeAD22035021), the Basic Ability Enhancement Program for Young and Middle-aged Teachers of Guangxi (2022KY0163), the National Natural Science Foundation of China (12261008), the Xiangsihu Young Scholars and Innovative Research Team of GXMZU (2022GXUNXSHQN02).
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Lan, Z., Liu, J. & Jiang, X. Convergence analysis of the largest and smallest H-eigenvalues for a class of tensor sequences. J. Appl. Math. Comput. (2024). https://doi.org/10.1007/s12190-024-02096-2
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DOI: https://doi.org/10.1007/s12190-024-02096-2