Abstract
In this chapter we discuss the computation of US-eigenpairs of complex symmetric tensors and U-eigenpairs of complex tensors. We derive an iterative algorithm for computing US-eigenpairs of complex symmetric tensors which is based on the Takagi factorization of complex symmetric matrices that are denoted as the QRCST Algorithm. We observe that multiple US-eigenpairs can be found from a local permutation heuristic, which is effectively a tensor similarity transformation, resulting in the permuted version of QRCST. We also present a generalization of their techniques to general complex tensors and derive a higher-order power type method for computing a US- or a U-eigenpair, similar to the higher-order power method for computing Z-eigenpairs of real symmetric tensors or a best rank-one approximation of real tensors. We illustrate the algorithms via numerical examples.
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References
G. Ni, L. Qi, M. Bai, Geometric measure of entanglement and U-eigenvalues of tensors. SIAM J. Matrix Anal. Appl. 35(1), 73–87 (2014)
T. Wei, P. Goldbart, Geometric measure of entanglement and applications to bipartite and multipartite quantum states. Phys. Rev. A 68(4), 042307 (2003)
S. Ragnarsson, C. Van Loan, Block tensors and symmetric embeddings. Linear Algebra Appl. 438(2), 853–874 (2013)
L. Autonne, Sur les matrices hypo-hermitiennes et sur les matrices unitaires. Ann. Univ. Lyon Nouv. Sér. I FASC 38, 1–77 (1915)
R. Horn, C. Johnson, Matrix Analysis, 2nd edn. (Cambridge University, Cambridge, 2014)
L. Hua, On the theory of automorphic functions of a matrix variable, II-the classification of hypercircles under the symplectic group. Am. J. Math. 66(4), 531–563 (1944)
L. Qi, Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40(6), 1302–1324 (2005)
L. Qi, Hankel tensors: Associated Hankel matrices and Vandermonde decomposition. Commun. Math. Sci. 13(1), 113–125 (2015)
J. Papy, L. De Lathauwer, S. Van Huffel, Exponential data fitting using multilinear algebra: the single-channel and multi-channel case. Numer. Linear Algebra Appl. 12(8), 809–826 (2005)
W. Ding, L. Qi, Y. Wei, Fast Hankel tensor-vector products and application to exponential data fitting. Numer. Linear Algebra Appl. 22(5), 814–832 (2015)
L. Sorber, M. Van Barel, L. De Lathauwer, Unconstrained optimization of real functions in complex variables. SIAM J. Optim. 22(3), 879–898 (2012)
S. Zhang, Y. Xia, J. Wang, A complex-valued projection neural network for constrained optimization of real functions in complex variables. IEEE Trans. Neural Netw. 26(12), 3227–3238 (2015)
S. Zhang, Y. Xia, W. X. Zheng, A complex-valued neural dynamical optimization approach and its stability analysis. Neural Netw. 61, 59–67 (2015)
E. Kofidis, P. Regalia, On the best rank-1 approximation of higher-order supersymmetric tensors. SIAM J. Matrix Anal. Appl. 23(3), 863–884 (2002)
L. De Lathauwer, B. De Moor, J. Vandewalle, On the best rank-1 and rank-(r 1, r 2, ..., r n) approximation of higher-order tensors. SIAM J. Matrix Anal. Appl. 21(4), 1324–1342 (2000)
M. Ishteva, P. Absil, P. Van Dooren, Jacobi algorithm for the best low multilinear rank approximation of symmetric tensors. SIAM J. Matrix Anal. Appl. 34(2), 651–672 (2013)
J. Nie, L. Wang, Semidefinite relaxations for best rank-1 tensor approximations. SIAM J. Matrix Anal. Appl. 35(3), 1155–1179 (2014)
A. Boralevi, J. Draisma, E. Horobet, E. Robeva, Orthogonal and Unitary Tensor Decomposition from an Algebraic Perspective (2014). ArXiv preprint:1512.08031v1
B. Jiang, Z. Li, S. Zhang, Characterizing real-valued multivariate complex polynomials and their symmetric tensor representations. SIAM J. Matrix Anal. Appl. 37(1), 381–408 (2016)
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Che, M., Wei, Y. (2020). US- and U-Eigenpairs of Complex Tensors. In: Theory and Computation of Complex Tensors and its Applications. Springer, Singapore. https://doi.org/10.1007/978-981-15-2059-4_7
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DOI: https://doi.org/10.1007/978-981-15-2059-4_7
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