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Neural Networks

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Theory and Computation of Complex Tensors and its Applications

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Abstract

We focus on the rank-one approximation problem of a tensor \(\mathcal {A}\in \mathbb {R}^{I_1\times I_2\times \dots \times I_N}\) by neural networks: finding a real scalar σ and N unit \({\mathbf {x}}_n\in \mathbb {R}^{I_n}\) to minimize

$$\displaystyle \sum _{i_1=1}^{I_1}\sum _{i_2=1}^{I_2}\dots \sum _{i_N=1}^{I_N}[a_{i_1i_2\dots i_N}-\sigma \cdot (x_{1,i_1}x_{2,i_2}\dots x_{N,i_N})]^2, $$

where \(x_{n,i_n}\) is the i nth element of \({\mathbf {x}}_n\in \mathbb {R}^{I_n}\) for all i n and n, and σ > 0 is a scaling factor.

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References

  1. A. Cichocki, R. Unbehauen, Neural Networks for Optimization and Signal Processing (Wiley, New York, 1993)

    MATH  Google Scholar 

  2. L. Liao, H. Qi, L. Qi, Neurodynamical optimization. J. Glob. Optim. 28(2), 175–195 (2004)

    Article  MathSciNet  Google Scholar 

  3. A. Rodriguezvazquez, R. Dominguezcastro, A. Rueda, J. Huertas, E. Sanchezsinencio, Nonlinear switched capacitor ‘neural’ networks for optimization problems. IEEE Trans. Circuits Syst. 37(3), 384–398 (1990)

    Article  MathSciNet  Google Scholar 

  4. A. Cichocki, Neural network for singular value decomposition. Electron. Lett. 28(8), 784–786 (1992)

    Article  Google Scholar 

  5. K. Diamantaras, S. Kung, Cross-correlation neural network models. IEEE Trans. Signal Process. 42(11), 3218–3223 (1994)

    Article  Google Scholar 

  6. D. Feng, Z. Bao, X. Zhang, A cross-associative neural network for SVD of non-squared data matrix in signal processing. IEEE Trans. Neural Netw. 12(5), 1215–1221 (2001)

    Article  Google Scholar 

  7. S. Fiori, Singular value decomposition learning on double stiefel manifold. Int. J. Neural Syst. 13(3), 155–170 (2003)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. L. Liu, W. Wu, Dynamical system for computing largest generalized eigenvalue, in Lecture Notes in Computer Science, vol. 3971 (Springer, Berlin, 2006), pp. 399–404

    Google Scholar 

  10. L. Liu, H. Shao, D. Nan, Recurrent neural network model for computing largest and smallest generalized eigenvalue. Neurocomputing 71(16), 3589–3594 (2008)

    Article  Google Scholar 

  11. Y. Tang, J. Li, Notes on “Recurrent neural network model for computing largest and smallest generalized eigenvalue”. Neurocomputing 73(4), 1006–1012 (2010)

    Article  MathSciNet  Google Scholar 

  12. S. Friedland, G. Ottaviani, The number of singular vector tuples and uniqueness of best rank-one approximation of tensors. Found. Comput. Math. 14(6), 1209–1242 (2014)

    Article  MathSciNet  Google Scholar 

  13. L. Lim, Singular values and eigenvalues of tensors: a variational approach, in IEEE CAMSAP 2005: First International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (IEEE, Piscataway, 2005), pp. 129–132

    Google Scholar 

  14. L. Wang, M. Chu, B. Yu, Orthogonal low rank tensor approximation: alternating least squares method and its global convergence. SIAM J. Matrix Anal. Appl. 36(1), 1–19 (2015)

    Article  MathSciNet  Google Scholar 

  15. 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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  17. S. Friedland, L. Lim, Nuclear norm of higher-order tensors. Math. Comput. 87, 1255–1281 (2018)

    Article  MathSciNet  Google Scholar 

  18. J. Nie, Symmetric tensor nuclear norms. SIAM J. Appl. Algebr. Geom. 1–1, 599–625 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  21. L. Qi, Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40(6), 1302–1324 (2005)

    Article  MathSciNet  Google Scholar 

  22. 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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  24. L. Ljung, Analysis of recursive stochastic algorithms. IEEE Trans. Autom. Control 22(4), 551–575 (1977)

    Article  MathSciNet  Google Scholar 

  25. M. Che, A. Cichocki, Y. Wei, Neural networks for computing best rank-one approximations of tensors and its applications. Neurocomputing 267(3), 114–133 (2017)

    Article  Google Scholar 

  26. N. Samardzija, R. Waterland, A neural network for computing eigenvectors and eigenvalues. Biol. Cybern. 65(4), 211–214 (1991)

    Article  MathSciNet  Google Scholar 

  27. J. Vegas, P. Zufiria, Generalized neural networks for spectral analysis: dynamics and liapunov function. Neural Netw. 17(16), 233–245 (2004)

    Article  Google Scholar 

  28. M. Che, X. Wang, Y. Wei, H. Yan, A collective neurodynamic optimization approach to the low multilinear rank approximation of tensors. Southwestern Univ. Financ. Econ. (2019). Preprint

    Google Scholar 

  29. X. Wang, M. Che, Y. Wei, Neural network for the computation of the best rank-one approximation of the fourth-order partially symmetric tensors. Numer. Math. Theory, Methods Appl. 11, 673–700 (2018)

    Google Scholar 

  30. M. Che, L. Qi, Y. Wei, Perturbation bounds of tensor eigenvalue and singular value problems with even order. Linear Multilinear Algebra 64(4), 622–652 (2016)

    Article  MathSciNet  Google Scholar 

  31. C. Van Loan, Generalizing the singular value decomposition. SIAM J. Numer. Anal. 13(1), 76–83 (1976)

    Article  MathSciNet  Google Scholar 

  32. Y. Luo, D. Tao, K. Ramamonhanarao, C. Xu, Y. Wen, Tensor canonical correlation analysis for multi-view dimension reduction. IEEE Trans. Knowl. Data Eng. 27(11), 3111–3124 (2015)

    Article  Google Scholar 

  33. G. Golub, C. Van Loan, Matrix Computations, 4th edn. (Johns Hopkins University Press, Baltimore, 2013)

    MATH  Google Scholar 

  34. K. Chang, K. Pearson, T. Zhang, On eigenvalue problems of real symmetric tensors. J. Math. Anal. Appl. 350(1), 416–422 (2009)

    Article  MathSciNet  Google Scholar 

  35. W. Ding, Y. Wei, Generalized tensor eigenvalue problems. SIAM J. Matrix Anal. Appl. 36(3), 1073–1099 (2015)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  39. D. Chu, L. De Lathauwer, B. De Moor, On the computation of the restricted singular value decomposition via the cosine-sine decomposition. SIAM J. Matrix Anal. Appl. 22(2), 580–601 (2000)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Economics Mathematics, Southwestern University of Finance and Economics, Chengdu, Sichuan, China

    Maolin Che

  2. School of Mathematical Sciences, Fudan University, Shanghai, China

    Yimin Wei

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  1. Maolin Che
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  2. Yimin Wei
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Che, M., Wei, Y. (2020). Neural Networks. In: Theory and Computation of Complex Tensors and its Applications. Springer, Singapore. https://doi.org/10.1007/978-981-15-2059-4_6

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