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A Unified Self-Stabilizing Neural Network Algorithm for Principal Takagi Component Extraction

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Abstract

In this paper, we develop efficient methods for the computation of the Takagi components and the Takagi subspaces of complex symmetric matrices via the complex-valued neural network models. Firstly, we present a unified self-stabilizing neural network learning algorithm for principal Takagi components and study the stability of the proposed unified algorithms via the fixed-point analysis method. Secondly, the unified algorithm for extracting principal Takagi components is generalized to compute the principal Takagi subspace. Thirdly, we prove that the associated differential equations will globally asymptotically converge to an invariance set and the corresponding energy function attains a unique global minimum if and only if its state matrices span the principal Takagi subspace. Finally, numerical simulations are carried out to illustrate the theoretical results.

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References

  1. Aizenberg I (2011) Complex-valued neural networks with multi-valued neurons, vol 253 of studies in computational intelligence. Springer, New York

    Google Scholar 

  2. Autonne L (1915) Sur les matrices hypohermitiennes et sur les matrices unitaires. Ann Univ Lyon 38:1–77

    Google Scholar 

  3. Bain AD (2003) Chemical exchange in NMR. Prog Nucl Magn Reson Spectrosc 43:63–103

    Google Scholar 

  4. Bohner M, Rao VSH, Sanyal S (2011) Global stability of complex-valued neural networks on time scales. Differ Equ Dyn Syst 19:3–11

    Google Scholar 

  5. Brandwood D (1983) A complex gradient operator and its application in adaptive array theory. IEE Proc F Commun Radar Signal Process 130:11–16

    Google Scholar 

  6. Bunse-Gerstner A, Gragg WB (1988) Singular value decompositions of complex symmetric matrices. J Comput Appl Math 21:41–54

    Google Scholar 

  7. Che M, Qi L, Wei Y (2017) Iterative algorithms for computing US- and U-eigenpairs of complex tensors. J Comput Appl Math 317:547–564

    Google Scholar 

  8. Che M, Qi L, Wei Y, Zhang G (2018) Geometric measures of entanglement in multipartite pure states via complex-valued neural networks. Neurocomputing 313:25–38

    Google Scholar 

  9. Che M, Qiao S, Wei Y (2018) Adaptive algorithms for computing the principal takagi vector of a complex symmetric matrix. Neurocomputing 317:79–87

    Google Scholar 

  10. Comon P (1994) Independent component analysis, a new concept? Signal Process 36:287–314

    Google Scholar 

  11. Golub GH, Van Loan CF (2013) Matrix Computations, fourth edn. Johns Hopkins University Press, Baltimore

    Google Scholar 

  12. Hirose A (2003) Complex-Valued Neural Networks: Theories and Applications, vol 5. World Scientific, Singapore

    Google Scholar 

  13. Hong C, Yu J, Chen X (2018) Image-based 3D human pose recovery with locality sensitive sparse retrieval. IEEE Trans Ind Electron 62:2103–2108

    Google Scholar 

  14. Hong C, Yu J, Wan J, Tao D, Wang M (2015) Multimodal deep autoencoder for human pose recovery. IEEE Trans Image Process 24:5659–5670

    Google Scholar 

  15. Hong C, Yu J, Zhang J, Jin X, Lee KH (2018) Multi-modal face pose estimation with multi-task manifold deep learning. IEEE Trans Ind Inform. https://doi.org/10.1109/TII.2018.2884211

    Google Scholar 

  16. Kong X, Hu C, Ma H, Han C (2012) A unified self-stabilizing neural network algorithm for principal and minor components extraction. IEEE Trans Neural Netw Learn Syst 23:185–198

    Google Scholar 

  17. Kreutz-Delgado K (2009) The complex gradient operator and the CR-calculus. ArXiv preprint arXiv:0906.4835

  18. Liu Y, Yang D, Li L, Yang J (2019) A split-complex valued gradient-based descent neuro-Fuzzy algorithm for ts system and its convergence. Neural Process Lett. https://doi.org/10.1007/s11063-018-9949-7

    Google Scholar 

  19. Ljung L (1977) Analysis of recursive stochastic algorithms. IEEE Trans Autom Control 22:551–575

    Google Scholar 

  20. Mathews JH, Fink KD (2004) Numerical Methods using MATLAB. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  21. Moller R (2004) A self-stabilizing learning rule for minor complement analysis. Int J Neural Syst 14:1–8

    Google Scholar 

  22. Nitta T (2009) Complex-Valued Neural Networks: Utilizing High-dimensional Parameters. Information Science Reference, Hershey

    Google Scholar 

  23. Remmert R (1991) Theory of complex functions, vol 122 of graduate texts in mathematics. Springer, New York

    Google Scholar 

  24. Schober G (1975) Univalent functions-selected topics, vol. 478 of Part of the Lecture Notes in Mathematics book series. Springer, New York

  25. Takagi T (1924) On an algebraic problem reluted to an analytic theorem of carathéodory and fejér and on an allied theorem of Landau. Jpn J Math Trans Abstr 1:83–93

    Google Scholar 

  26. Trefethen LN (1981) Near-circularity of the error curve in complex Chebyshev approximation. J Approx Theory 31:344–367

    Google Scholar 

  27. Van Den Bos A (1994) Complex gradient and Hessian. IEEE Proc Vis Image Signal Process 141:380–383

    Google Scholar 

  28. Van Huffel S (1993) Enhanced resolution based on minimum variance estimation and exponential data modeling. Signal Process 33:333–355

    Google Scholar 

  29. Veloz A, Salas R, Allendecid H, Allende H, Moraga C (2016) Identification of lags in nonlinear autoregressive time series using a flexible Fuzzy model. Neural Process Lett 43:641–666

    Google Scholar 

  30. Wang D, Quek C, Ng GS (2004) Novel self-organizing Takagi Sugeno Kang Fuzzy neural networks based on ART-like clustering. Neural Process Lett 20:39–51

    Google Scholar 

  31. Wang G, Wei Y, Qiao S (2018) Generalized Inverses: Theory and Computations. Springer, Science Press, Singapore, Beijing

    Google Scholar 

  32. Wang H, Duan S, Huang T, Wang L, Li C (2017) Exponential stability of complex-valued memristive recurrent neural networks. IEEE Trans Neural Netw Learn Syst 28:766–771

    Google Scholar 

  33. Wang X, Che M, Wei Y (2017) Complex-valued neural networks for the takagi vector of complex symmetric matrices. Neurocomputing 223:77–85

    Google Scholar 

  34. Wang Z, Guo Z, Huang L, Liu X (2017) Dynamical behavior of complex-valued Hopfield neural networks with discontinuous activation functions. Neural Process Lett 45:1039–1061

    Google Scholar 

  35. Wei T, Goldbart PM (2003) Geometric measure of entanglement and applications to bipartite and multipartite quantum states. Phys Rev A 68:042307

    Google Scholar 

  36. Xu W, Qiao S (2008) A divide-and-conquer method for the Takagi factorization. SIAM J Matrix Anal Appl 30:142–153

    Google Scholar 

  37. Xu W, Qiao S (2009) A twisted factorization method for symmetric SVD of a complex symmetric tridiagonal matrix. Numer Linear Algebra Appl 16:801–815

    Google Scholar 

  38. Yu J, Hong C, Rui Y, Tao D (2018) Multitask autoencoder model for recovering human poses. IEEE Trans Industr Electron 65:5060–5068

    Google Scholar 

  39. Yu J, Rui Y, Tang YY, Tao D (2014) High-order distance-based multiview stochastic learning in image classification. IEEE Trans Syst Man Cybern 44:2431–2442

    Google Scholar 

  40. Yu J, Rui Y, Tao D (2014) Click prediction for web image reranking using multimodal sparse coding. IEEE Trans Image Process 23:2019–2032

    Google Scholar 

  41. Yu J, Tao D, Wang M, Rui Y (2015) Learning to rank using user clicks and visual features for image retrieval. IEEE Trans Syst Man Cybern 45:767–779

    Google Scholar 

  42. Yu J, Zhu C, Zhang J, Huang Q, Tao D (2018) Spatial pyramid enhanced NetVLAD with weighted triplet loss for place recognition. IEEE Trans Neural Netw Learn Syst. https://doi.org/10.1109/TNNLS.2019.2908982

    Google Scholar 

  43. Zhang J, Yu J, Tao D (2018) Local deep-feature alignment for unsupervised dimension reduction. IEEE Trans Image Process 27:2420–2432

    Google Scholar 

  44. Zhang S, Xia Y, Wang J (2015) A complex-valued projection neural network for constrained optimization of real functions in complex variables. IEEE Trans Neural Networks 26:3227–3238

    Google Scholar 

  45. Zhang S, Xia Y, Zheng WX (2015) A complex-valued neural dynamical optimization approach and its stability analysis. Neural Netw 61:59–67

    Google Scholar 

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Acknowledgements

The authors would like to thank the Editor and two anonymous reviewers for their careful and detailed comments on our paper.

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Authors and Affiliations

  1. School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, People’s Republic of China

    Maolin Che

  2. School of Mathematics and Statistics, Hexi University, Zhangye, 734000, People’s Republic of China

    Xuezhong Wang

  3. School of Mathematical Sciences and Shanghai Key Laboratory of Contemporary Applied Mathematics, Fudan University, Shanghai, 200433, People’s Republic of China

    Yimin Wei

Authors
  1. Maolin Che
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  2. Xuezhong Wang
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  3. Yimin Wei
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Correspondence to Yimin Wei.

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Maolin Che: This author is supported by the National Natural Science Foundation of China under grant 11901471. Xuezhong Wang: Partial work is finished when the author visited Shanghai Key Laboratory of Contemporary Applied Mathematics in 2019 and supported by the National Natural Science Foundation of China under Grant 11771099. Yimin Wei: This author is supported by Innovation Program of Shanghai Municipal Education Commission.

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Che, M., Wang, X. & Wei, Y. A Unified Self-Stabilizing Neural Network Algorithm for Principal Takagi Component Extraction. Neural Process Lett 51, 591–610 (2020). https://doi.org/10.1007/s11063-019-10109-6

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  • DOI: https://doi.org/10.1007/s11063-019-10109-6

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