Skip to main content
SpringerLink
Log in

Discrete-Time Recurrent Neural Network for Solving Multi-linear \(\mathcal {M}\)-tensor Equation

  • Conference paper
  • First Online:
Advances in Computational Intelligence Systems (UKCI 2021)

Part of the book series: Advances in Intelligent Systems and Computing ((AISC,volume 1409))

Included in the following conference series:

  • 750 Accesses

Abstract

Multi-linear \(\mathcal {M}\)-tensor equation is a complex system of linear equations, whose coefficient is a higher-order tensor. Recurrent neural network (RNN) has great advantages in the processing of sequence data, especially in the processing of nonlinear systems, showing a very potent ability. In this paper, solving the multi-linear \(\mathcal {M}\)-tensor equation is transformed into an error function from the perspective of control science. By controlling the error to zero, the real solution to the error function is approached to the theoretical solution. The model is discretized for the convenience of implementation in computers. Moreover, the relevant theoretical analyses are provided, and the effectiveness and advantages of the proposed method are proved by comparing it with the Newton iteration (NI) neural algorithm.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 249.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Zahra, R., Mohammad, M.H.: Tens-embedding: a tensor-based document embedding method. Expert Syst. Appl. 162, 113770 (2020). https://doi.org/10.1016/j.eswa.2020.113770

    Article  Google Scholar 

  2. Luo, X., Wu, H., Yuan, H.Q., Zhou, M.C.: Temporal pattern-aware QoS prediction via biased non-negative latent factorization of tensors. IEEE Trans. Cybern. 50(5), 1798–1809 (2020). https://doi.org/10.1109/TCYB.2019.2903736

    Article  Google Scholar 

  3. Wu, D., et al.: A deep latent factor model for high-dimensional and sparse matrices in recommender systems. IEEE Trans. Syst. Man Cybern. Syst. 51(7), 4285–4296 (2021). https://doi.org/10.1109/TSMC.2019.2931393

    Article  Google Scholar 

  4. Luo, X., Zhou, M.C., Li, S., Wu, D., Liu, Z.G., Shang, M.S.: Algorithms of unconstrained non-negative latent factor analysis for recommender systems. IEEE Trans. Big Data. 7(1), 227–240 (2021). https://doi.org/10.1109/TBDATA.2019.2916868

    Article  Google Scholar 

  5. Xie, Z.T., Jin, L., Luo, X., Li, S., Xiao, X.C.: A data-driven cyclic-motion generation scheme for kinematic control of redundant manipulators. IEEE Trans. Control Syst. Technol. 29(1), 53–63 (2021). https://doi.org/10.1109/TCST.2019.2963017

    Article  Google Scholar 

  6. Qi, L.Q.: Eigenvalues of a real supersymmetric tensor. J. Symb. Comput. 40(6), 1302–1324 (2005). https://doi.org/10.1016/j.jsc.2005.05.007

    Article  MathSciNet  MATH  Google Scholar 

  7. Luo, X., Liu, Z.G., Li, S., Shang, M.S., Wang, Z.D.: A fast non-negative latent factor model based on generalized momentum method. IEEE Trans. Syst. Man Cybern. Syst. 51(1), 610–620 (2021). https://doi.org/10.1109/TSMC.2018.2875452

    Article  Google Scholar 

  8. Song, Y., Li, M., Luo, X., Yang, G.S., Wang, C.J.: Improved symmetric and nonnegative matrix factorization models for undirected, sparse and large-scaled networks: a triple factorization-based approach. IEEE Trans. Ind. Inform. 16(5), 3006–3017 (2020). https://doi.org/10.1109/TII.2019.2908958

    Article  Google Scholar 

  9. Qi, L.Q.: Symmetric nonnegative tensors and copositive tensors. Linear Alg. Appl. 439(1), 228–238 (2013). https://doi.org/10.1016/j.laa.2013.03.015

    Article  MathSciNet  MATH  Google Scholar 

  10. Ding, W.Y., Qi, L.Q., Wei, Y.M.: \(\cal{M}\)-tensors and nonsingular \(\cal{M}\)-tensors. Linear Alg. Appl. 439(10), 3264–3278 (2013). https://doi.org/10.1016/j.laa.2013.08.038

    Article  MathSciNet  MATH  Google Scholar 

  11. Zhang, Y.X., Liu, Q.L., Chen, Z.: Preconditioned Jacobi type method for solving multi-linear systems with \(\cal{M}\)-tensors. Appl. Math. Lett. 104, 106287 (2020). https://doi.org/10.1016/j.aml.2020.106287

    Article  MathSciNet  MATH  Google Scholar 

  12. Liu, D.D., Li, W., Vong, S.W.: The tensor splitting with application to solve multi-linear systems. J. Comput. Appl. Math. 330, 75–94 (2018). https://doi.org/10.1016/j.cam.2017.08.009

    Article  MathSciNet  MATH  Google Scholar 

  13. Han, L.X.: A homotopy method for solving multilinear systems with \(\cal{M}\)-tensors. Appl. Math. Lett. 69, 49–54 (2017). https://doi.org/10.1016/j.aml.2017.01.019

    Article  MathSciNet  MATH  Google Scholar 

  14. Wang, X.Z., Che, M.L., Wei, Y.M.: Neural network approach for solving nonsingular multi-linear tensor systems. J. Comput. Appl. Math. 368, 112569 (2020). https://doi.org/10.1016/j.cam.2019.112569

    Article  MathSciNet  MATH  Google Scholar 

  15. Liu, M., Li, H.W., Li, Y., Jin, L., Huang, Z.G.: From WASD to BLS with application to pattern classification. Appl. Soft. Comput. 108, 107455 (2021). https://doi.org/10.1016/j.asoc.2021.107455

    Article  Google Scholar 

  16. Yang, C.G., Peng, G.Z., Li, Y.A., Cui, R.X., Cheng, L., Li, Z.J.: Neural networks enhanced adaptive admittance control of optimized robot-environment interaction. IEEE. Trans. Syst. Cybern. 49(7), 2568–2579 (2019). https://doi.org/10.1109/TCYB.2018.2828654

    Article  Google Scholar 

  17. Wei, L., Jin, L., Yang, C.G., Chen, K., Li, W.B.: New noise-tolerant neural algorithms for future dynamic nonlinear optimization with estimation on Hessian matrix inversion. IEEE Trans. Syst. Man Cybern. Syst. 51(4), 2611–2623 (2021). https://doi.org/10.1109/TSMC.2019.2916892

    Article  Google Scholar 

  18. Xie, Z.T., Jin, L., Du, X.J., Xiao, X.C., Li, H.X., Li, S.: On generalized RMP scheme for redundant robot manipulators aided with dynamic neural networks and nonconvex bound constraints. IEEE Trans. Ind. Inform. 15(9), 5172–5181 (2019). https://doi.org/10.1109/TII.2019.2899909

    Article  Google Scholar 

  19. Liu, M., Peng, B., Shang, M.: Lower limb movement intention recognition for rehabilitation robot aided with projected recurrent neural network. Complex Intell. Syst. 1–12 (2021). https://doi.org/10.1007/s40747-021-00341-w

  20. Jin, L., Yan, J.K., Du, X.J., Xiao, X.C., Fu, D.Y.: RNN for solving time-variant generalized Sylvester equation with applications to robots and acoustic source localization. IEEE Trans. Ind. Inform. 16(10), 6359–6369 (2020). https://doi.org/10.1109/TII.2020.2964817

    Article  Google Scholar 

  21. Lu, H.Y., Jin, L., Luo, X., Liao, B.L., Guo, D.S., Xiao, L.: RNN for solving perturbed time-varying underdetermined linear system with double bound limits on residual errors and state variables. IEEE Trans. Ind. Inform. 15(11), 5931–5942 (2019). https://doi.org/10.1109/TII.2019.2909142

    Article  Google Scholar 

  22. Xie, Z.T., Jin, L., Luo, X., Sun, Z.B., Liu, M.: RNN for repetitive motion generation of redundant robot manipulators: an orthogonal projection-based scheme. IEEE Trans. Neural Netw. Learn. Syst. https://doi.org/10.1109/TNNLS.2020.3028304

  23. Qi, Y.M., Jin, L., Li, H.X., Li, Y.M., Liu, M.: Discrete computational neural dynamics models for solving time-dependent Sylvester equations with applications to robotics and MIMO systems. IEEE Trans. Ind. Inform. 16(10), 6231–6241 (2020). https://doi.org/10.1109/TII.2020.2966544

    Article  Google Scholar 

  24. Jin, L., et al.: Perturbed manipulability optimization in a distributed network of redundant robots. IEEE Trans. Ind. Electron. 68(8), 7209–7220 (2021). https://doi.org/10.1109/TIE.2020.3007099

    Article  Google Scholar 

  25. Jin, L., Li, S., Hu, B., Liu, M., Yu, J.G.: A noise-suppressing neural algorithm for solving the time-varying system of linear equations: a control-based approach. IEEE Trans. Ind. Inform. 15(1), 236–246 (2019). https://doi.org/10.1109/TII.2018.2798642

    Article  Google Scholar 

  26. Zhang, J.Z., Jin, L., Cheng, L.: RNN for perturbed manipulability optimization of manipulators based on a distributed scheme: a game-theoretic perspective. IEEE Trans. Neural Netw. Learn. Syst. 31(12), 5116–5126 (2020). https://doi.org/10.1109/TNNLS.2020.2963998

    Article  MathSciNet  Google Scholar 

  27. Qi, Y.M., Jin, L., Wang, Y.N., Xiao, L., Zhang, J.L.: Complex-valued discrete-time neural dynamics for perturbed time-dependent complex quadratic programming with applications. IEEE Trans. Neural Netw. Learn. Syst. 31(9), 3555–3569 (2020). https://doi.org/10.1109/TNNLS.2019.2944992

    Article  MathSciNet  Google Scholar 

  28. Jin, L., Xie, Z.T., Liu, M., Chen, K., Li, C.X., Yang, C.G.: Novel joint-drift-free scheme at acceleration level for robotic redundancy resolution with tracking error theoretically eliminated. IEEE/ASME Trans. Mechatron. 26(1), 90–101 (2021). https://doi.org/10.1109/TMECH.2020.3001624

    Article  Google Scholar 

  29. Lu, H.Y., Jin, L., Zhang, J.L., Sun, Z.N., Li, S., Zhang, Z.J.: New joint-drift-free scheme aided with projected ZNN for motion generation of redundant robot manipulators perturbed by disturbances. IEEE Trans. Syst. Man Cybern. Syst. https://doi.org/10.1109/TSMC.2019.2956961

  30. Wang, X.Z., Che, M.L., Wei, Y.M.: Neural networks based approach solving multi-linear systems with \(\cal{M}\)-tensors. Neurocomputing 351, 33–42 (2019). https://doi.org/10.1016/j.neucom.2019.03.025

    Article  Google Scholar 

  31. Ding, W., Wei, Y.: Solving multi-linear systems with \(\cal{M}\)-tensors. J. Sci. Comput. 68(2), 689–715 (2016). https://doi.org/10.1007/s10915-015-0156-7

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This work was supported in part by the Team Project of Natural Science Foundation of Qinghai Province, China, under Grant 2020-ZJ-903, in part by the Key Laboratory of IoT of Qinghai under Grant 2020-ZJ-Y16, in part by the Natural Science Foundation of Gansu Province, China, under Grant 20JR10RA639, in part by the Fundamental Research Funds for the Central Universities under Grant lzujbky-2019-89, lzujbky-2021-it35 and lzujbky-2021-it36.

Author information

Authors and Affiliations

  1. School of Information Science and Engineering, Lanzhou University, Lanzhou, 730000, China

    Huanmei Wu & Mei Liu

  2. Academy of Plateau Science and Sustainability, Xining, 810016, China

    Huanmei Wu, Shuqiao Wang, Xiujuan Du & Mei Liu

  3. Collage of Computer, Qinghai Normal University, Xining, 810008, China

    Shuqiao Wang & Xiujuan Du

Authors
  1. Huanmei Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shuqiao Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xiujuan Du
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Mei Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mei Liu .

Editor information

Editors and Affiliations

  1. Department of Computer Science, Aberystwyth University, Aberystwyth, UK

    Thomas Jansen

  2. Department of Computer Science, Aberystwyth University, Aberystwyth, UK

    Richard Jensen

  3. Department of Computer Science, Aberystwyth University, Aberystwyth, UK

    Neil Mac Parthaláin

  4. Department of Electrical Engineering, Yuan Ze University , Taoyuan, Taiwan

    Chih-Min Lin

Rights and permissions

Reprints and permissions

Copyright information

© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Wu, H., Wang, S., Du, X., Liu, M. (2022). Discrete-Time Recurrent Neural Network for Solving Multi-linear \(\mathcal {M}\)-tensor Equation. In: Jansen, T., Jensen, R., Mac Parthaláin, N., Lin, CM. (eds) Advances in Computational Intelligence Systems. UKCI 2021. Advances in Intelligent Systems and Computing, vol 1409. Springer, Cham. https://doi.org/10.1007/978-3-030-87094-2_12

Download citation

Publish with us

Policies and ethics

Search

Navigation