Skip to main content
SpringerLink
Log in

Design of semi-tensor product-based kernel function for SVM nonlinear classification

  • Research Article
  • Published:
Control Theory and Technology Aims and scope Submit manuscript

This article has been updated

Abstract

The kernel function method in support vector machine (SVM) is an excellent tool for nonlinear classification. How to design a kernel function is difficult for an SVM nonlinear classification problem, even for the polynomial kernel function. In this paper, we propose a new kind of polynomial kernel functions, called semi-tensor product kernel (STP-kernel), for an SVM nonlinear classification problem by semi-tensor product of matrix (STP) theory. We have shown the existence of the STP-kernel function and verified that it is just a polynomial kernel. In addition, we have shown the existence of the reproducing kernel Hilbert space (RKHS) associated with the STP-kernel function. Compared to the existing methods, it is much easier to construct the nonlinear feature mapping for an SVM nonlinear classification problem via an STP operator.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Change history

  • 12 December 2022

    The author portraits have been interchanged. This has been corrected.

References

  1. Cortes, C., & Vapnik, V. (1995). Support-vector networks. Machine learning, 20(3), 273–297.

  2. Chapelle, O., Haffner, P., & Vapnik, V. N. (1999). Support vector machines for histogram-based image classification. IEEE Transactions on Neural Networks, 10(5), 1055–1064.

    Article  Google Scholar 

  3. DeCoste, D., & Schölkopf, B. (2002). Training invariant support vector machines. Machine Learning, 46(1), 161–190.

    Article  MATH  Google Scholar 

  4. Pradhan, S., Hacioglu, K., Krugler, V., Ward, W., Martin, J. H., & Jurafsky, D. (2005). Support vector learning for semantic argument classification. Machine Learning, 60(1), 11–39.

    Article  Google Scholar 

  5. Statnikov, A., Hardin, D., & Aliferis, C. (2006). Using svm weight-based methods to identify causally relevant and non-causally relevant variables. Sign, 1(4), 474–484.

    Google Scholar 

  6. Maity, A. (2016). Supervised classification of radarsat-2 polarimetric data for different land features. arXiv preprint. arXiv:1608.00501.

  7. Wenzel, F., Galy-Fajou, T., Deutsch, M., & Kloft, M. (2017). Bayesian nonlinear support vector machines for big data. In: Joint European Conference on Machine Learning and Knowledge Discovery in Databases, pp. 307–322. Springer.

  8. Zhou, C., Chase, J. G., & Rodgers, G. W. (2021). Support vector machines for automated modelling of nonlinear structures using health monitoring results. Mechanical Systems and Signal Processing, 149, 107201.

    Article  Google Scholar 

  9. Tanveer, M., Rajani, T., Rastogi, R., Shao, Y.-H., & Ganaie, M. (2022). Comprehensive review on twin support vector machines. Annals of Operations Research. https://doi.org/10.1007/s10479-022-04575-w

  10. Vapnik, V. N. (1999). An overview of statistical learning theory. IEEE Transactions On Neural Networks, 10(5), 988–999.

    Article  Google Scholar 

  11. Cheng, D. (2016). On equivalence of matrices. arXiv preprint. arXiv:1605.09523.

  12. Cheng, D., & Qi, H. (2009). Controllability and observability of boolean control networks. Automatica, 45(7), 1659–1667.

    Article  MathSciNet  MATH  Google Scholar 

  13. Li, L., Chen, Y., Peng, H., & Yang, Y. (2020). Chaotic deep network for mobile d2d communication. IEEE Internet of Things Journal, 8(10), 8078–8096.

    Article  Google Scholar 

  14. Cheng, D., Zhao, Y., & Mu, Y. (2010). Strategy optimization with its application to dynamic games. In: The 49th IEEE Conference on Decision and Control (CDC), pp. 5822–5827. IEEE.

  15. Feng, J.-E., Lv, H., & Cheng, D. (2013). Multiple fuzzy relation and its application to coupled fuzzy control. Asian Journal of Control, 15(5), 1313–1324.

    MathSciNet  MATH  Google Scholar 

  16. Cheng, D., Qi, H., & Zhao, Y. (2012). An Introduction to Semi-tensor Product of Matrices and Its Applications. Singapore: World Scientific.

    Book  MATH  Google Scholar 

  17. Gao, Z., Fang, S.-C., Luo, J., & Medhin, N. (2021). A kernel-free double well potential support vector machine with applications. European Journal of Operational Research, 290(1), 248–262.

    Article  MathSciNet  MATH  Google Scholar 

  18. Berlinet, A., & Thomas-Agnan, C. (2011). Reproducing Kernel Hilbert Spaces in Probability and Statistics. New York: Springer.

  19. Cucker, F., & Zhou, D. X. (2007). Learning Theory: an Approximation Theory Viewpoint (Vol. 24). Cambridge, MA: Cambridge University Press.

    Book  MATH  Google Scholar 

  20. Mohammed, J., & Meira, W., Jr. (2020). Data Mining and Machine Learning: Fundamental Concepts and Algorithms. Cambridge, UK: Cambridge University.

    MATH  Google Scholar 

  21. Tauber, S. (1963). On multinomial coefficients. The American Mathematical Monthly, 70(10), 1058–1063.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We thank Prof. Daizhan Cheng for giving some valuable suggestions, and also the anonymous reviewers for their helpful comments.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Yulin University, Yulin, 719000, Shaanxi, China

    Shengli Xue

  2. School of Marine Science and Technology, Northwestern Polytechnical University, Xi’an, 710000, Shaanxi, China

    Lijun Zhang & Zeyu Zhu

Authors
  1. Shengli Xue
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lijun Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zeyu Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lijun Zhang.

Additional information

This work was supported by the National Natural Science Foundation of China (61573288), the Key Programs in Shaanxi Province of China (2021JZ-12) and the Yulin Science and Technology Bureau project (2019-89-2).

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xue, S., Zhang, L. & Zhu, Z. Design of semi-tensor product-based kernel function for SVM nonlinear classification. Control Theory Technol. 20, 456–464 (2022). https://doi.org/10.1007/s11768-022-00120-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11768-022-00120-z

Keywords

Search

Navigation