An Adaptive Construction Test Method Based on Geometric Calculation for Linearly Separable Problems

  • Shuiming ZhongEmail author
  • Xiaoxiang Lu
  • Meng Li
  • Chengguang Liu
  • Yong Cheng
  • Victor S. Sheng
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11068)


The linearly separable problem is a fundamental problem in pattern classification. Firstly, from the perspective of spatial distribution, this paper focuses on the linear separability of a region dataset at the distribution level instead of the linearly separable issue between two datasets at the traditional category level. Firstly, the former can reflect the spatial distribution of real data, which is more helpful to its application in pattern classification. Secondly, based on spatial geometric theory, an adaptive construction method for testing the linear separability of a region dataset is demonstrated and designed. Finally, the corresponding computer algorithm is designed, and some simulation verification experiments are carried out based on some manual datasets and benchmark datasets. Experimental results show the correctness and effectiveness of the proposed method.


Pattern classification Linear separability Region datasets Geometric calculation 



This work was supported by the National Natural Science Foundation of China (71373131, 61402236, 61572259 and U1736105), Training Program of the Major Research Plan of the National Science Foundation of China (91546117).


  1. 1.
    Rosenblatt, F.: The perceptron: a probabilistic model for information storage and organization in the brain. Psychol. Rev. 65(6), 386–408 (1958)CrossRefGoogle Scholar
  2. 2.
    Minsky, M.L., Papert, S.: Perceptrons: An Introduction to Computational Geometry. The MIT Press, Cambridge (1969)zbMATHGoogle Scholar
  3. 3.
    Rumelhart, D.E., Hinton, G. E., Williams, R.J.: Learning representations by back-propagating errors. Neurocomputing: foundations of research. MIT Press (1988)Google Scholar
  4. 4.
    Cortes, C., Vapnik, V.: Support vector network. Mach. Learn. 20(3), 273–297 (1995)zbMATHGoogle Scholar
  5. 5.
    Kuhn, H.W.: Solvability and consistency for linear equations and inequalities. Am. Math. Mon. 63(4), 217–232 (1956)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bazaraa, M.S., Jarvis, J.J., Sherali, H.D.: Linear programming and network flows. J. Oper. Res. Soc. 29(5), 510 (1978)CrossRefGoogle Scholar
  7. 7.
    Tajine, M., Elizondo, D.: New methods for testing linear separability. Neurocomputing 47(1), 161–188 (2002)CrossRefGoogle Scholar
  8. 8.
    McCulloch, W.S., Pitts, W.: A logical calculus of the ideas immanent in nervous activity. Bull. Math. Biol. 52(1–2), 99–115 (1990)CrossRefGoogle Scholar
  9. 9.
    Mullin, A.A., Rosenblatt, F.: Principles of neurodynamics. Cybern. Syst. Anal. 11(5), 841–842 (1962)Google Scholar
  10. 10.
    Pang, S., Kim, D., Bang, S.Y.: Face membership authentication using SVM classification tree generated by membership-based LLE data partition. IEEE Trans. Neural Netw. 16(2), 436 (2005)CrossRefGoogle Scholar
  11. 11.
    Elizondo, D.: The linear separability problem: some testing methods. IEEE Trans. Neural Netw. 17(2), 330 (2006)CrossRefGoogle Scholar
  12. 12.
    Rao, Y., Zhang, X.: Characterization of linearly separable boolean functions: a graph-theoretic perspective. IEEE Trans. Neural Netw. Learn. Syst. 28(7), 1542–1549 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hochbaum, D.S., Shanthikumar, J.G.: Convex separable optimization is not much harder than linear optimization. J. ACM 37(4), 843–862 (1990)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Bobrowski, L.: Induction of linear separability through the ranked layers of binary classifiers. In: Iliadis, L., Jayne, C. (eds.) AIAI/EANN -2011. IAICT, vol. 363, pp. 69–77. Springer, Heidelberg (2011). Scholar
  15. 15.
    Abd, E.K.M.S., Abo-Bakr, R.M.: Linearly and quadratically separable classifiers using adaptive approach. In: Computer Engineering Conference, vol. 26, pp. 89–96. IEEE (2011)Google Scholar
  16. 16.
    Ben-Israel, A., Levin, Y.: The geometry of linear separability in data sets. Linear Algebra Appl. 416(1), 75–87 (2006)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Bauman, E., Bauman, K.: One-class semi-supervised learning: detecting linearly separable class by its mean (2017)Google Scholar
  18. 18.
    Elizondo, D.: Searching for linearly separable subsets using the class of linear separability method. In: IEEE International Joint Conference on Neural Networks, Proceedings, vol. 2, pp. 955–959. IEEE (2004)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Shuiming Zhong
    • 1
    Email author
  • Xiaoxiang Lu
    • 1
  • Meng Li
    • 1
  • Chengguang Liu
    • 1
  • Yong Cheng
    • 1
  • Victor S. Sheng
    • 2
  1. 1.School of Computer and SoftwareNanjing University of Information Science and TechnologyNanjingChina
  2. 2.Department of Computer ScienceUniversity of Central ArkansasConwayUSA

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