Approximate Convex Hulls Family for One-Class Classification

  • Pierluigi Casale
  • Oriol Pujol
  • Petia Radeva
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6713)


In this work, a new method for one-class classification based on the Convex Hull geometric structure is proposed. The new method creates a family of convex hulls able to fit the geometrical shape of the training points. The increased computational cost due to the creation of the convex hull in multiple dimensions is circumvented using random projections. This provides an approximation of the original structure with multiple bi-dimensional views. In the projection planes, a mechanism for noisy points rejection has also been elaborated and evaluated. Results show that the approach performs considerably well with respect to the state the art in one-class classification.


Convex Hull Random Projections One-Class Classification 


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  1. 1.
    Bhattacharya, B.K.: Application of computational geometry to pattern recognition problems. PhD thesis (1982)Google Scholar
  2. 2.
    Toussaint, G.T.: The convex hull as a tool in pattern recognition. In: AFOSR Workshop in Communication Theory and Applications (1978)Google Scholar
  3. 3.
    Bennett, K.P., Bredensteiner, E.J.: Duality and geometry in svm classifiers. In: ICML, pp. 57–64 (2000)Google Scholar
  4. 4.
    Bi, J., Bennett, K.P.: Duality, geometry, and support vector regression. In: NIPS, pp. 593–600 (2002)Google Scholar
  5. 5.
    Takahashi, T., Kudo, M.: Margin preserved approximate convex hulls for classification. In: ICPR (2010)Google Scholar
  6. 6.
    Pal, S., Bhattacharya, S.: IEEE Transactions on Neural Networks 18(2), 600–605 (2007)CrossRefGoogle Scholar
  7. 7.
    Mavroforakis, M.E., Theodoridis, S.: A geometric approach to support vector machine (svm) classification. Neural Networks, IEEE Transactions on 17, 671–682 (2006)CrossRefGoogle Scholar
  8. 8.
    Zhou, X., Jiang, W., Tian, Y., Shi, Y.: Kernel subclass convex hull sample selection method for SVM on face recognition, vol.73 (2010)Google Scholar
  9. 9.
    Japkowicz, N.: Concept-learning in the absence of counter-examples: An autoassociation-based approach to classification. IJCAI, 518–523 (1995)Google Scholar
  10. 10.
    Tax, D.M.J.: One-class classification. PhD thesis (2001)Google Scholar
  11. 11.
    Preparata, F.P., Shamos, M.I.: Computational geometry: an introduction. Springer, New York.Inc (1985)CrossRefzbMATHGoogle Scholar
  12. 12.
    Johnson, W., Lindenstauss, J.: Extensions of lipschitz maps into a hilbert space. Contemporary Mathematics (1984)Google Scholar
  13. 13.
    Vempala, S.: The Random Projection Method. AMS (2004)Google Scholar
  14. 14.
    Blum, A.: Random projection, margins, kernels, and feature-selection. LNCS (2005)Google Scholar
  15. 15.
    Rahimi, A., Recht, B.: Weighted sums of random kitchen sinks: Replacing minimization with randomization in learning. In: NIPS, pp. 1313–1320 (2008)Google Scholar
  16. 16.
    Breiman, L.: Bagging predictors. Machine Learning 24, 123–140 (1996)zbMATHGoogle Scholar
  17. 17.
    Frank, A., Asuncion, A.: UCI machine learning repository (2010)Google Scholar
  18. 18.
    Juszczak, P., Paclik, P., Pekalska, E., de Ridder, D., Tax, D., Verzakov, S., Duin, R.: A Matlab Toolbox for Pattern Recognition, PRTools4.1 (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Pierluigi Casale
    • 1
    • 2
  • Oriol Pujol
    • 1
    • 2
  • Petia Radeva
    • 1
    • 2
  1. 1.Computer Vision CenterBarcelonaSpain
  2. 2.Dept. Applied Mathematics and AnalysisUniversity of BarcelonaBarcelonaSpain

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