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A New Algorithm for Computing the Minimal Enclosing Sphere in Feature Space

  • Chonghui Guo
  • Mingyu Lu
  • Jiantao Sun
  • Yuchang Lu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3614)

Abstract

The problem of computing the minimal enclosing sphere (MES) of a set of points in the high dimensional kernel-induced feature space is considered. In this paper we develop an entropy-based algorithm that is suitable for any Mercer kernel mapping. The proposed algorithm is based on maximum entropy principle and it is very simple to implement. The convergence of the novel algorithm is analyzed and the validity of this algorithm is confirmed by preliminary numerical results.

Keywords

Support Vector Feature Space Quadratic Programming Quadratic Programming Problem Maximum Entropy Principle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Berg, M.: Computational geometry: algorithms and application. Springer, New york (1997)Google Scholar
  2. 2.
    Elzinga, D.J., Hearn, D.W.: The minimum covering sphere problem. Management Science 19, 96–104 (1972)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Schölkopf, B., Burges, C., Vapnik, V.: Extracting support data for a given task. In: Fayyad, U.M., Uthurusamy, R. (eds.) Proceedings First International Conference on Knowledge Discovery & Data Mining, pp. 252–257. AAAI Press, Menlo Park (1995)Google Scholar
  4. 4.
    Megiddo, N.: Linear-time algorithms for linear programming in R 3 and related problems. SIAM J. Comput. 12, 759–776 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Welzl, E.: Smallest enclosing disks (balls and ellipses). In: Maurer, H. (ed.) New Results and New Trends in Computer Science. LNCS, vol. 555, pp. 359–370. Springer, Heidelberg (1991)CrossRefGoogle Scholar
  6. 6.
    Gärtner, B.: Fast and robust smallest enclosing balls. In: Nešetřil, J. (ed.) ESA 1999. LNCS, vol. 1643, pp. 325–338. Springer, Heidelberg (1999)Google Scholar
  7. 7.
    Vapnik, V.: Statistical learning theory. John wiley & Sons, New York (1998)zbMATHGoogle Scholar
  8. 8.
    Ben-Hur, A., Horn, D., Siegelmann, H.T., Vapnik, V.: Support vector clustering. Journal of Machine Learning Research 2, 125–137 (2001)CrossRefGoogle Scholar
  9. 9.
    Horn, D.: Clustering via Hibert space. Physica A 302, 70–79 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Platt, J.: Fast training of support vector machines using sequential minimal optimization. In: Scholkopf, B., Burges, C.J.C., Smola, A.J. (eds.) Advances in Kernel Methods — Support Vector Learning, pp. 185–208. MIT Press, Cambridge (1999)Google Scholar
  11. 11.
    Li, X.S.: An information entropy for optimization problems. Chinese Journal of Operations Research 8(1), 759–776 (1989)Google Scholar
  12. 12.
    Templeman, A.B., Li, X.S.: An maximum entropy approach to constrained non-linear programming. Engineering Optimization 12, 191–205 (1987)CrossRefGoogle Scholar
  13. 13.
    Kress, R.: Numerical analysis. Springer, New York (1998)zbMATHGoogle Scholar
  14. 14.
    Blake, C.L., Merz, C.J.: UCI repository of machine learning databases, available at (1998), at http://www.ics.uci.edu/~mlearn/MLRepository.html
  15. 15.
    Polak, E., Royset, J.O., Womersley, R.S.: Algorithms for adaptive smoothing for finite minimax problem. Journal of Optimization Theory and Applications 119(3), 459–484 (2003)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Chonghui Guo
    • 1
  • Mingyu Lu
    • 2
  • Jiantao Sun
    • 2
  • Yuchang Lu
    • 2
  1. 1.Department of Applied MathematicsDalian University of TechnologyDalianChina
  2. 2.Department of Computer ScienceTsinghua UniversityBeijingChina

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