Advertisement

Journal of Global Optimization

, Volume 16, Issue 1, pp 23–32 | Cite as

k-Plane Clustering

  • P.S. Bradley
  • O.L. Mangasarian
Article

Abstract

A finite new algorithm is proposed for clustering m given points in n-dimensional real space into k clusters by generating k planes that constitute a local solution to the nonconvex problem of minimizing the sum of squares of the 2-norm distances between each point and a nearest plane. The key to the algorithm lies in a formulation that generates a plane in n-dimensional space that minimizes the sum of the squares of the 2-norm distances to each of m1 given points in the space. The plane is generated by an eigenvector corresponding to a smallest eigenvalue of an n × n simple matrix derived from the m1 points. The algorithm was tested on the publicly available Wisconsin Breast Prognosis Cancer database to generate well separated patient survival curves. In contrast, the k-mean algorithm did not generate such well-separated survival curves.

Clustering k-Mean Linear regression 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Anderberg, M.R., (1973), Cluster Analysis for Applications. Academic Press, New York.Google Scholar
  2. 2.
    Andrews, H.C., (1972), Introduction to Mathematical Techniques in Pattern Recognition. John Wiley & Sons, New York.Google Scholar
  3. 3.
    Berry, M.W., Dumais, S.T. and O'Brein, G.W., (1995), Using linear algebra for intelligent information retrieval. SIAM Review, 37:573–595. http://www.cs.utk.edu/~berry.Google Scholar
  4. 4.
    Bradley, P.S., Mangasarian, O.L. and Street, W.N., (1997), Clustering via concave minimization. In:M.C. Mozer, M.I. Jordan, and T. Petsche, editors, Advances in Neural Information Processing Systems-9-, pages 368–374, Cambridge, MA. MIT Press. ftp://ftp.cs.wisc.edu/mathprog/tech-reports/96-03.ps.Z.Google Scholar
  5. 5.
    Cavalier T. and Melloy, B., (1995), An iterative linear programming solution to the Euclideanregression model. Computers and Operations Research, 28:781–793.Google Scholar
  6. 6.
    Celeux, G. and Govaert, G., (1995), Gaussian parsimonious clustering models. Pattern Recognition, 28:781–793.Google Scholar
  7. 7.
    Fisher, D., (1987), Knowledge acquisition via incremental conceptual clustering. Machine Learning, 2:139–172.Google Scholar
  8. 8.
    Hassoun, M.H., (1995), Fundamentals of Artificial Neural Networks. MIT Press, Cambridge, MA.Google Scholar
  9. 9.
    Jain, A.K. and Dubes, R.C., (1988), Algorithms for Clustering Data. Prentice-Hall, Inc, Englewood Cliffs, NJ.Google Scholar
  10. 10.
    Kaplan, E.L. and Meier, P., (1958), Nonparametric estimation from incomplete observations. Journal of the American Statistical Association, 53:457–481.Google Scholar
  11. 11.
    Kleinbaum, David G., (1996), Survival Analysis. Springer-Verlag, New York.Google Scholar
  12. 12.
    Mangasarian, O.L., (1999), Arbitrary-norm separating plane. Operations Research Letters, 24(1–2). ftp://ftp.cs.wisc.edu/math-prog/tech-reports/97–07.ps.Z.Google Scholar
  13. 13.
    Murphy, P.M. and Aha, D.W., (1992), UCI repository of machine learning databases. Technical report, Department of Information and Computer Science, University of California, Irvine www.ics.uci.edu/~mlearn/MLRepository.html.Google Scholar
  14. 14.
    Noble, B. and Daniel, J.W., (1988), Applied Linear Algebra. Prentice Hall, Englewood Cliffs, New Jersey, third edition.Google Scholar
  15. 15.
    Rao, M.R., (1971), Cluster analysis and mathematical programming. Journal of the American Statistical Association, 66:622–626.Google Scholar
  16. 16.
    Selim, S.Z. and Ismail, M.A., (1984), K-Means-Type algorithms: a generalized convergence theorem and characterization of local optimality. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-6:81–87.Google Scholar
  17. 17.
    Späth, H., (1992), Mathematical Algorithms for Linear Regression. Academic Press, San Diego.Google Scholar
  18. 18.
    Stone, M., (1974), Cross-validatory choice and assessment of statistical predictions. Journal of the Royal Statistical Society, 36:111–147.Google Scholar
  19. 19.
    Strang, G., (1993), Introduction to Linear Algebra. Wellesley-Cambridge Press, Wellesley, MA.Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • P.S. Bradley
    • 1
    • 2
  • O.L. Mangasarian
    • 1
    • 2
  1. 1.Microsoft ResearchRedmondUSA
  2. 2.Computer Sciences DepartmentUniversity of WisconsinMadisonUSA

Personalised recommendations