A New Unsupervised Learning for Clustering Using Geometric Associative Memories

  • Benjamín Cruz
  • Ricardo Barrón
  • Humberto Sossa
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5856)


Associative memories (AMs) have been extensively used during the last 40 years for pattern classification and pattern restoration. A new type of AMs have been developed recently, the so-called Geometric Associative Memories (GAMs), these make use of Conformal Geometric Algebra (CGA) operators and operations for their working. GAM’s, at the beginning, were developed for supervised classification, getting good results. In this work an algorithm for unsupervised learning with GAMs will be introduced. This new idea is a variation of the k-means algorithm that takes into account the patterns of the a specific cluster and the patterns of another clusters to generate a separation surface. Numerical examples are presented to show the functioning of the new algorithm.


Associative Memory Unsupervised Learn Geometric Algebra Separation Surface Conformal Geometric Algebra 
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.


  1. 1.
    Clifford, W.: Applications of Grassmann’s Extensive Algebra. Am. J. of Math. 1(4), 350–358 (1878)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Cruz, B., Barrón, R., Sossa, H.: Geometric Associative Memories and their Applications to Pattern Classification. In: Bayro-Corrochano, E., Sheuermann, G. (eds.) Geometric Algebra Computing for Computing Science and Engineering. Springer, London (2009) (to be published)Google Scholar
  3. 3.
    Dorst, L., Fontijne, D.: 3D Euclidean Geometry Through Conformal Geometric Algebra (a GAViewer Tutorial). University of Amsterdam Geometric Algebra Website (2005),
  4. 4.
    Duda, R., HArt, P., Stork, D.: Pattern Classification. John Wiley & Sons, New York (2001)zbMATHGoogle Scholar
  5. 5.
    Hestenes, D., Sobczyk, G.: Clifford Algebra to Geometric Calculus. Kluwer/Springer (1984)Google Scholar
  6. 6.
    Hestenes, D., Li, H., Rockwood, A.: New Algebraic Tools for Classical Geometry. In: Sommer, G. (ed.) Geometric Computing with Clifford Algebras, vol. 40, pp. 3–23. Springer, Heildeberg (2001)Google Scholar
  7. 7.
    Hildebrand, D.: Geometric Computing in Computer Graphics Using Conformal Geometric Algebra. Tutorial, TU Darmstadt, Germany. Interact. Graph. Syst. Group (2005)Google Scholar
  8. 8.
    Hitzer, E.: Euclidean Geometric Objects in the Clifford Geometric Algebra of Origin, 3-Space. Infinity. Bull. of the Belgian Math. Soc. 11(5), 653–662 (2004)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Jain, A.: Data Clustering: 50 Years Beyond K-means. In: Daelemans, W., Goethals, B., Morik, K. (eds.) ECML PKDD 2008, Part I. LNCS (LNAI), vol. 5211, pp. 3–4. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  10. 10.
    Li, H., Hestenes, D., Rockwood, A.: Generalized Homogeneous Coordinates for Computational Geometry. In: Sommer, G. (ed.) Geometric Computing with Clifford Algebras, vol. 40, pp. 27–52. Springer, Heildeberg (2001)Google Scholar
  11. 11.
    MacQueen, J.: Some Methods for Classification and Analysis of Multivariate Observations. In: Proc. of 5-th Berkeley Symp. on Math. Stats. and Prob., vol. 1, pp. 281–297. Unversity of California Press, Berkeley (1967)Google Scholar
  12. 12.
    Inaba, M., Katoh, N., Imai, H.: Applications of weighted Voronoi diagrams and randomization to variance-based k-clustering. In: Proceedings of 10th ACM Symposium on Computational Geometry, pp. 332–339 (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Benjamín Cruz
    • 1
  • Ricardo Barrón
    • 1
  • Humberto Sossa
    • 1
  1. 1.Center for Computing ResearchNational Polytechnic InstituteMéxico CityMéxico

Personalised recommendations