Classification and Clustering of Spatial Patterns with Geometric Algebra

  • Minh Tuan PhamEmail author
  • Kanta Tachibana
  • Eckhard M. S. Hitzer
  • Tomohiro Yoshikawa
  • Takeshi Furuhashi


In fields of classification and clustering of patterns most conventional methods of feature extraction do not pay much attention to the geometric properties of data, even in cases where the data have spatial features. This paper proposes to use geometric algebra to systematically extract geometric features from data given in a vector space. We show the results of classification of handwritten digits and those of clustering of consumers’ impression with the proposed method.


Feature Extraction Gaussian Mixture Model Expectation Maximization Algorithm Unlabeled Data Kernel Matrix 
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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  1. 1.
    Hu, M.K.: Visual pattern recognition by moment invariants. IRE Trans. Inf. Theory 8(2), 179–187 (1962) CrossRefGoogle Scholar
  2. 2.
    Mukundan, R., Ramakrishman, K.R.: Moment Functions in Image Analysis, Theory and Application. World Scientific, Singapore (1998) Google Scholar
  3. 3.
    Doran, C., Lasenby, A.: Geometric Algebra for Physicists. Cambridge University Press, Cambridge (2003) zbMATHGoogle Scholar
  4. 4.
    Hestenes, D.: New Foundations for Classical Mechanics. Springer, Dordrecht (1986) zbMATHGoogle Scholar
  5. 5.
    Dorst, L., Fontijne, D., Mann, S.: Geometric Algebra for Computer Science: An Object-oriented Approach to Geometry. Morgan Kaufmann Series in Computer Graphics. Morgan Kaufmann, San Mateo (2007) Google Scholar
  6. 6.
    Sekita, I., Kurita, T., Otsu, N.: Complex autoregressive model for shape recognition. IEEE Trans. Pattern Anal. Mach. Intell. 14(4), 489–496 (1992) CrossRefGoogle Scholar
  7. 7.
    Hirose, A.: Complex-Valued Neural Networks: Theories and Applications. Series on Innovative Intelligence, vol. 5. World Scientific, Singapore (2006) Google Scholar
  8. 8.
    Matsui, N., Isokawa, T., Kusamichi, H., Peper, F., Nishimura, H.: Quaternion neural network with geometrical operators. J. Intell. Fuzzy Syst. 15(3–4), 149–164 (2004) zbMATHGoogle Scholar
  9. 9.
    Buchholz, S., Le Bihan, N.: Optimal separation of polarized signals by quaternionic neural networks. In: 14th European Signal Processing Conference, EUSIPCO 2006, September 4–8, Florence, Italy (2006) Google Scholar
  10. 10.
    Nitta, T.: An extension of the back-propagation algorithm to complex numbers. Neural Netw. 10(8), 1391–1415 (1997) CrossRefGoogle Scholar
  11. 11.
    Hildenbrand, D., Hitzer, E.: Analysis of point clouds using conformal geometric algebra. In: 3rd International Conference on Computer Graphics Theory and Applications, Funchal, Madeira, Portugal (2008) Google Scholar
  12. 12.
    Hitzer, E.: Quaternion Fourier transform on quaternion fields and generalizations. Adv. Appl. Clifford Algebr. 17(3), 497–517 (2007) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Sommer, G.: Geometric Computing with Clifford Algebras. Springer, Berlin (2001) zbMATHGoogle Scholar
  14. 14.
    Dempster, A., Laird, N., Rubin, D.: Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. Ser. B 39(1), 1–38 (1977) zbMATHMathSciNetGoogle Scholar
  15. 15.
    Asuncion, A., Newman, D.J.: UCI Machine Learning Repository. University of California, School of Information and Computer Science, Irvine (2007) Google Scholar
  16. 16.
    Tipping, M.E., Bishop, C.M.: Mixtures of probabilistic principal component analysers. Neural Comput. 11, 443–482 (1999) CrossRefGoogle Scholar
  17. 17.
    Zhu, X., Lafferty, J., Ghahramani, Z.: Combining active learning and semi-supervised learning using Gaussian fields and harmonic functions. In: ICML 2003 Workshop on the Continuum from Labeled to Unlabeled Data in Machine Learning and Data Mining (2003) Google Scholar
  18. 18.
    Cristianini, N., Kandola, J., Elisseeff, A., Shawe-Taylor, J.: On kernel target alignment. J. Mach. Learn. Res. (2002) Google Scholar
  19. 19.
    Meinicke, P., Ritter, H.: Resolution-based complexity control for Gaussian mixture models. Neural Comput. 13(2), 453–475 (2001) zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London 2010

Authors and Affiliations

  • Minh Tuan Pham
    • 1
    Email author
  • Kanta Tachibana
    • 2
  • Eckhard M. S. Hitzer
    • 3
  • Tomohiro Yoshikawa
    • 1
  • Takeshi Furuhashi
    • 1
  1. 1.Nagoya UniversityNagoyaJapan
  2. 2.Kogakuin UniversityTokyoJapan
  3. 3.Department of Applied PhysicsUniversity of FukuiFukuiJapan

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