Skip to main content
Springer Nature Link
Log in

Geometric neural networks

  • Visual and Motor Signal Neurocomputation
  • Conference paper
  • First Online:
Algebraic Frames for the Perception-Action Cycle (AFPAC 1997)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1315))

Abstract

The representation of the external world in biological creatures appears to be defined in terms of geometry. This suggests that researchers should look for suitable mathematical systems with powerful geometric and algebraic characteristics. In such mathematical context the design and implementation of neural networks will be certainly more advantageous. This paper presents the generalization of feedforward neural networks in the Clifford or geometric algebra framework. The efficiency of the geometric neural nets indicate a step forward in the design of algorithms for multidimensional artificial learning.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Arena P., Caponetto R., Fortuna L., Muscato G. and Xibilia M.G. 1996. Quaternionic multilayer perceptrons for chaotic time series prediction. IEICE Trans. Fundamentals. Vol. E79-A. No. 10 October, pp. 1–6.

    Google Scholar 

  2. Bayro-Corrochano E., Buchholz S., Sommer G. 1996. Selforganizing Clifford neural network IEEE ICNN'96 Washington, DC, June, pp. 120–125.

    Google Scholar 

  3. Bayro-Corrochano E., Lasenby J., Sommer G. Geometric Algebra: A framework for computing point and line correspondences and projective structure using nuncalibrated cameras IEEE Proceedings of ICPR'96 Viena, Austria, Vol. 1, pages 334–338, August, 1996.

    Google Scholar 

  4. Bayro-Corrochano E., Daniilidis K. and Sommer G. 1997. Hand-eye calibration in terms of motion of lines using Geometric Algebra. To appear in SCIA'97, June 9–11,Lappeenranta, Finland.

    Google Scholar 

  5. Cybenko G. Approximation by superposition of a sigmoidal function. Mathematics of control, signals and systems, Vol. 2, 303:314, 1989.

    Google Scholar 

  6. Doran C.J.L. 1994. Geometric algebra and its applications to mathematical physics. Ph.D. Thesis, University of Cambridge.

    Google Scholar 

  7. Georgiou G. M. and Koutsougeras C. Complex domain backpropagation. IEEE Trans. on Circuits and Systems, 330:334, 1992.

    Google Scholar 

  8. Hamilton W.R. 1853. Lectures on Quaternions. Hodges and Smith, Dublin.

    Google Scholar 

  9. Hestenes D. 1966. Space-Time Algebra. Gordon and Breach.

    Google Scholar 

  10. Hestenes D. and Sobczyk G. 1984. Clifford Algebra to Geometric Calculus: A unified language for mathematics and physics. D. Reidel, Dordrecht.

    Google Scholar 

  11. Hestenes D. 1993. Invariant body kinematics I: Saccadic and compensatory eye movements. Neural Networks, Vol. 7, 65–77.

    Article  Google Scholar 

  12. Hestenes D. 1993. Invariant body kinematics II: Reaching and neurogeometry. Neural Networks, Vol. 7, 79–88.

    Article  Google Scholar 

  13. Hornik K. 1989. Multilayer feedforward networks are universal approximators. Neural Networks, Vol. 2, pp. 359–366.

    Article  Google Scholar 

  14. Koenderink J. J. 1990. The brain a geometry engine. Psychological Research, Vol. 52, pp. 122–127.

    Article  PubMed  Google Scholar 

  15. Pearson J. K. and Bisset D.L.. 1992. Back Propagation in a Clifford Algebra. Artificial Neural Networks, 2, I. Aleksander and J. Taylor (Ed.), 413:416.

    Google Scholar 

  16. Pellionisz A. and Llinás R. 1980. Tensorial approach to the geometry of brain function: cerebellar coordination via a metric tensor. Neuroscience Vol. 5, pp. 1125–1136

    Article  PubMed  Google Scholar 

  17. Pellionisz A. and Llinás R. 1985. Tensor network theory of the metaorganization of functional geometries in th central nervous system. Neuroscience Vol. 16, No. 2, pp. 245–273.

    Article  PubMed  Google Scholar 

  18. Perantonis S. J. and Lisboa P. J. G. 1992. Translation, rotation, and scale invariant pattern recognition by high-order neural networks and moment classifiers. IEEE Trans. on Neural Networks, Vol. 3, No. 2, March, pp 241–251.

    Article  Google Scholar 

  19. Poggio T. and Girosi F. Networks for approximation and learning. IEEE Proc., Vol. 78, No. 9, 1481:1497, Sept. 1990.

    Article  Google Scholar 

  20. Porteous I.R. 1995. Clifford Algebras and the Classical Groups. Cambridge University Press, Cambridge.

    Google Scholar 

  21. Sommer G., Bayro-Corrochano E. and Billow T. 1997. Geometric algebra as a framework for the perception-action cycle. Workshop on Theoretical Foundation of Computer Vision, Dagstuhl, March 13–19 1996, Springer Wien.

    Google Scholar 

  22. Rumelhart D.E. and McClelland J. L.. 1986. Parallel Distributed Processing: Explorations in the Microestructure of Cognition. 2 Vols. Cambridge: MIT Press.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computer Science Institute, Cognitive Systems Group, Christian Albrechts University, Kiel, Germany

    Eduardo Bayro-Corrochano & Sven Buchholz

Authors
  1. Eduardo Bayro-Corrochano
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sven Buchholz
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Gerald Sommer Jan J. Koenderink

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Bayro-Corrochano, E., Buchholz, S. (1997). Geometric neural networks. In: Sommer, G., Koenderink, J.J. (eds) Algebraic Frames for the Perception-Action Cycle. AFPAC 1997. Lecture Notes in Computer Science, vol 1315. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0017879

Download citation

  • DOI: https://doi.org/10.1007/BFb0017879

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-63517-8

  • Online ISBN: 978-3-540-69589-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics