Abstract
In the last period there were introduced some new types of neural networks based on a more general type of algebra. They obey the multidimensional neural activities compared to the one-dimensional neural activity within the standard neural network framework. Instead of using basically the product of two scalar values they utilise some special algebraic product of two multidimensional quantities. Most of them can be considered as a special type of geometric (Clifford) algebra [1], [2] based neural networks (GANNs).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Hestenes D. and Sobczyk G. Clifford Algebra to Geometric Calculus. D. Reidel Publishing Company, 1984.
Hestenes D. New Foundations for Classical Mechanics. D. Reidel Publishing Company, 1986.
Arena P., Fortuna L., Re R., and Xibilia MG. On the capability of neural networks with complex neurons in complex-valued function approximation. In Proceedings IEEE International Conference on Circuit and Systems, pages 2168–2171, Chicago, 1993.
Benvenuto N., Marchesi M., Piazza F., and Uncini A. A comparison between real and complex-valued neural networks in communication applications. In Kohonen et al., editor, Artificial Neural Networks, pages 1177–1180. Elsevier Science Publishers B.V. ( North-Holland ), 1991.
Birx DL. and Pipenberg SJ. A complex mapping network for phase sensitive classification. IEEE Transactions on Neural Networks, 4 (1): 127–135, 1993.
Hirose A. Dynamics of fully complex-valued neural networks. Electronic Letters, 28 (16): 1492–1494, 1992.
Hirose A. Continuous complex-valued back-propagation learning. Electronic Letters, 28 (20): 1854–1855, 1992.
Leung H. and Haikyn S. The complex backpropagation algorithm. IEEE Transactions on Signal Processing, 39 (9): 2101–2104, 1991.
Arena P., Fortuna L., Muscato G., and Xibilia MG. Multilayer perceptrons to approximate quaternion-valued functions. Neural Networks, 9(6): 1–8, 6 1996.
Masters T. Signal and Image Processing with Neural Networks: a C++ Sourcebook. John Wiley Sons, 1994.
Blow Th. and Sommer G. Real multidimensional fourier transform. In SCIA ‘87, 1997. submitted.
Chudÿ L. and Chudÿ V. Why complex-valued neural networks? (on the possible role of geometric algebra framework for the neural network computation). In Proceedings of IEEE Workshop on Computer-Intensive Methods in Control and Dignal Processing, pages 109–114, Prague, 1996.
Kalman BL. and Kwasny SC. Why tanh? choosing a sigmoidal function. In Proceedings of Int. Joint Conference on Neural Networks, Baltimore, MD, 1991.
Friedman JH. An overwiev of predictive learning and function approximation. In V. Cherkassky, J. Friedman, and H. Wechsler, editors, From statistics to neural networks., NATO ASI Series, pages 1–62. Springer-Verlag, 1994.
Hertz J. and Krogh A. Palmer RG. Introduction to the theory of neural computation. Addison-Wesley, 1991.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer-Verlag London Limited
About this chapter
Cite this chapter
Kárný, M., Warwick, K., Kůrková, V. (1998). Geometric Algebra Based Neural Networks. In: Kárný, M., Warwick, K., Kůrková, V. (eds) Dealing with Complexity. Perspectives in Neural Computing. Springer, London. https://doi.org/10.1007/978-1-4471-1523-6_10
Download citation
DOI: https://doi.org/10.1007/978-1-4471-1523-6_10
Publisher Name: Springer, London
Print ISBN: 978-3-540-76160-0
Online ISBN: 978-1-4471-1523-6
eBook Packages: Springer Book Archive