Abstract
In this paper we propose a new type of neuron developed in the framework of Clifford algebra. It is shown how this novel Clifford neuron covers complex and quaternionic neurons and that it can compute orthogonal transformations very efficiently. The introduced framework can also be used for neural computation of non–linear geometric transformations which makes it very promising for applications. As an example we develop a Clifford neuron that computes the cross-ratio via the corresponding Möobius transformation. Experimental results for the proposed novel neural models are reported.
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References
Arena, P., Fortuna, L., Muscato, G., Xibilia, M.G.: Neural Networks in Multidimensional Domains. LNCIS, vol. 234. Springer, Heidelberg (1998)
Buchholz, S., Sommer, G.: Introduction to Neural Computation in Clifford Algebra. In: Sommer, G. (ed.) Geometric Computing with Clifford Algebra, ch. 13. Springer, Heidelberg (2000) (to appear)
Caelli, T., Squire, D., Wild, T.: Model–Based Neural Networks. Neural Networks 6, 613–625 (1993)
Cybenko, G.: Approximation by superposition of a sigmoidal function. Mathematics of Control, Signals and Systems 2, 303–314 (1989)
Ferraro, M., Caelli, T.: Neural computation of algebraic and geometrical structures. Neural Networks 11, 669–707 (1998)
Geman, S., Bienenstock, E., Doursat, R.: Neural networks and the bias/variance dilemma. Neural Computation 4, 1–58 (1992)
Georgiou, G., Koutsougeras, C.: Complex domain backpropagation. IEEE Trans. Circ. and Syst. II 39, 330–334 (1992)
Lounesto, P.: Clifford Algebras and Spinors. Cambridge University Press, Cambridge (1997)
Masters, T.: Signal and Image Processing with Neural Networks. John Willey and Sons, West Sussex (1994)
McCulloch, W.S., Pitts, W.: A Logical Calculus of Ideas Immanent in Nervous Activity. Bulletin of Mathematical Biophysics 5, 115–133 (1943)
Needham, T.: Visual Complex Analysis. Clarendon Press, Oxford (1997)
Porteous, I.R.: Clifford Algebras and the Classical Groups. Cambridge University Press, Cambridge (1995)
Rumelhart, D.E., Hinton, G.E., Williams, R.J.: Learning internal representations by error propagation. In: Parallel Distributed Processing, vol. 1. MIT Press, Cambridge (1986)
Vidyasagar, M.: A Theory of Learning and Generalization. Springer, Heidelberg (1997)
Voss, K., Süsse, H.: Adaptive Modelle und Invarianten für zweidimensionale Bilder. Verlag Shaker, Aachen (1995)
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Buchholz, S., Sommer, G. (2000). Learning Geometric Transformations with Clifford Neurons. In: Sommer, G., Zeevi, Y.Y. (eds) Algebraic Frames for the Perception-Action Cycle. AFPAC 2000. Lecture Notes in Computer Science, vol 1888. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10722492_9
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DOI: https://doi.org/10.1007/10722492_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-41013-3
Online ISBN: 978-3-540-45260-7
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