Abstract
Recently, models of neural networks in the real domain have been extended into the high dimensional domain such as the complex and quaternion domain, and several high-dimensional models have been proposed. These extensions are generalized by introducing Clifford algebra (geometric algebra). In this paper we extend conventional real-valued models of recurrent neural networks into the domain defined by Clifford algebra and discuss their dynamics. We present models of fully connected recurrent neural networks, which are extensions of the real-valued Hopfield type neural networks to the domain defined by Clifford algebra. We study dynamics of the models from the point view of existence conditions of an energy function. We derive existence conditions of an energy function for two classes of the Hopfield type Clifford neural networks.
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.
Similar content being viewed by others
References
Hirose, A. (ed.): Complex-Valued Neural Networks Theories and Applications. World Scientific (2003)
Nitta, T. (ed.): Complex-Valued Neural Networks Utilizing High-Dimensional Parameters. IGI Global (2009)
Buchholz, S.: A Theory of Neural Computation with Clifford Algebra. Ph.D. Thesis, University. of Kiel (2005)
Lounesto, P.: Clifford Algebras and Spinors, 2nd edn. Cambridge Univ. Press (2001)
Perwass, C.: Geometric Algebra with Applications in Engineering. Springer, Heidelberg (2009)
Hopfield, J.J.: Neurons with graded response have collective computational properties like those of two-state neurons. Proc. Natl. Acad. Sci. USA 81, 3088–3092 (1984)
Hopfield, J.J., Tank, D.W.: Neural’ computation of decisions in optimization problems. Biol. Cybern. 52, 141–152 (1985)
Tank, D.W., Hopfield, J.J.: Simple ‘neural’ optimization networks: an A/D converter, signal decision circuit, and a linear programming circuit. IEEE Trans. Circuits Syst. 33, 533–541 (1986)
Hashimoto, N., Kuroe, Y., Mori, T.: On Energy Function for Complex-Valued Neural Networks. Trans. of the Institute of System, Control and Information Engineers 15(10), 559–565 (2002)
Kuroe, Y., Yoshida, M., Mori, T.: On Activation Functions for Complex-Valued Neural Networks - Existence of Energy Functions. In: Kaynak, O., Alpaydın, E., Oja, E., Xu, L. (eds.) ICANN 2003 and ICONIP 2003. LNCS, vol. 2714, pp. 985–992. Springer, Heidelberg (2003)
Yoshida, M., Kuroe, Y., Mori, T.: Models of Hopfield-Type Quaternion Neural Networks and Their Energy Functions. International Journal of Neural Systems 15(1&2), 129–135 (2005)
Kuroe, Y.: Models of Clifford Recurrent Neural Networks and Their Dynamics. In: Proceedings of 2011 International Joint Conference on Neural Networks, pp. 1035–1041 (2011)
Dorst, L., Fontijne, D., Mann, S.: Geometric Algebra for Computer Science An object-oriented Approach to Geometry. Morgan Kaufmann (2007)
Bayro-Corrochano, E., Scheuermann, G. (eds.): Geometric Algebra Computing in Engineering and Computer Science. Springer, Heidelberg (2010)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kuroe, Y., Tanigawa, S., Iima, H. (2011). Models of Hopfield-Type Clifford Neural Networks and Their Energy Functions - Hyperbolic and Dual Valued Networks -. In: Lu, BL., Zhang, L., Kwok, J. (eds) Neural Information Processing. ICONIP 2011. Lecture Notes in Computer Science, vol 7062. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24955-6_67
Download citation
DOI: https://doi.org/10.1007/978-3-642-24955-6_67
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-24954-9
Online ISBN: 978-3-642-24955-6
eBook Packages: Computer ScienceComputer Science (R0)