High Dimensional Neurocomputing pp 79-103 | Cite as
Higher-Order Computational Model for Novel Neurons
- 3 Citations
- 960 Downloads
Abstract
Artificial neural network (ANN) has attracted a tremendous amount of interest for the solution of many complicated engineering and real-life problems. A small complexity, quick convergence, and robust performance are vital for its extensive applications. These features are pertinent upon the architecture of the basic working unit or neuron model, used in neural network. The computational capability of a neuron governs the architectural complexity of its neural network, which in turn defines the number of nodes and connections. Therefore, it is imperative to look for some neuron models, which yield ANN having small complexity in terms of network topology, number of learning parameters (connection weights) and at the same time they should possess fast learning, and superior functional capabilities. The conventional artificial neurons compute its internal state as the sum of contributions (aggregation) from impinging signals. For a neuron to respond strongly toward correlation among inputs, one must include higher-order relation among a set of inputs in their aggregation. A wide survey into design of artificial neurons brings out the fact that a higher-order neuron may generate an ANN which can have better classification and functional mapping capabilities with comparatively less number of neurons. Adequate functionality of ANN in a complex domain has also been observed in recent researches. This chapter presents higher-order computational models for novel neurons with well-defined learning procedures. Their implementation in a complex domain will provide a powerful scheme for learning input/output mapping in complex as well as in real domain along with better accuracy in wide spectrum of applications. The real domain implementation may be realized as its special case. The purpose of investigation in this chapter is to present the suitability and sustainability of higher-order neurons for readers, which can serve as a basis of the formulation for powerful ANN.
Keywords
Artificial Neural Network Hide Layer Neuron Model Synaptic Input Aggregation FunctionReferences
- 1.Koch, C., Poggio, T.: Multiplying with synapses and neurons. In: McKenna, T., Davis, J., Zornetzer, S.F. (eds.) Single Neuron Computation, pp. 315–345. Academic, Boston, MA (1992)Google Scholar
- 2.Mel, B.W.: Information processing in dendritic trees. Neural Comput. 6, 1031–1085 (1995)CrossRefGoogle Scholar
- 3.Arcas, B.A., Fairhall, A.L., Bialek, W.: What can a single neuron compute? In: Leen, T., Dietterich, T., Tresp, V. (eds.) Advances in Neural Information Processing, pp. 75–81. MIT press, Cambridge (2001)Google Scholar
- 4.McCulloch, W.S., Pitts, W.: A logical calculation of the ideas immanent in nervous activity. Bull. Math. Biophys. 5, 115–133 (1943)CrossRefzbMATHMathSciNetGoogle Scholar
- 5.Koch, C.: Biophysics of Computation: Information Processing in Single Neurons. Oxford University Press, New York (1999)Google Scholar
- 6.Mel, B.W., Koch, C.: Sigma-pi learning : on radial basis functions and cortical associative learning. In: Touretzky, D.S. (ed.) Advances in Neural Information Processing Systems 2, pp. 474–481. Morgan-Kaufmann, San Mateo, CA (1990)Google Scholar
- 7.Durbin, R., Rumelhart, R.: Product units: a computationally powerful and biologically plausible extension to backpropagation networks. Neural Comput. 1, 133–142 (1989)CrossRefGoogle Scholar
- 8.Bukovsky, I., Bila, J., Gupta, M.M., Hou, Z.G., Homma, N.: Foundation and classification of nonconventional neural units and paradigm of nonsynaptic neural interaction. In: Wang, Y. (ed.) (University of Calgary, Canada) Discoveries and Breakthroughs in Cognitive Informatics and Natural Intelligence (in the ACINI book series). IGI, Hershey PA, USA (ISBN: 978-1-60566-902-1) (2009)Google Scholar
- 9.Taylor, J.G., Commbes, S.: Learning higher order correlations. Neural Netw. 6, 423–428 (1993)CrossRefGoogle Scholar
- 10.Cotter, N.E.: The Stone-Weierstrass theorem and its application to neural networks. IEEE Trans. Neural Netw. 1, 290–295 (1990)Google Scholar
- 11.Shin, Y., Ghosh, J.: The Pi-sigma Network: an efficient higher-order neural network for pattern classification and function approximation. Proceedings of the International Joint Conference on Neural Networks, pp. 13–18 (1991)Google Scholar
- 12.Heywood, M., Noakes, P.: A framework for improved training of Sigma-Pi networks. IEEE Trans. Neural Netw. 6, 893–903 (1996)Google Scholar
- 13.Chen, M.S., Manry, M.T.: Conventional modeling of the multilayer perceptron using polynomial basis functions. IEEE Trans. Neural Netw. 4, 164–166 (1993)Google Scholar
- 14.Anthony, A., Holden, S.B.: Quantifying generalization in linearly weighted neural networks. Complex Syst. 18, 91–114 (1994)MathSciNetGoogle Scholar
- 15.Chen, S., Billings, S.A.: Neural networks for nonlinear dynamic system modeling and identification. Int. J. Contr. 56(2), 319–346 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
- 16.Schmidt, W., Davis, J.: Pattern recognition properties of various feature spaces for higher order neural networks. IEEE Trans. Pattern Anal. Mach. Intell. 15, 795–801 (1993)CrossRefGoogle Scholar
- 17.Kosmatopoulos, E., Polycarpou, M., Christodoulou, M., Ioannou, P.: High-order neural network structures for identification of dynamical systems. IEEE Trans. Neural Netw. 6(2), 422–431 (1995)Google Scholar
- 18.Liu, G.P., Kadirkamanathan, V., Billings, S.A.: On-line identification of nonlinear systems using volterra polynomial basis function neural networks. Neural Netw. 11(9), 1645–1657 (1998)CrossRefGoogle Scholar
- 19.Elder, J.F., Brown D.E.: Induction and polynomial networks. In: Fraser, M.D. (ed.) Network Models for Control and Processing, pp. 143–198. Intellect Books, Exeter, UK (2000)Google Scholar
- 20.Bukovsky, I., Redlapalli, S., Gupta, M.M.: Quadratic and cubic neural units for identification and fast state feedback control of unknown non-linear dynamic systems. Fourth International Symposium on Uncertainty Modeling and Analysis ISUMA 2003, pp. 330–334 (ISBN 0-7695-1997-0). IEEE Computer Society, Maryland, USA (2003)Google Scholar
- 21.Hou, Z.G., Song, K.Y., Gupta, M.M., Tan, M.: Neural units with higher-order synaptic operations for robotic image processing applications. Soft Comput. 11(3), 221–228 (2007)CrossRefGoogle Scholar
- 22.Nikolaev, N.Y., Iba, H.: Adaptive Learning of Polynomial Networks: Genetic Programming, Backpropagation and Bayesian Methods (ISBN: 0-387-31239-0, series: Genetic and Evolutionary Computation), vol. XIV, p. 316. Springer, New York (2006)Google Scholar
- 23.Zhang, M. (ed.) (Christopher Newport University): Artificial Higher Order Neural Networks for Economics and Business (ISBN: 978-1-59904-897-0). IGI-Global, Hershey, USA (2008)Google Scholar
- 24.Rosenblatt, F.: The perceptron : a probabilistic model for information storage and organization in the brain. Psychol. Rev. 65, 231–237 (1958)MathSciNetGoogle Scholar
- 25.Tripathi, B.K., Kalra, P.K.: On efficient learning machine with root power mean neuron in complex domain 22(5), 727–738, (ISSN: 1045-9227). IEEE Trans. Neural Netw. (2011)Google Scholar
- 26.Tripathi, B.K., Kalra, P.K.: Complex generalized-mean neuron model and its applications. Appl. Soft Comput. 11(1), 768–777 (Elsevier Science) (2011)Google Scholar
- 27.Tripathi, B.K., Kalra, P.K.: Functional mapping with complex higher order compensatory neuron model. World Congress on Computational Intelligence (WCCI-2010), July 18–23. IEEE Xplore, Barcelona, Spain (2010). Proc. IEEE Int. Joint Conf. Neural Netw. 22(5), 727–738 (ISSN: 1098–7576) (2011)Google Scholar
- 28.Hirose, A.: Complex-Valued Neural Networks. Springer, New York (2006)CrossRefzbMATHGoogle Scholar
- 29.Piazza, F., Benvenuto, N.: On the complex backpropagation algorithm. IEEE Trans. Signal Process. 40(4), 967–969 (1992)Google Scholar
- 30.Kim, T., Adali, T.: Approximation by fully complex multilayer perceptrons. Neural Comput. 15, 1641–1666 (2003)CrossRefzbMATHGoogle Scholar
- 31.Shin, Y., Jin, K.-S., Yoon, B.-Y.: A complex pi-sigma network and its application to equalization of nonlinear satellite channels. In: Proceedings of the IEEE International Conference on Neural Networks (1997)Google Scholar
- 32.Nitta, T.: An extension of the back-propagation algorithm to complex numbers. Neural Netw. 10(8), 1391–1415 (1997)CrossRefGoogle Scholar
- 33.Nitta, T.: An analysis of the fundamental structure of complex-valued neurons. Neural Process. Lett. 12, 239–246 (2000)CrossRefzbMATHGoogle Scholar
- 34.Tripathi, B.K., Kalra, P.K.: The novel aggregation function based neuron models in complex domain. Soft Comput. 14(10), 1069–1081 (Springer) (2010)Google Scholar
- 35.Piegat, A.: Fuzzy Modeling and Control. Springer, Heidelberg (2001)CrossRefzbMATHGoogle Scholar
- 36.Dyckhoff, H., Pedrycz, W.: Generalized means as model of compensative connectives. Fuzzy Sets Syst. 14, 143–154 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
- 37.Lee, C.C., Chung, P.C., Tsai, J.R., Chang, C.I.: Robust radial basis function neural network. IEEE Trans. Syst. Man Cybern. B Cybern. 29(6), 674–685 (1999)Google Scholar
- 38.Dubois, D., Prade, H.: A review of fuzzy set aggregation connectives. Inf. Sci. 36(1–2), 85–121 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
- 39.Kolmogoroff, A.N.: Sur la notion de la moyenne. Acad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. 12, 388–391 (1930)zbMATHGoogle Scholar
- 40.Nagumo, M.: Uber eine Klasse der Mittelwerte. Japan. J. Math. 7, 71–79 (1930)zbMATHGoogle Scholar
- 41.Schmitt, M.: On the complexity of computing and learning with multiplicative neural networks. Neural Comput. 14, 241–301 (2001)Google Scholar
- 42.Shiblee, Md., Chandra, B., Kalra, P.K.: New neuron model for blind source separation. In: Proceedings of the International Conference on Neural Information Processing, November 25–28 (2008)Google Scholar
- 43.Georgiou, G.M.: Exact interpolation and learning in quadratic neural networks. In Proceedings IJCNN, Vancouver, BC, Canada, July 16–21 (2006)Google Scholar
- 44.Foggel, D.B.: An information criterion for optimal neural network selection. IEEE Trans. Neural Netw. 2(5), 490–497 (1991)Google Scholar
- 45.Blake, C.L., Merz, C.J.: UCI repository of machine learning database. http://www.ics.uci.edu/mealrn/MLRepository.html. University of California, Department of Information and Computer Science (1998)
- 46.Daugman, J.G.: Entropy reduction and decorrelation in visual coding by oriented neural receptive fields. IEEE Trans. Biomed. Eng. 36(1), 107–114 (1989)CrossRefGoogle Scholar
- 47.Shin, Y., Ghosh, J.: Ridge polynomial networks. IEEE Trans. Neural Netw. 6(3), 610–622 (1995)Google Scholar
- 48.Li, C.-K.: A sigma-pi-sigma neural network. Neural Process. Lett. 17, 1–19 (2003)CrossRefGoogle Scholar