Abstract
We discuss properties of high order neurons in competitive learning. In such neurons, geometric shapes replace the role of classic ‘point’ neurons in neural networks. Complex analytical shapes are modeled by replacing the classic synaptic weight of the neuron by high-order tensors in homogeneous coordinates. Such neurons permit not only mapping of the data domain but also decomposition of some of its topological properties, which may reveal symbolic structure of the data. Moreover, eigentensors of the synaptic tensors reveal the coefficients of polynomial rules that the network is essentially carrying out. We show how such neurons can be formulated to follow the maximum-correlation activation principle and permit simple local Hebbian learning. We demonstrate decomposition of spatial arrangements of data clusters including very close and partially overlapping clusters, which are difficult to separate using classic neurons.
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
Abe, S., Thawonmas, R.: A fuzzy classifier with ellipsoidal regions. IEEE Trans. OnFuzzy Systems 5(3), 358–368 (1997)
Anderson, E.: The Irises of the Gaspe Peninsula. Bulletin of the American IRIS Society 59, 2–5 (1939)
Bishop, C.M.: Neural Networks for Pattern Recognition. Clarendon press, Oxford (1997)
Blatt, M., Wiseman, S., Domany, E.: Superparamagnetic clustering of data. Physical Review Letters 76(18), 3251–3254 (1996)
Davé, R.N.: Use of the adaptive fuzzy clustering algorithm to detect lines in digital images. In: Proc. SPIE, Conf. Intell. Robots and Computer Vision, SPIE, vol. 1192(2), pp. 600–611 (1989)
Duda, R.O., Hart, P.E.: Pattern classification and scene analysis. Wiley, New York (1973)
Faux, I.D., Pratt, M.J.: Computational Geometry for Design and Manufacture. John Wiley & Sons, Chichester (1981)
Frigui, H., Krishnapuram, R.: A comparison of fuzzy shell-clustering methods for the detection of ellipses. IEEE Transactions on Fuzzy Systems 4(2), 193–199 (1996)
Mclachlan, G.J., Krishnan, T.: The EM algorithm and extensions. Wiley-Interscience, New York (1997)
Gnanadesikan, R.: Methods for statistical data analysis of multivariate observations. Wiley, New York (1977)
Graham, A.: Kronecker products and Matrix Calculus: with Applications. Wiley, Chichester (1981)
Gustafson, E.E., Kessel, W.C.: Fuzzy clustering with fuzzy covariance matrix. In: Proc. IEEE CDC, San Diego, CA, pp. 761–766 (1979)
Haykin, S.: Neural Networks, A comprehensive foundation. Prentice Hall, New Jersey (1994)
Kavuri, S.N., Venkatasubramanian, V.: Using fuzzy clustering with ellipsoidal units in neural networks for robust fault classification. Computers Chem. Eng. 17(8), 765–784 (1993)
Kohonen, T.: Self organizing maps. Springer, Berlin (1997)
Krishnapuram, R., Frigui, H., Nasraoui, O.: Fuzzy and probabilistic shell clustering algorithms and their application to boundary detection and surface approximation - Parts I and II. IEEE Transactions on Fuzzy Systems 3(1), 29–60 (1995)
Lipson, H., Siegelmann, H.T.: Clustering Irregular Shapes Using High-Order Neurons. Neural Computation (1999) (accepted for publication)
Mao, J., Jain, A.: A self-organizing network for hyperellipsoidal clustering (HEC). IEEE Transactions on Neural Networks 7(1), 16–29 (1996)
Pal, N., Bezdek, J.C., Tsao, E.C.-K.: Generalized clustering networks and Kohonen’s self-organizing scheme. IEEE Transactions on Neural Networks 4(4), 549–557 (1993)
Pan, J.S., McInnes, F.R., Jack, M.A.: Fast clustering algorithms for vector quantization. Pattern Recognition 29(3), 511–518 (1996)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Lipson, H., Siegelmann, H.T. (2000). High Order Eigentensors as Symbolic Rules in Competitive Learning. In: Wermter, S., Sun, R. (eds) Hybrid Neural Systems. Hybrid Neural Systems 1998. Lecture Notes in Computer Science(), vol 1778. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10719871_20
Download citation
DOI: https://doi.org/10.1007/10719871_20
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-67305-7
Online ISBN: 978-3-540-46417-4
eBook Packages: Springer Book Archive