Abstract
We shall now demonstrate a very important phenomenon that apparently has a close relationship to the brain maps discussed in Sect. 2.15 and occurs in certain spatially interacting neural networks. While being categorizable as a special kind of adaptation, this phenomenon is also related to regression. In regression, some simple mathematical function is usually fitted to the distribution of sample values of input data. The “nonparametric regression” considered in this chapter, however, involves fitting a number of discrete, ordered reference vectors, similar to the codebook vectors discussed in Sect. 1.5, to the distribution of vectorial input samples. In order to approximate continuous functions, the reference vectors are here made to define the nodes of a kind of hypothetical “elastic network,” whereby the topological order characteristic of this mapping, and a certain degree of regularity of the neighboring reference vectors ensue from their local interactions, reflecting a kind of “elasticity.” One possibility to implement such an “elasticity” would be to define the local interactions between the nodes in the signal space [3.1-7], whereas more realistic spatial interactions, from a neural modeling point of view, are definable between the neurons along the neural network. The latter approach is mainly made in this text.
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References
R. Durbin, D. Willshaw: Nature 326, 689 (1987)
R. Durbin, G. Mitchison: Nature 343, 644 (1990)
G. Goodhill, D. Willshaw: Network 1, 41 (1990)
G. Goodhill: In Advances in Neural Information Processing Systems 5, ed. by L. Giles, S. Hanson, J. Cowan (Morgan Kaufmann, San Mateo, CA 1993) p. 985
J. A. Rangas, T. K. Kohonen, J. T. Laaksonen: IEEE Trans. Neural Networks 1, 93 (1990)
T. Martinetz: PhD Thesis (Technische Universität München, München, Germany 1992)
T. Martinetz: In Proc. ICANN’93, Int. Conf. on Artificial Neural Networks, ed. by S. Gielen, B. Kappen (Springer, London, UK 1993) p. 427
Y. Cheng: Neural Computation 9(8), 1667 (1997)
U. Grenander. Private communication, 1981
S. Orey: Limit Theorems for Markov Chain Transition Probabilities (Van Nostrand, London, UK 1971)
M. Cottrell, J.-C. Fort: Annales de l’Institut Henri Poincaré 23, 1 (1987)
T. Kohonen: Biol. Cyb. 44, 135 (1982)
T. Kohonen, E. Oja: Report TKK-F-A474 (Helsinki University of Technology, Espoo, Finland 1982)
T. Martinetz, K. Schulten: Neural Networks 7 (1994)
T. Kohonen: In Symp. on Neural Networks; Alliances and Perspectives in Senri (Senri Int. Information Institute, Osaka, Japan 1992)
T. Kohonen: In Proc. ICNN’93, Int. Conf. on Neural Networks (IEEE Service Center, Piscataway, NJ 1993) p. 1147
F. Mulier, V. Cherkassky: In Proc. 12 ICPR, Int. Conf. on Pattern Recognition (IEEE Service Center, Piscataway, NJ 1994) p. II–224
E. Erwin, K. Obermayer, K. Schulten: Biol. Cyb. 67, 35 (1992)
S. P. Luttrell: Technical Report 4669 (DRA, Malvern, UK 1992)
T. M. Heskes, B. Kappen: In Proc. ICNN’93, Int. Conf. on Neural Networks (IEEE Service Center, Piscataway, NJ 1993) p. III–1219
H. Ritter, K. Schulten: Biol. Cyb. 54, 99 (1986)
H. Ritter: IEEE Trans, on Neural Networks 2, 173 (1991)
D. R. Dersch, P. Tavan: IEEE Trans, on Neural Networks 6, 230 (1995)
S. P. Luttrell: IEEE Trans, on Neural Networks 2, 427 (1991)
S. P. Luttrell: Memorandum 4669 (Defense Research Agency, Mahern, UK, 1992)
T. Kohonen, J. Hynninen, J. Kangas, J. Laaksonen: Technical Report A31 (Helsinki University of Technology, Laboratory of Computer and Information Science, Helsinki 1996)
T. Kohonen, J. Hynninen, J. Kangas, J. Laaksonen, K. Torkkola: Technical Report A30 (Helsinki University of Technology, Laboratory of Computer and Information Science, Helsinki 1996)
H.-U. Bauer, K. R. Pawelzik: IEEE Trans, on Neural Networks 3 570 (1992)
S. Zrehen: In Proc. ICANN’93, Int. Conf. on Artificial Neural Networks (Springer-Verlag, London 1993) p. 609
T. Villmann, R. Der, T. Martinetz: In Proc. ICNN’94, IEEE Int. Conf. on Neural Networks (IEEE Service Center, Piscataway, NJ 1994), p. 645
K. Kiviluoto: In Proc. ICNN’96, IEEE Int. Conf. on Neural Networks (IEEE Service Center, Piscataway, NJ 1996), p. 294
S. Kaski, K. Lagus: In Lecture Notes in Computer Science, vol. 1112, ed. by C. v. d. Malsburg, W. von Seelen, J. C. Vorbrüggen, B. Sendhoff (Springer, Berlin 1996) p. 809
T. Samad, S. A. Harp: In Proc. IJCNN’91, Int. Joint Conf. on Neural Networks (IEEE Service Center, Piscataway, NJ 1991) p. II–949
T. Samad, S. A. Harp: Network: Computation in Neural Systems 3, 205 (1992)
A. Ultsch, H. Siemon: Technical Report 329 (Univ. of Dortmund, Dortmund, Germany 1989)
M. A. Kraaijveld, J. Mao, A. K. Jain: In Proc. 111CPR, Int. Conf. on Pattern Recognition (IEEE Comput. Soc. Press, Los Alamitos, CA 1992) p. 41
P. Koikkalainen: In Proc. ECAI 94, 11th European Conf. on Artificial Intelligence, ed.by A. Cohn (Wiley, New York, NY 1994) p. 211
P. Koikkalainen: In Proc. ICANN, Int. Conf. on Artificial Neural Networks (Paris, France 1995) p. II–63
T. Kohonen: Report A33 (Helsinki University of Technology, Laboratory of Computer and Information Science, Espoo 1996)
J. S. Rodrigues, L. B. Almeida: In Proc. INNC’90, Int. Neural Networks Conference (Kluwer, Dordrecht, Netherlands 1990) p. 813
J. S. Rodrigues, L. B. Almeida: In Neural Networks: Advances and Applications, ed. by E. Gelenbe (North-Holland, Amsterdam, Netherlands 1991) p. 63
B. Fritzke: In Proc. IJCNN’91, Int. Joint Conf. on Neural Networks (IEEE Service Center, Piscataway, NJ 1991) p. 531
B. Fritzke: In Artificial Neural Networks, ed. by T. Kohonen, K. Mäkisara, O. Simula, J. Kangas (North-Holland, Amsterdam, Netherlands 1991) p. I–403
B. Fritzke: Arbeitsbericht des IMMD, Universität Erlangen-Nürnberg 25, 9 (1992)
B. Fritzke: In Artificial Neural Networks, 2, ed. by I. Aleksander, J. Taylor (North-Holland, Amsterdam, Netherlands 1992) p. II–1051
B. Fritzke: PhD Thesis (Technische Fakultät, Universität Erlangen-Nürnberg, Erlangen, Germany 1992)
B. Fritzke: In Advances in Neural Information Processing Systems 5, ed. by L. Giles, S. Hanson, J. Cowan (Morgan Kaufmann, San Mateo, CA 1993) p. 123
B. Fritzke: In Proc. 1993 IEEE Workshop on Neural Networks for Signal Processing (IEEE Service Center, Piscataway, NJ 1993)
B. Fritzke: Technical Report TR-93-026 (Int. Computer Science Institute, Berkeley, CA 1993)
B. Fritzke: In Proc. ICANN’93, Int. Conf. on Artificial Neural Networks, ed. by S. Gielen, B. Kappen (Springer, London, UK 1993) p. 580
J. Blackmore, R. Miikkulainen: Technical Report TR AI92-192 (University of Texas at Austin, Austin, TX 1992)
J. Blackmore, R. Miikkulainen: In Proc. ICNN’93, Int. Conf. on Neural Networks (IEEE Service Center, Piscataway, NJ 1993) p. I–450
C. Szepesvári, A. Lőrincz: In Proc. ICANN-93, Int. Conf. on Artificial Neural Networks, ed. by S. Gielen, B. Kappen (Springer, London, UK 1993) p. 678
C. Szepesvári, A. Lőrincz: In Proc. WCNN’93, World Congress on Neural Networks (INNS, Lawrence Erlbaum, Hillsdale, NJ 1993) p. II–497
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Kohonen, T. (2001). The Basic SOM. In: Self-Organizing Maps. Springer Series in Information Sciences, vol 30. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56927-2_3
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DOI: https://doi.org/10.1007/978-3-642-56927-2_3
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