Abstract
Self-organizing map (SOM) is a famous type of artificial neural network, which was first developed by Kohonen (1997). The SOM algorithm is vary practical and has many useful applications, such as semantic map, diagnosis of speech voicing, solving combinatorial optimization problem, and so on. However, its theoretical and mathematical structure is not clear. In this chapter, we discuss a special property, i.e., monotonicity, of model functions in fundamental SOM with one-dimensional array of nodes and real-valued nodes. Firstly, so-called quasiconvexity and quasiconcavity for model functions have been suggested. Then it has been shown that the renewed model function of a quasiconvex (quasiconcave) model function is also quasiconvex (quasiconcave), and quasiconvex states or quasiconcave states of a model function appear in the previous stage of the monotonic states.
This chapter is organized as follows. Section 5.1 gives a simple review of neural network and self-organizing map and introduces our motivation for the research. Section 5.2 presents the basic concept and algorithm of the SOM. Section 5.3 gives the main theoretical results and detail proof. In Section 5.4 numerical examples are given to illustrate the properties of the SOM. Concluding remarks are given in the final section.
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Acknowledgements
This work was a cooperation between researchers including Dr. Mituhiro Hoshino, Dr. Yutaka Kimura and Dr. Kaku. Their contribution is very much appreciated.
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Yin, Y., Kaku, I., Tang, J., Zhu, J. (2011). Neural Network and Self-organizing Maps. In: Data Mining. Decision Engineering. Springer, London. https://doi.org/10.1007/978-1-84996-338-1_5
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DOI: https://doi.org/10.1007/978-1-84996-338-1_5
Publisher Name: Springer, London
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