A mathematical analysis on the distribution-preserving property of Self-Organizing Map (SOM)

Self-organizing maps (SOM) are among the more popular neural network models first studied by Teuvo Kohonen in the early 1980\u27s. In this model, learning is unsupervised and the units organize themselves in a way that reflects the inter-relationships of the inputs they represent. Kohonen proposed an algorithm for th SOM using a highly simplified learning rule. This work reviews the original analysis done for the simple model. Several simulations on variants of the learning rule are conducted. From these simulations, a learning rule that approximates a Mexican Hat function is chosen. This is formally analyzed using the expectation values of the units in the map. Further simulations are conducted to validate the mathematical analysis done on the variant. The distribution-preserving property of 1-dimensional SOM\u27s is carefully analyzed and illustrated using simulations. This analysis is extended to 2-dimentional SOM\u27s