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Self-organisation in Kohonen's SOM.

John A. Flanagan1

  • 1Swiss Federal Institute of Technology, Switzerland

Neural Networks : the Official Journal of the International Neural Network Society
|October 1, 1996
PubMed
Summary

This study proves that neuron weights in self-organising maps (SOMs) converge to an organized state with probability one, even in higher dimensions. A modified algorithm ensures this convergence for improved self-organisation analysis.

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Analysing a Self-organising Algorithm.

Neural networks : the official journal of the International Neural Network Society·1997
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Area of Science:

  • Artificial Intelligence
  • Computational Neuroscience
  • Machine Learning

Background:

  • Self-organising maps (SOMs) are unsupervised learning algorithms crucial for data visualization and dimensionality reduction.
  • Existing analyses primarily focus on one-dimensional SOMs, leaving higher-dimensional behavior less understood.
  • The dynamic behavior of neuron weights in SOMs is often modeled as a stochastic process.

Purpose of the Study:

  • To demonstrate probability one convergence of neuron weights to an organized configuration in both one- and higher-dimensional SOMs.
  • To provide a theoretical framework for understanding self-organisation in SOMs beyond one dimension.
  • To analyze the convergence properties of modified SOM algorithms.

Main Methods:

  • Modeling neuron weights as a Markov process to analyze self-organisation dynamics.
  • Developing a proof for probability one convergence in one-dimensional SOMs for general probability distributions.
  • Introducing a modified SOM algorithm with an absorbing organized configuration.
  • Analyzing the convergence and first entry time into organized states for higher-dimensional SOMs.

Main Results:

  • A proof of self-organisation is established for one-dimensional SOMs under general conditions.
  • A modified SOM algorithm guarantees an absorbing organized configuration, even in higher dimensions.
  • Probability one convergence to this organized configuration is demonstrated for the modified algorithm.
  • For higher-dimensional SOMs, the first entry time into a predefined organized state is shown to be finite with probability one under certain conditions.

Conclusions:

  • The study provides robust theoretical evidence for the self-organisation of neuron weights in SOMs, extending beyond one dimension.
  • The modified SOM algorithm offers a reliable method for achieving and analyzing organized configurations in complex, high-dimensional data.
  • These findings contribute to a deeper understanding of SOM dynamics and their convergence properties, enhancing their applicability in machine learning and data analysis.

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