Self Organizing Map algorithm and distortion measure

TL;DR

This paper analyzes the statistical properties of the Self Organizing Map (SOM) algorithm, focusing on the relationship between its equilibrium points and the minima of the distortion measure, and proves consistency under certain conditions.

Contribution

It establishes the strong consistency of the SOM map minimizing empirical distortion and clarifies the mismatch between theoretical minima and algorithm equilibria.

Findings

SOM map is strongly consistent in compact Euclidean spaces.

Equilibrium points of SOM do not generally coincide with minima of the distortion measure.

Illustrative example demonstrates the discrepancy between minima and equilibria.

Abstract

We study the statistical meaning of the minimization of distortion measure and the relation between the equilibrium points of the SOM algorithm and the minima of distortion measure. If we assume that the observations and the map lie in an compact Euclidean space, we prove the strong consistency of the map which almost minimizes the empirical distortion. Moreover, after calculating the derivatives of the theoretical distortion measure, we show that the points minimizing this measure and the equilibria of the Kohonen map do not match in general. We illustrate, with a simple example, how this occurs.