Asymptotic Level Density of the Elastic Net Self-Organizing Feature Map

TL;DR

This paper analyzes the asymptotic density behavior of the Elastic Net self-organizing map, revealing a universal magnification law that depends solely on local stimulus density, contrasting with the Kohonen map's power law.

Contribution

It provides an analytical and numerical study showing the Elastic Net's unique universal magnification law for one-dimensional maps, differing from the Kohonen map.

Findings

Elastic Net exhibits a universal magnification law

Density depends only on local stimulus density

Contrasts with Kohonen map's power law

Abstract

Whileas the Kohonen Self Organizing Map shows an asymptotic level density following a power law with a magnification exponent 2/3, it would be desired to have an exponent 1 in order to provide optimal mapping in the sense of information theory. In this paper, we study analytically and numerically the magnification behaviour of the Elastic Net algorithm as a model for self-organizing feature maps. In contrast to the Kohonen map the Elastic Net shows no power law, but for onedimensional maps nevertheless the density follows an universal magnification law, i.e. depends on the local stimulus density only and is independent on position and decouples from the stimulus density at other positions.