Time behavior of topological ordering in self-organizing feature mapping

Abstract

The dynamics of the feature maps created by Kohonen's algorithm is studied by analyzing the wave number and frequency-dependent spectral density of synaptic fluctuations. This so-called dynamical spectral density, which is well known in nonequilibrium statistical physics, constitutes a complete record of the time and length scales involved in the evolution of the map, the pertinent information being of much practical interest for the study of the convergence properties and the design of effective parameter cooling strategies. We derive explicit theoretical expressions for the dynamical spectral density based on the Fokker-Planck description of the stochastic process of learning and study in some detail the folding phenomena observed in the feature map as a consequence of a dimensional conflict between input and output space. By comparisons with extensive numerical simulations the Fokker-Planck picture is found to describe both the space and the time behavior of the map very well as soon as the dimensional conflict is well below a certain critical value. Results for the time and length scales involved in the evolution of the map are given both below and above the critical value of the dimensional conflict. Moreover exploiting a certain analogy of the feature map with an elastic net we propose a new quantitative criterion measuring the topographic (neighborhood preserving) properties of the map in terms of the spectral density of the elastic tensions in the net. By way of examples we demonstrate how topological defects such as twists and kinks lead to characteristic elastic tensions that are revealed immediately by the spectral analysis.