TIME BEHAVIOR OF TOPOLOGICAL ORDERING IN SELF-ORGANIZING FEATURE MAPPING

The dynamics of the feature maps created by Kohonen's algorithm is stu died by analyzing the wave number and frequency-dependent spectral den sity of synaptic fluctuations. This so-called dynamical spectral densi ty, which is well known in nonequilibrium statistical physics, constit utes a complete record of the time and length scales involved in the e volution of the map, the pertinent information being of much practical interest for the study of the convergence properties and the design o f effective parameter cooling strategies. We derive explicit theoretic al 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 con sequence of a dimensional conflict between input and output space. By comparisons with extensive numerical simulations the Fokker-Planck pic ture 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 cert ain 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 analo gy of the feature map with an elastic net we propose a new quantitativ e criterion measuring the topographic (neighborhood preserving) proper ties of the map in terms of the spectral density of the elastic tensio ns in the net. By way of examples we demonstrate how topological defec ts such as twists and kinks lead to characteristic elastic tensions th at are revealed immediately by the spectral analysis.