Geodesic topographic product: An improvement to measure topology preservation of self-organizing neural networks

Abstract

Self-organizing neural networks endeavour to preserve the topology of an input space by means of competitive learning. There are diverse measures that allow to quantify how good is this topology preservation. However, most of them are not applicable to measure non-linear input manifolds, since they don't consider the topology of the input space in their calculation. In this work, we have modified one of the most employed measures, the topographic product, incorporating the geodesic distance as distance measure among the reference vectors of the neurons. Thus, it is possible to use it with non-lineal input spaces. This improvement allows to extend the studies realized with the original topographic product focused to the representation of objects by means of self-organizing neural networks. It would be also useful to determine the right dimensionality that a network must have to adapt correctly to an input manifold. © Springer-Verlag Berlin Heidelberg 2004.