TOPOLOGY PRESERVATION IN SELF-ORGANIZING FEATURE MAPS - EXACT DEFINITION AND MEASUREMENT

The neighborhood preservation of self-organizing feature maps like the Kohonen map is an important property which is exploited in many appli cations. However, if a dimensional conflict arises this property Is lo st, Various qualitative and quantitative approaches are known for meas uring the degree of topology preservation, They are based on using the locations of the synaptic weight vectors. These approaches, however, may fail in case of nonlinear data manifolds, To overcome this problem , in this paper we present an approach which uses what we call the ind uced receptive fields for determining the degree of topology preservat ion, We first introduce a precise definition of topology preservation and then propose a tool for measuring it, the topographic function, Th e topographic function vanishes if and only if the map is topology pre serving, We demonstrate the power of this tool for various examples of data manifolds.