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The neighborhood preservation of self-organizing feature maps like the Kohonen map is an important property which is exploited in many applications. However, if a dimensional conflict arises this property is lost. Various qualitative and quantitative approaches are known for measuring 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 induced receptive fields for determining the degree of topology preservation. We first introduce a precise definition of topology preservation and then propose a tool for measuring it, the topographic function. The topographic function vanishes if and only if the map is topology preserving. We demonstrate the power of this tool for various examples of data manifolds.

An overview is given of prototype-based models in machine learning. In this framework, observations, i.e., data, are stored in terms of typical representatives. Together with a suitable measure of similarity, the systems can be employed in the context of unsupervised and supervised analysis of potentially high-dimensional, complex datasets. We discuss basic schemes of competitive vector quantization as well as the so-called neural gas approach and Kohonen's topology-preserving self-organizing map. Supervised learning in prototype systems is exemplified in terms of learning vector quantization. Most frequently, the familiar Euclidean distance serves as a dissimilarity measure. We present extensions of the framework to nonstandard measures and give an introduction to the use of adaptive distances in relevance learning.

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a b s t r a c tWe present an extension of the recently introduced Generalized Matrix Learning Vector Quantization algorithm. In the original scheme, adaptive square matrices of relevance factors parameterize a discriminative distance measure. We extend the scheme to matrices of limited rank corresponding to low-dimensional representations of the data. This allows to incorporate prior knowledge of the intrinsic dimension and to reduce the number of adaptive parameters efficiently.In particular, for very large dimensional data, the limitation of the rank can reduce computation time and memory requirements significantly. Furthermore, two-or three-dimensional representations constitute an efficient visualization method for labeled data sets. The identification of a suitable projection is not treated as a pre-processing step but as an integral part of the supervised training. Several real world data sets serve as an illustration and demonstrate the usefulness of the suggested method.