The self-organizing map (SOM) and neural gas (NG) constitute popular algorithms to represent data by means of prototypes arranged on a topographic map. Both methods rely on the Euclidean metric, hence clusters are isotropic. In this contribution, we extend prototype-based clustering algorithms such as NG and SOM towards a metric which is given by a full adaptive matrix such that ellipsoidal clusters are accounted for. We derive batch optimization learning rules for prototype and matrix adaptation based on a general cost function for NG and SOM and we show convergence of the algorithm. It can be seen that matrix learning implicitly performs minor local principal component analysis (PCA) and the local eigenvectors correspond to the main axes of the ellipsoidal clusters. We demonstrate the behavior in several examples.
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