Local matrix adaptation in topographic neural maps

The self-organizing map (SOM) and neural gas (NG) and generalizations thereof such as the generative topographic map constitute popular algorithms to represent data by means of prototypes arranged on a (hopefully) topology representing map. However, most standard methods rely on the Euclidean metric, hence the resulting clusters are isotropic and they cannot account for local distorsions or correlations of data. In this contribution, we extend prototype-based clustering algorithms such as NG and SOM towards a more general metric which is given by a full adaptive matrix such that ellipsoidal clusters are accounted for. Thereby, the approach relies on a natural extension of the standard cost functions of NG and SOM (in the form of Heskes) and is conceptually intuitive. We derive batch optimization learning rules for prototype and matrix adaptation based on these generalized cost functions and we show convergence of the algorithm. Thereby, it can be seen that matrix learning implicitly performs local principal component analysis (PCA) and the local eigenvectors correspond to the main axes of the ellipsoidal clusters. Thus, the proposal also provides a cost function associated to alternative proposals in the literature which combine SOM or NG with local PCA models. We demonstrate the behavior of the proposed model in several benchmark examples and in an application to image compression.