Non-Euclidean Self-Organizing Maps

Self-Organizing Maps (SOMs, Kohonen networks) belong to neural network models of the unsupervised class. Most data analysts take it for granted to use some subregions of a flat space as their data model. However, assuming that the underlying geometry is non-Euclidean, we obtain a new degree of freedom for the techniques that translate the similarities into spatial neighborhood relationships. In this presentation, we introduce the generalized setup for non-Euclidean SOMs. We will present the theoretical contribution (usage of quotient spaces; geometry-related adjustments in the dispersion function) and experimental verification. Our quantitative analysis shows that the shape of data matters for Self-Organizing Maps. Not only do we get more regular and visually appealing results than the previous setups, but our experiments also show that using our dispersion function is superior to the traditional Gaussian one. Our proposition can be successfully applied to dimension reduction, clustering, or finding similarities in big data (both hierarchical and non-hierarchical).