Non-Euclidean Self-Organizing Maps

TL;DR

This work generalizes Self-Organizing Maps to non-Euclidean data spaces, enabling neighborhood relationships to reflect hyperbolic, spherical, and quotient-space geometries. It introduces a geometry-aware dispersion mechanism and leverages Goldberg-Coxeter tessellations to scale templates, while using closed manifolds (e.g., elliptic plane, tori, Hurwitz surfaces) to mitigate boundary effects. The authors provide a comprehensive experimental framework across diverse manifolds, along with topology-preserving and embedding-quality measures (including a Villmann-based topology metric) and a public codebase. The results demonstrate improved topology preservation and more interpretable visualizations, particularly for hierarchical or polarization-prone data, suggesting non-Euclidean SOMs as both stand-alone tools and auxiliary components in broader models. This approach broadens SOM applicability to complex data with inherent non-Euclidean structure and offers practical guidance for manifold choice, distance definitions, and tessellation strategies.

Abstract

Self-Organizing Maps (SOMs, Kohonen networks) belong to neural network models of the unsupervised class. In this paper, we present the generalized setup for non-Euclidean SOMs. Most data analysts take it for granted to use some subregions of a flat space as their data model; however, by the assumption 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. We improve the traditional SOM algorithm by introducing topology-related extensions. Our proposition can be successfully applied to dimension reduction, clustering or finding similarities in big data (both hierarchical and non-hierarchical).

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