General Riemannian SOM

TL;DR

This paper extends Self-Organizing Maps to Riemannian geometries, providing a new framework for data reduction in non-Euclidean spaces with analysis of stability limits.

Contribution

It introduces the General Riemannian SOM (GRiSOM), including implementation for constant curvature geometries and stability analysis across different tessellations.

Findings

Analytic and numerical stability limits are consistent for larger neighborhood ranges.

Deviations occur in Euclidean maps with very small neighborhoods.

The framework supports non-Euclidean data analysis.

Abstract

Kohonen's Self-Organizing Maps (SOMs) have proven to be a successful data-reduction method to identify the intrinsic lower-dimensional sub-manifold of a data set that is scattered in the higher-dimensional feature space. Motivated by the possibly non-Euclidian nature of the feature space and of the intrinsic geometry of the data set, we extend the definition of classic SOMs to obtain the General Riemannian SOM (GRiSOM). We additionally provide an implementation as a proof-of-concept for geometries with constant curvature. We furthermore perform the analytic and numerical analysis of the stability limits of certain (GRi)SOM configurations covering the different possible regular tessellation of the map space in each geometry. A deviation between the numerical and analytic stability limit has been observed for the square and hexagonal Euclidean maps for very small neighbourhoods in the map space as well as agreement in case of longer-ranged relations between the map nodes.