A method deciding topological relationship for self-organizing maps by persistent homology analysis

Self-organizing maps (SOM) are an effective method to quantize data space into several subspaces so-called centroids based on the probability distribution over input data space. The two-dimensional grid is widely used in SOM, yet several recent studies have introduced non-grid relationship, such as spherical or torus, which fit well depending on the topological structure among datapoints. However, there has been no definitive method to decide appropriate Topological Relationship. Traditionally, topological relationship had to be determined by looking into raw high-dimensional data, which requires highly non-trivial skills of experts. This paper proposes a systematic procedure to decide the suitable topological relationship of SOM via Persistence Homology analysis. Experimental results revealed that the proposed method enables to reduce the quantization error of SOM, to be a mapping in accordance with probability distribution, and to be a mapping which maintains similarity.