Self-Organizing Homotopy Network

Abst ract— In this paper, we propose a conceptual learning algorithm called the ‘self-organizing homotopy (SOH)’ together with an implementation thereof. As in the case of the SOM, our SOH organizes a homotopy in a selforganizing manner by giving a set of data episodes. Thus it is an extension of the SOM, moving from a ‘map’ to a ‘homotopy’. From a geometrical viewpoint, the SOH represents a set of (i.e. multiple) data distributions by a fiber bundle, whereas the SOM represents a single data distribution by a manifold. Therefore, this paper also proposes the concept of ‘fiber bundle learning” as an extension of manifold learning. One of the solutions to the SOH is SOM 2 , in which every reference vector unit of the conventional SOM is itself replaced by an SOM. Consequently SOM 2 has the ability to represent a fiber bundle, i.e. a product manifold, by using a product space of SOM SOM. It is also possible to design SOM n to represent higher order fiber bundles. It is expected that SOHs will play important roles in the fields of pattern recognition, adaptive functions, context understanding, and others, in which nonlinear manifolds and the homotopy play crucial roles.