Interweaving Kohonen Maps of Different Dimensions to Handle Measure Zero Constraints on Topological Mappings

The usual 'Kohonen' algorithm uses samples of points in a domain to develop a topological correspondence between a grid of 'neurons' and a continuous domain. Topological' means that near points are mapped to near points. However, for many applications there are additional constraints, which are given by sets of measure zero, which are not preserved by this method, because of insufficient sampling. In particular, boundary points do not typically map to boundary points because in general the likelihood of a sample point from a two-dimensional domain falling on the boundary is typically zero for continuous data, and extremely small for numerical data. A specific application, (assigning meshes for the finite element method), was recently solved by interweaving a two-dimensional Kohonen mapping on the entire grid with a one-dimensional Kohonen mapping on the boundary. While the precise method of interweaving was heuristic, the underlying rationale seems widely applicable. This general method is problem independent and suggests a direct generalization to higher dimensions as well.