Competitive Hebbian Learning Rule Forms Perfectly Topology Preserving Maps

Abstract

The problem of fOnning perfectly topology preserving maps of feature manifolds is studied. First, through introducing ''masked Voronoi polyhedra" as a geometrical construct for detennining neighborhood on manifolds, a rigorous definition of the term "topology preserving feature map" is given. Starting from this definition, it is shown that a network G of neural units i, i = 1, ... ,N has to have a lateral connectivity structure A, Ai; e {O, I}, i,j = 1, ... , N which corresponds to the "induced Delaunay triangulation" of the synaptic weight vectors Wi e lRD in order to form a perfectly topology preserving map of a given manifold M ~ !R D of features v eM. The lateral connections detennine the neighborhood relations between the units in the network, which have to match the neighborhood relations of the features on the manifold. If all the weight vectors Wi are distributed over the given feature manifold M, and if this distribution resolves the shape of M, it can be shown that Hebbian learning with competition leads to lateral connections i-j (Ai; = 1) that correspond to the edges of the "induced Delaunay trianguJation" and, hence, leads to a network structure that forms a perfectly topology preserving map of M, independent of M's topology. This yields a means for constructing perfectly topology preserving maps of arbitrarily structured feature manifolds.