Topology representing networks

This competitive Hebbian rule provides a novel approach to the problem of constructing topology preserving feature maps and representing intricately structured manifolds and makes this novel approach particularly useful in all applications where neighborhood relations have to be exploited or the shape and topology of submanifolds have to been take into account.

Abstract

A Hebbian adaptation rule with winner-take-all like competition is introduced. It is shown that this competitive Hebbian rule forms so-called Delaunay triangulations, which play an important role in computational geometry for efficiently solving proximity problems. Given a set of neural units i, i = 1,…, N, the synaptic weights of which can be interpreted as pointers wi, i = 1,…, N in RD, the competitive Hebbian rule leads to a connectivity structure between the units i that corresponds to the Delaunay triangulation of the set of pointers wi. Such competitive Hebbian rule develops connections (Cij > 0) between neural units i, j with neighboring receptive fields (Voronoi polygons) Vi, Vj, whereas between all other units i, j no connections evolve (Cij = 0). Combined with a procedure that distributes the pointers wi over a given feature manifold M, for example, a submanifold M ⊂ RD, the competitive Hebbian rule provides a novel approach to the problem of constructing topology preserving feature maps and representing intricately structured manifolds. The competitive Hebbian rule connects only neural units, the receptive fields (Voronoi polygons) Vi, Vj of which are adjacent on the given manifold M. This leads to a connectivity structure that defines a perfectly topology preserving map and forms a discrete, path preserving representation of M, also in cases where M has an intricate topology. This makes this novel approach particularly useful in all applications where neighborhood relations have to be exploited or the shape and topology of submanifolds have to be take into account.