Modeling Connectionist Network Structures: Some Geometric and Categorical Aspects

Abstract

This contribution deals with an approach for mathematical modeling of the network structures of a certain connectionist network paradigm. Analysis of the structure of an artificial neural network (ANN) in that class of networks shows a possibility to introduce geometric and categorical modeling methods. This can be described briefly as follows. A (noncommutative) geometric space can be interpreted as a so-called geometric net. To a given ANN a corresponding geometric net can be associated. Geometric spaces form a category. Consequently, one obtains a category of geometric nets with a suitable notion of morphism. Then it is natural to interpret a learning step of an ANN as a morphism, thus learning corresponds to a finite sequence of morphisms (the associated networks are the objects). An associated (“local”) geometric net is less complex than the original ANN, but it contains all necessary information about the network structure. The association process together with learning (expressed by morphisms) leads to a commutative diagram corresponding to a suitable natural transformation, in terms of category theory. Commutativity of the diagram can be exploited to make learning “cheaper”. The simplified mathematical network model was used in ANN simulation applied in an industrial project on quality control. The “economy” of the model could be observed in a considerable increase of performance and decrease of production costs. Some prospects on the role of group operations that are induced by the regular structure of the underlying networks conclude the article.