A Note on Modeling Connectionist Network Structures: Geometric and Categorical Aspects

Abstract

This position paper proposes a mathematical modeling approach for a certain class of connectionist network structures. Investigation of the structure of an artificial neural network (ANN) in that class (paradigm) suggested the use of geometric and categorical modeling methods in the following sense. 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. 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. Commutativity 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.