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.
Similar content being viewed by others
References
J. Adámek, H. Herrlich and G.E. Strecker, Abstract and Concrete Categories (Wiley, New York, 1990).
J. André, Endliche nichtkommutative Geometrie, Annales Univ. Saraviensis, Ser. Math. 2(1) (1988) 1-136.
J. André, Configurational conditions and digraphs, Journal of Geometry 43 (1992) 22-29.
J. André, On non-commutative geometry, Annales Univ. Saraviensis, Ser. Math. 4(2) (1993) 93-129.
R. Eckmiller, Concerning the emerging role of geometry in neuroinformatics, in: Parallel Processing in Neural Systems and Computers, eds. R. Eckmiller, G. Hartmann and G. Hauske (North-Holland, Amsterdam, 1990).
H. Geiger, Optical quality control with self-learning systems using a combination of algorithmic and neural network approaches, in: Proceedings of the Second European Congress on Intelligent Techniques and Soft Computing, EUFIT'94, Aachen, September 20-23 (1994).
H. Geiger and J. Pfalzgraf, Modeling a connectionist network paradigm: Geometric and categorical perspectives, in preparation.
H. Geiger and J. Pfalzgraf, Quality control connectionist networks supported by a mathematical model, in: Proceedings of the International Conference on Engineering Applications of Artificial Neural Networks (EANN'95), 21-23 August 1995, Helsinki, eds. A.B. Bulsari and S. Kallio (Finnish AI Society, 1995).
S. Mac Lane, Categories for the Working Mathematician, Graduate Texts in Mathematics, Vol. 5, 2nd ed. (Springer, New York, 1998).
F.W. Lawvere and S.H. Schanuel, Conceptual Mathematics: A First Introduction to Categories (Cambridge University Press, Cambridge, 2000).
D. Nauck, F. Klawonn and R. Kruse, Neuronale Netze und Fuzzy-Systeme (Vieweg Verlag, 1994).
J. Pfalzgraf, On a model for non-commutative geometric spaces, Journal of Geometry 25 (1985) 147-163.
J. Pfalzgraf, On geometries associated with group operations, Geometriae Dedicata 21 (1986) 193-203.
J. Pfalzgraf, A note on simplices as geometric configurations, Archiv der Mathematik 49 (1987) 134-140.
J. Pfalzgraf, On a general notion of a hull, in: Automated Practical Reasoning, eds. J. Pfalzgraf and D. Wang, Texts and Monographs in Symbolic Computation (Springer, Wien, New York, 1994).
J. Pfalzgraf, Graph products of groups and group spaces, Journal of Geometry 53 (1995) 131-147.
J. Pfalzgraf, On a category of geometric spaces and geometries induced by group actions, Ukrainian Journal of Physics 43(7) (1998) 847-856.
B.C. Pierce, Basic Category Theory for Computer Scientists (The MIT Press, Cambridge, 1991).
R. Rojas, Theorie der neuronalen Netze, Springer-Lehrbuch (Springer, Berlin, 1993).
J. Sixt, Design of an artificial neural network simulator and its integration with a robot simulation environment, Diploma Thesis, RISC-Linz, University of Linz (1994).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Pfalzgraf, J. Modeling Connectionist Network Structures: Some Geometric and Categorical Aspects. Annals of Mathematics and Artificial Intelligence 36, 279–301 (2002). https://doi.org/10.1023/A:1016094912264
Issue Date:
DOI: https://doi.org/10.1023/A:1016094912264