On partial groupoids associated with the composition of multilayer feedforward neural networks

A. V. Litavrin   ; T. V. Moiseenkova

The Bulletin of Irkutsk State University. Series Mathematics, Tome 50 (2024), pp. 101-115

Résumé

In this work, partial groupoids are constructed associated with compositions of multilayer neural networks of direct signal distribution (hereinafter simply neural networks). The elements of these groupoids are tuples of a special type. Specifying such a tuple determines the structure (i.e., architecture) of the neural network. Each such tuple can be associated with a mapping that will implement the operation of the neural network as a computational circuit. Thus, in this work, the neural network is identified primarily with its architecture, and its work is implemented by a mapping that is built using an artificial neuron model. The partial operation in the constructed groupoids is designed in such a way that the result of its application (if defined) to a pair of neural networks gives a neural network that, on each input signal, acts in accordance with the principle of composition of neural networks (i.e., the output signal of one network is sent to the input second network). It is established that the constructed partial groupoids are semigroupoids (i.e. partial groupoids with the condition of strong associativity). Some endomorphisms of the indicated groupoids are constructed, which make it possible to change the threshold values and activation functions of the neurons of the specified population. Transformations of the constructed partial groupoids are studied, which allow changing the weights of synoptic connections from a given set of synoptic connections. In the general case, these transformations are not endomorphisms. A partial groupoid was constructed for which this transformation is an endomorphism (the support of this partial groupoid is a subset in the support of the original partial groupoid).

Détail

Keywords: semigroupoids, multilayer feedforward neural network.
Mots-clés : partial groupoid, endomorphism of partial groupoid

Bibliographie

[1] Arapina-Arapova E.S., “Partial groupoids in relation to information systems”, News of the Southern Federal University. Technical science, 2013, no. 2, 46–51 (in Russian)

[2] Glushkov V.M., “Abstract theory of automata”, UMN, 16:5 (1961), 3–62 (in Russian) | MR

[3] Gluskin L.M., “Research on general algebra in Saratov”, Izv. Vyssh. Uchebn. Zaved. Mat., 1970, no. 4, 3–16 (in Russian) | MR | Zbl

[4] Golovko V.A., Krasnoproshin V.V., Neural network technologies for data processing, Publ. house Belarus State University, Minsk, 2017, 263 pp. (in Russian)

[5] Gorban' A.N., “Generalized approximation theorem and computational capabilities of neural networks”, Sib. zhurn. calculated mathematics, 1:1 (1998), 11–24 (in Russian) | MR

[6] Litavrin A.V., Moiseenkova T.V., “About one groupoid associated with the composition of multilayer feedforward neural networks”, Zhurnal Srednevolzhskogo matematicheskogo obshchestva, 26:2 (2024), 111–122 (in Russian) | DOI | MR | Zbl

[7] Litavrin A.V., “On endomorphisms of the additive monoid of subnets of a two-layer neural network”, The Bulletin of Irkutsk State University. Series Mathematics, 39 (2022), 111–126 | DOI | MR | Zbl

[8] Lyapin E.S., “Partial groupoids that can be obtained from semigroups by restrictions and homomorphisms”, Izv. Vyssh. Uchebn. Zaved. Mat., 33:10 (1989), 37–45 | MR

[9] Slepovichev I.I., “Algebraic properties of abstract neural networks”, Izvestiya Saratov Univ. New series. Series Mathematics. Mechanics. Informatics, 16:1 (2016), 96–103 (in Russian) | DOI | MR | Zbl

[10] Sozykin A.V., “An overview of methods for deep learning in neural networks”, Vestn. YuUrGU. Ser. Vych. Matem. Inform., 6:3 (2017), 28–59 (in Russian) | DOI