List of issues > Series «Mathematics». 2024. Vol 50
On Partial Groupoids Associated with
the Composition of Multilayer Feedforward
Neural Networks
Author(s)
Andrey V. Litavrin1
,
Tatyana V. Moiseenkova1


1Siberian Federal University, Krasnoyarsk, Russian Federation
Abstract
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).
About the Authors
Andrey V. Litavrin, Cand. Sci. (Phys.–Math.), Assoc. Prof., Siberian Federal University, Krasnoyarsk, 660041, Russian Federation, anm11@rambler.ru
Tatyana V. Moiseenkova, Cand. Sci. (Phys.–Math.), Assoc. Prof., Siberian Federal University, Krasnoyarsk, 660041, Russian Federation, tanya-mois11@yandex.ru
For citation
Litavrin A. V., Moiseenkova T. V. On partial groupoids associated with the composition of multilayer feedforward neural networks. The Bulletin of Irkutsk State University. Series Mathematics, 2024, vol. 50, pp. 101–115.
https://doi.org/10.26516/1997-7670.2024.50.101
Keywords
partial groupoid, semigroupoids, endomorphism of partial groupoid, multilayer feedforward neural network
UDC
512.577+519.68:007.5
MSC
08A35, 08A62, 68Q06
DOI
https://doi.org/10.26516/1997-7670.2024.50.101
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