Тип публикации: статья из журнала
Год издания: 2021
Идентификатор DOI: 10.21538/0134-4889-2021-27-1-130-145
Ключевые слова: groupoid endomorphism, multilayer feedforward neural networks, multilayer neural network subnet
Аннотация: In this paper, we introduce commutative, but generally not associative, groupoids AGS(N) consisting of idempotents. The groupoid (AGS(N), +) is closely related to the multilayer feedforward neural networks JV (hereinafter just a neural network). It turned In this paper, we introduce commutative, but generally not associative, groupПоказать полностьюoids AGS(N) consisting of idempotents. The groupoid (AGS(N),+) is closely related to the multilayer feedforward neural networks N (hereinafter just a neural network). It turned out that in such neural networks, specifying a subnet of a fixed neural network is tantamount to specifying some special tuple composed of finite sets of neurons in the original network. All special tuples defining some subnet of the neural network N are contained in the set AGS(N). The rest of the tuples from AGS(N) also have a neural network interpretation. Thus, AGS(N) = F1 ⊂ F2, where F1 is the set of tuples that induce subnets and F2 is the set of other tuples. If two subnets of a neural network are specified, then two cases arise. In the first case, a new subnet can be obtained from these subnets by merging the sets of all neurons of these subnets. In the second case, such a merger is impossible due to neural network reasons. The operation (+) for any tuples from AGS(N) returns a tuple that induces a subnet or returns a neutral element that does not induce subnets. In particular, if for two elements from F1 the operation (+) returns a neutral element, then the subnets induced by these elements cannot be combined into one subnet. For any two elements from AGS(N), the operation has a neural network interpretation. In this paper, we study the algebraic properties of the groupoids AGS(N) and construct some classes of endomorphisms of such groupoids. It is shown that every subnet N' of the net N defines a subgroupoid T in the groupoid AGS(N) isomorphic to AGS(N'). It is proved that for every finite monoid G there is a neural network N such that G is isomorphically embeddable into the monoid of all endomorphisms End(AGS(N)). This statement is the main result of the work. © 2021 Krasovskii Institute of Mathematics and Mechanics. All Rights Reserved.
Журнал: TRUDY INSTITUTA MATEMATIKI I MEKHANIKI URO RAN
Выпуск журнала: Vol. 27, Is. 1
Номера страниц: 130-145
ISSN журнала: 01344889
Место издания: YEKATERINBURG
Издатель: KRASOVSKII INST MATHEMATICS & MECHANICS URAL BRANCH RUSSIAN ACAD SCIENCES