Convolutional neural networks from the perspective of the problems of multivalued logic

Abstract

Abstract It is shown that it is possible to reduce the operations performed by convolutional neural networks to algebraic, and then to operations of multivalued logic. The tool for this is signal models, which are functions that take values in Galois fields and algebraic rings. This approach, among other things, allows us to switch to the use of partial convolutions, which significantly simplify the analysis of digital signals and images. Such convolutions are performed directly in terms of Galois fields, which makes it possible to use a digital analogue of the classical convolution theorem. The advantage in this regard is the possibility of using a digital analogue of transfer functions, because in terms of Fourier-Galois spectra, the convolution operation is reduced to the multiplication operation by the transfer function. This, in turn, allows us to exhaustively describe a convolutional neural network through a set of transfer functions that take values in Galois fields. The close connection between Galois fields and multivalued logics makes it possible to describe the functioning of convolutional neural networks using variables of multivalued logic.