International Journal of Electronics and Telecommunications

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.

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Peculiarities of Applying Partial Convolutions to the Computation of Reduced Numerical Convolutions

Suleimenov,

Kadyrzhan,

Matrassulova

et al. 2024

Applied Sciences

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A method is proposed that reduces the computation of the reduced digital convolution operation to a set of independent convolutions computed in Galois fields. The reduced digital convolution is understood as a modified convolution operation whose result is a function taking discrete values in the same discrete scale as the original functions. The method is based on the use of partial convolutions, reduced to computing a modulo integer q0, which is the product of several prime numbers: q0=p1p2…pn. It is shown that it is appropriate to use the expansion of the number q0, to q=p0p1p2…pn, where p0 is an additional prime number, to compute the reduced digital convolution. This corresponds to the use of additional digits in the number system used to convert to partial convolutions. The inverse procedure, i.e., reducing the result of calculations modulo q to the result corresponding to calculations modulo q0, is provided by the formula that used only integers proved in this paper. The possibilities of practical application of the obtained results are discussed.

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International Journal of Electronics and Telecommunications

Cited by 3 publications

References 26 publications

Algorithm for Analyzing Rotating Images Based on the Fourier-Galois Transform

Algorithm for Analyzing Rotating Images Based on the Fourier-Galois Transform

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

Peculiarities of Applying Partial Convolutions to the Computation of Reduced Numerical Convolutions

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