Prospects for the use of algebraic rings to describe the operation of convolutional neural networks

A new type of number systems (integer coding systems) is used. In the system a set of digits, each of which corresponds to a certain prime number, is used instead of digits corresponding to the powers of a certain integer (for example, ten), All the prime numbers corresponding to different digits are different. Such an encoding of integers corresponds to a discrete signal model, in which the function corresponding to this model takes values in some algebraic ring. The advantage of such an encoding is the independent multiplication of numbers corresponding to different digits, which provides a significant simplification of calculations, including calculation of convolutions of signals presented in a discrete form. It is shown that in this case the convolution operation can be reduced to a situation where the convolution is calculated in Galois fields. In this case, the convolution operations carried out for the signals presented in proposed number system are carried out independently for each digit. A specific algorithm that implements this approach is proposed and its advantages for describing convolutional neural networks are proved. A specific example demonstrating these advantages is considered.