# Peculiarities of Applying Partial Convolutions to the Computation of Reduced Numerical Convolutions
## 2 Related Works  
This work lies in the general trend of modern research aimed at improving the efficiency of computational systems, including those focused on the use of convolutional neural networks. References to specific works are given in Table 1.  
One of the important directions here is related to the use of RNS (modular arithmetic), which can be used for neural networks too [37]. Such systems are closely related to fast arithmetic [38; 39], i.e., an area of research that has received increasing attention over the last decades [40]. This work solves the problem of bringing the result of convolution operation to the original discretization scale, while preserving the possibility of computation in terms of RNS, i.e., it responds to the above trend. It is of importance that the latter approach allows for generalization, since RNS can be considered as a special case of finite algebraic rings [24]. A wide variety of rings can be used for representation any set containing a finite number of elements, including non-associative ones [41; 42]. This, in particular, justifies the use of the term \"partial convolutions\", which implies the possibility of a subsequent transition to the use of finite algebraic rings of various types. A concrete example of using non-trivial algebraic rings for digital signal processing was given in [18]. The generalization of the RNS approach obviously creates additional opportunities for digital signal processing. Further, some subsets of an algebraic ring used to represent sets containing a finite number of elements in many cases (including the case considered in this paper) can be put in correspondence with Galois fields. Such fields are also increasingly used in information technology, in particular, in cryptography to improve the reliability of encoding [43; 44]. Computations in such fields can be realized directly in the form of electronic circuits, and research in this direction is also actively carried out at present [45; 46]. More precisely, computations of this kind can be realized on the basis of chips with tunable logic structure, which are also actively used nowadays [47; 48]. Reduced convolution operation is therefore of interest, also from the point of view of implementing the corresponding computations (including the implementation of convolutional neural networks) directly by specially programmed chips. An example in this respect is a very definite problem arising in the processing of satellite video information intended for remote sensing of the Earth. Communication channels have limited bandwidth, so it is important to exclude images in those situations when the Earth surface is covered by clouds [49; 50]. To solve this problem, it is expedient to use on-board means of information pre-processing, which can use convolutional neural networks. This particular example demonstrates the importance of the problem solved in this paper. A sufficiently coarse sampling scale can be used to exclude frames with cloud cover. Applying the convolution operation without reduction to the original sampling scale will not only correspond to the excess of accuracy, but will also require additional computational resources. It becomes especially important to overcome this difficulty when the corresponding operations are performed directly by the onboard computer. The transition to computations in Galois fields has one more aspect. Namely, in this case it is possible to operate with functions taking values on variables of multivalued logic [23; 24]. This aspect is important in those cases when, for example, we analyze the relationships between the preceding and following values of time series data, which are also used in different sciences. Provided the system is linear, it can often be assumed that such relationships are given by a convolution operation. The transition to the use of multivalued logics in the future allows us to raise the question of identifying true logical relationships in prediction, but the convolution operation itself should be reduced to logical