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Researchers are actively trying to gain better insights into therepresentational properties of convolutional neural networks for guiding betternetwork designs and for interpreting a network's computational nature. Gainingsuch insights can be an arduous task due to the number of parameters in anetwork and the complexity of a network's architecture. Current approaches ofneural network interpretation include Bayesian probabilistic interpretationsand information theoretic interpretations. In this study, we take a differentapproach to studying convolutional neural networks by proposing an abstractalgebraic interpretation using finite transformation semigroup theory.Specifically, convolutional layers are broken up and mapped to a finite space.The state space of the proposed finite transformation semigroup is then definedas a single element within the convolutional layer, with the acting elementsdefined by surrounding state elements combined with convolution kernelelements. Generators of the finite transformation semigroup are defined tocomplete the interpretation. We leverage this approach to analyze the basicproperties of the resulting finite transformation semigroup to gain insights onthe representational properties of convolutional neural networks, includinginsights into quantized network representation. Such a finite transformationsemigroup interpretation can also enable better understanding outside of theconfines of fixed lattice data structures, thus useful for handling data thatlie on irregular lattices. Furthermore, the proposed abstract algebraicinterpretation is shown to be viable for interpreting convolutional operationswithin a variety of convolutional neural network architectures.

Seeing Convolution Through the Eyes of Finite Transformation Semigroup Theory: An Abstract Algebraic Interpretation of Convolutional Neural Networks