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

Abstract: Researchers are actively trying to gain better insights into the representational properties of convolutional neural networks for guiding better network designs and for interpreting a network's computational nature. Gaining such insights can be an arduous task due to the number of parameters in a network and the complexity of a network's architecture. Current approaches of neural network interpretation include Bayesian probabilistic interpretations and information theoretic interpretations. In this study, we take a different approach to studying convolutional neural networks by proposing an abstract algebraic 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 defined as a single element within the convolutional layer, with the acting elements defined by surrounding state elements combined with convolution kernel elements. Generators of the finite transformation semigroup are defined to complete the interpretation. We leverage this approach to analyze the basic properties of the resulting finite transformation semigroup to gain insights on the representational properties of convolutional neural networks, including insights into quantized network representation. Such a finite transformation semigroup interpretation can also enable better understanding outside of the confines of fixed lattice data structures, thus useful for handling data that lie on irregular lattices. Furthermore, the proposed abstract algebraic interpretation is shown to be viable for interpreting convolutional operations within a variety of convolutional neural network architectures.