Tropical Expressivity of Neural Networks

Abstract

We propose an algebraic geometric framework to study the expressivity of linear activation neural networks. A particular quantity of neural networks that has been actively studied is the number of linear regions, which gives a quantification of the information capacity of the architecture. To study and evaluate information capacity and expressivity, we work in the setting of tropical geometry - a combinatorial and polyhedral variant of algebraic geometry - where there are known connections between tropical rational maps and feedforward neural networks. Our work builds on and expands this connection to capitalize on the rich theory of tropical geometry to characterize and study various architectural aspects of neural networks. Our contributions are threefold: we provide a novel tropical geometric approach to selecting sampling domains among linear regions; an algebraic result allowing for a guided restriction of the sampling domain for network architectures with symmetries; and a new open source OSCAR library to analyze neural networks symbolically using their tropical representations, where we present a new algorithm that computes the exact number of their linear regions. We provide a comprehensive set of proof-of-concept numerical experiments demonstrating the breadth of neural network architectures to which tropical geometric theory can be applied to reveal insights on expressivity characteristics of a network. Our work provides the foundations for the adaptation of both theory and existing software from computational tropical geometry and symbolic computation to neural networks and deep learning