Tropical Expressivity of Neural Networks

We propose an algebraic geometric framework to study the expressivity of linear activation neural networks. A particular quantity that has been actively studied in the field of deep learning is the number of linear regions, which gives an estimate 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 an open source library to analyze neural networks as tropical Puiseux rational maps. 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 deep learning.