A Computational Tropical Geometry Framework for Neural Networks

We propose a computational tropical geometry framework for the symbolic analysis of neural networks with tropical activations. The number of linear regions of a neural network has been actively studied as a measure of the expressivity of a given architecture. To study these, 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. We expand this connection by developing concrete computational tools for studying the linear regions of neural networks. We present an algorithm, together with a proof of correctness, which computes the linear regions of a neural network as explicit unions of polyhedra. We further relate the computation of the number of linear regions of a tropical expression to the number of monomials that appear in it, and show how tropical expressions can often be pruned to remove redundant monomials. We introduce the Hoffman constant of a neural network's tropical expression, a geometric quantity that controls the distance from any point in the input space to the farthest linear region. We provide the open source Julia library TropicalNN.jl, which is built on top of the OSCAR computer algebra system and implements the algorithms mentioned above to analyze neural networks symbolically using their tropical representations. We present a set of proof-of-concept computational examples to demonstrate how our tropical geometric theory can be applied to reveal insights on the expressivity of a network architecture.