The Real Tropical Geometry of Neural Networks

TL;DR

This paper explores the tropical geometric structure of neural network decision boundaries, revealing new subdivisions of parameter space and their implications for understanding neural network behavior.

Contribution

It introduces a novel tropical geometric framework for neural networks, analyzing parameter space subdivisions and their relation to decision boundary combinatorics.

Findings

Parameter space subdivided into semialgebraic sets with fixed decision boundary types

Decision boundary sublevel sets can be disconnected

Classification fan described via normal fan of activation polytope

Abstract

We consider a binary classifier defined as the sign of a tropical rational function, that is, as the difference of two convex piecewise linear functions. The parameter space of ReLU neural networks is contained as a semialgebraic set inside the parameter space of tropical rational functions. We initiate the study of two different subdivisions of this parameter space: a subdivision into semialgebraic sets, on which the combinatorial type of the decision boundary is fixed, and a subdivision into a polyhedral fan, capturing the combinatorics of the partitions of the dataset. The sublevel sets of the 0/1-loss function arise as subfans of this classification fan, and we show that the level-sets are not necessarily connected. We describe the classification fan i) geometrically, as normal fan of the activation polytope, and ii) combinatorially through a list of properties of associated bipartite graphs, in analogy to covector axioms of oriented matroids and tropical oriented matroids. Our findings extend and refine the connection between neural networks and tropical geometry by observing structures established in real tropical geometry, such as positive tropicalizations of hypersurfaces and tropical semialgebraic sets.