The Real Tropical Geometry of Neural Networks
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. In particular, we consider binary classifications through ReLU neural networks, whose parameter space 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.
Submission Length: Long submission (more than 12 pages of main content)
Assigned Action Editor: ~Jeremias_Sulam1
Submission Number: 2838
Loading