The Real Tropical Geometry of Neural Networks
This work provides foundational insights into neural network theory for researchers in machine learning and mathematics, though it is incremental in refining existing connections to tropical geometry.
The paper tackles the problem of understanding the parameter space of ReLU neural networks by modeling it within tropical geometry, showing that the 0/1-loss function's level-sets are not necessarily connected and describing the classification fan geometrically and combinatorially.
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.