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We algorithmically determine the regions and facets of all dimensions of thecanonical polyhedral complex, the universal object into which a ReLU networkdecomposes its input space. We show that the locations of the vertices of thecanonical polyhedral complex along with their signs with respect to layer mapsdetermine the full facet structure across all dimensions. We present analgorithm which calculates this full combinatorial structure, making use of ourtheorems that the dual complex to the canonical polyhedral complex is cubicaland it possesses a multiplication compatible with its facet structure. Theresulting algorithm is numerically stable, polynomial time in the number ofintermediate neurons, and obtains accurate information across all dimensions.This permits us to obtain, for example, the true topology of the decisionboundaries of networks with low-dimensional inputs. We run empirics on suchnetworks at initialization, finding that width alone does not increase observedtopology, but width in the presence of depth does. Source code for ouralgorithms is accessible online at https://github.com/mmasden/canonicalpoly.

Algorithmic Determination of the Combinatorial Structure of the Linear Regions of ReLU Neural Networks