ReLU Neural Networks, Polyhedral Decompositions, and Persistent Homolog
This provides a novel method for topological data analysis that leverages existing neural networks, potentially benefiting researchers in machine learning and computational topology.
The paper tackles the problem of detecting topological features in data using ReLU neural networks, showing that the networks' polyhedral decompositions can be combined with persistent homology to identify homological signals of manifolds from samples, even when the networks are trained for unrelated tasks.
A ReLU neural network leads to a finite polyhedral decomposition of input space and a corresponding finite dual graph. We show that while this dual graph is a coarse quantization of input space, it is sufficiently robust that it can be combined with persistent homology to detect homological signals of manifolds in the input space from samples. This property holds for a variety of networks trained for a wide range of purposes that have nothing to do with this topological application. We found this feature to be surprising and interesting; we hope it will also be useful.