  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.
