  We discover that deep ReLU neural network classifiers can see a
low-dimensional Riemannian manifold structure on data. Such structure comes via
the local data matrix, a variation of the Fisher information matrix, where the
role of the model parameters is taken by the data variables. We obtain a
foliation of the data domain and we show that the dataset on which the model is
trained lies on a leaf, the data leaf, whose dimension is bounded by the number
of classification labels. We validate our results with some experiments with
the MNIST dataset: paths on the data leaf connect valid images, while other
leaves cover noisy images.
