Model-Centric Data Manifold: The Data Through the Eyes of the Model

We show that deep ReLU neural network classifiers can see a low-dimensional Riemannian manifold structure on data. Such structure comes via the \sl 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 \sl 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.