Model-centric data manifold: the data through the eyes of the model

Luca Grementieri, Rita Fioresi

Keywords: Deep Learning, Information Geometry, Data Manifold, Fisher matrix

Abstract: 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.

One-sentence Summary: We discover that deep ReLU neural network classifiers can see a low-dimensional Riemannian manifold structure on data.

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