Model-centric Data Manifold: the Data Through the Eyes of the Model

TL;DR

This paper reveals that deep ReLU classifiers perceive a low-dimensional manifold structure in data, where the data lies on a leaf of the manifold bounded by the number of classes, validated through experiments on MNIST.

Contribution

It introduces the concept of a data manifold seen through the model's perspective, linking data geometry to neural network parameters via a local data matrix.

Findings

Data lies on a low-dimensional leaf of the manifold.

Paths on the leaf connect valid images.

Other leaves correspond to noisy images.

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.