Geometry of Singular Foliations and Learning Manifolds in ReLU Networks via the Data Information Matrix

TL;DR

This paper introduces a geometric framework using the Data Information Matrix to analyze the distribution of data in high-dimensional spaces via ReLU networks, revealing foliation structures and aiding in dataset comparison.

Contribution

It proposes a novel geometric perspective on data distribution in neural networks using the DIM, uncovering foliation structures and enabling dataset similarity measurement.

Findings

Singular foliation points are measure zero

Local regular foliation exists almost everywhere

Data correlates with foliation leaves

Abstract

Understanding how real data is distributed in high dimensional spaces is the key to many tasks in machine learning. We want to provide a natural geometric structure on the space of data employing a ReLU neural network trained as a classifier. Through the Data Information Matrix (DIM), a variation of the Fisher information matrix, the model will discern a singular foliation structure on the space of data. We show that the singular points of such foliation are contained in a measure zero set, and that a local regular foliation exists almost everywhere. Experiments show that the data is correlated with leaves of such foliation. Moreover we show the potential of our approach for knowledge transfer by analyzing the spectrum of the DIM to measure distances between datasets.