Manifold Learning via Foliations, and Knowledge Transfer

Eliot Tron, Rita Fioresi

Keywords: Foliation Theory, Riemaniann Geometry, Fisher Information, Manifold Learning, Knowledge Transfer

TL;DR: This study explores deep ReLU neural networks and manifold learning, uncovering a foliation structure that correlates with real data in high-dimensional spaces and shows potential for knowledge transfer.

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 deep 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.

Primary Area: unsupervised, self-supervised, semi-supervised, and supervised representation learning

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Submission Number: 8124