Understanding Deep Learning via Notions of Rank

TL;DR

This paper explores how notions of rank can provide a theoretical framework for understanding deep learning, focusing on generalization, expressiveness, and the implicit low-rank regularization induced by gradient-based training.

Contribution

It introduces a rank-based perspective for analyzing neural networks, linking neural network behavior to tensor factorizations and demonstrating implications for regularization and data preprocessing.

Findings

Gradient-based training induces implicit low-rank regularization.

Low-rank phenomena help explain generalization on natural data.

Rank characterizations apply to graph neural networks and quantum entanglement.

Abstract

Despite the extreme popularity of deep learning in science and industry, its formal understanding is limited. This thesis puts forth notions of rank as key for developing a theory of deep learning, focusing on the fundamental aspects of generalization and expressiveness. In particular, we establish that gradient-based training can induce an implicit regularization towards low rank for several neural network architectures, and demonstrate empirically that this phenomenon may facilitate an explanation of generalization over natural data (e.g., audio, images, and text). Then, we characterize the ability of graph neural networks to model interactions via a notion of rank, which is commonly used for quantifying entanglement in quantum physics. A central tool underlying these results is a connection between neural networks and tensor factorizations. Practical implications of our theory for designing explicit regularization schemes and data preprocessing algorithms are presented.