Dimensionality compression and expansion in Deep Neural Networks

TL;DR

This paper investigates how deep neural networks learn low-dimensional manifolds from high-dimensional data, revealing a two-phase process of expansion and compression that relates to their generalization ability.

Contribution

It introduces a novel analysis of neural network training dynamics, showing how intrinsic dimensionality evolves and influences generalization, supported by new dimensionality estimation techniques.

Findings

Neural networks undergo a two-phase process: expansion then compression of data representations.

Noise from stochastic gradient descent acts as a regularizer balancing representation dimensionality.

Low-dimensional representations are linked to better generalization performance.

Abstract

Datasets such as images, text, or movies are embedded in high-dimensional spaces. However, in important cases such as images of objects, the statistical structure in the data constrains samples to a manifold of dramatically lower dimensionality. Learning to identify and extract task-relevant variables from this embedded manifold is crucial when dealing with high-dimensional problems. We find that neural networks are often very effective at solving this task and investigate why. To this end, we apply state-of-the-art techniques for intrinsic dimensionality estimation to show that neural networks learn low-dimensional manifolds in two phases: first, dimensionality expansion driven by feature generation in initial layers, and second, dimensionality compression driven by the selection of task-relevant features in later layers. We model noise generated by Stochastic Gradient Descent and show how this noise balances the dimensionality of neural representations by inducing an effective regularization term in the loss. We highlight the important relationship between low-dimensional compressed representations and generalization properties of the network. Our work contributes by shedding light on the success of deep neural networks in disentangling data in high-dimensional space while achieving good generalization. Furthermore, it invites new learning strategies focused on optimizing measurable geometric properties of learned representations, beginning with their intrinsic dimensionality.