  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.
