Intrinsic dimension of data representations in deep neural networks

TL;DR

This paper investigates the intrinsic dimensionality of data representations in deep neural networks, revealing that trained networks produce low-dimensional, non-linear manifolds that correlate with classification performance.

Contribution

It introduces a method to measure the intrinsic dimensionality of neural representations and shows how this measure varies across layers and correlates with accuracy.

Findings

Intrinsic dimensionality is much smaller than layer size.

ID first increases then decreases in deeper layers.

Last layer ID predicts test accuracy.

Abstract

Deep neural networks progressively transform their inputs across multiple processing layers. What are the geometrical properties of the representations learned by these networks? Here we study the intrinsic dimensionality (ID) of data-representations, i.e. the minimal number of parameters needed to describe a representation. We find that, in a trained network, the ID is orders of magnitude smaller than the number of units in each layer. Across layers, the ID first increases and then progressively decreases in the final layers. Remarkably, the ID of the last hidden layer predicts classification accuracy on the test set. These results can neither be found by linear dimensionality estimates (e.g., with principal component analysis), nor in representations that had been artificially linearized. They are neither found in untrained networks, nor in networks that are trained on randomized labels. This suggests that neural networks that can generalize are those that transform the data into low-dimensional, but not necessarily flat manifolds.