Complexity for deep neural networks and other characteristics of deep feature representations

TL;DR

This paper introduces measures of complexity and effective dimension for deep neural networks, revealing their dynamics during training and potential links to dataset structure, with implications for neuroscience and physics.

Contribution

It defines new complexity and dimension measures for neural networks and explores their scaling and interpretability during training.

Findings

Power law scaling during training

Complexity relates to dataset structure

Measures applicable to biological neural systems

Abstract

We define a notion of complexity, which quantifies the nonlinearity of the computation of a neural network, as well as a complementary measure of the effective dimension of feature representations. We investigate these observables both for trained networks for various datasets as well as explore their dynamics during training, uncovering in particular power law scaling. These observables can be understood in a dual way as uncovering hidden internal structure of the datasets themselves as a function of scale or depth. The entropic character of the proposed notion of complexity should allow to transfer modes of analysis from neuroscience and statistical physics to the domain of artificial neural networks. The introduced observables can be applied without any change to the analysis of biological neuronal systems.