Complexity of Linear Regions in Deep Networks

TL;DR

This paper develops a mathematical framework to analyze the number of linear regions in piecewise linear neural networks, revealing that most networks have far fewer regions than the exponential maximum, both at initialization and after training.

Contribution

It introduces a method to count linear regions and measure boundary volume, showing the practical expressivity of networks is much lower than theoretical bounds.

Findings

Average number of regions grows linearly with neurons at initialization

Distance to boundary scales inversely with number of neurons

Number of regions remains far below exponential after training

Abstract

It is well-known that the expressivity of a neural network depends on its architecture, with deeper networks expressing more complex functions. In the case of networks that compute piecewise linear functions, such as those with ReLU activation, the number of distinct linear regions is a natural measure of expressivity. It is possible to construct networks with merely a single region, or for which the number of linear regions grows exponentially with depth; it is not clear where within this range most networks fall in practice, either before or after training. In this paper, we provide a mathematical framework to count the number of linear regions of a piecewise linear network and measure the volume of the boundaries between these regions. In particular, we prove that for networks at initialization, the average number of regions along any one-dimensional subspace grows linearly in the total number of neurons, far below the exponential upper bound. We also find that the average distance to the nearest region boundary at initialization scales like the inverse of the number of neurons. Our theory suggests that, even after training, the number of linear regions is far below exponential, an intuition that matches our empirical observations. We conclude that the practical expressivity of neural networks is likely far below that of the theoretical maximum, and that this gap can be quantified.