On the Number of Linear Regions of Convolutional Neural Networks

TL;DR

This paper investigates the expressivity of convolutional neural networks by analyzing the maximal and average number of linear regions they can form, revealing that deeper CNNs are more expressive than shallower ones and more efficient than fully-connected networks per parameter.

Contribution

The paper provides mathematical bounds on the number of linear regions in CNNs, advancing understanding of their expressive power compared to fully-connected networks.

Findings

Deeper CNNs have more linear regions than shallower ones.

CNNs are more expressive than fully-connected NNs per parameter.

Derived bounds for linear regions in multi-layer CNNs.

Abstract

One fundamental problem in deep learning is understanding the outstanding performance of deep Neural Networks (NNs) in practice. One explanation for the superiority of NNs is that they can realize a large class of complicated functions, i.e., they have powerful expressivity. The expressivity of a ReLU NN can be quantified by the maximal number of linear regions it can separate its input space into. In this paper, we provide several mathematical results needed for studying the linear regions of CNNs, and use them to derive the maximal and average numbers of linear regions for one-layer ReLU CNNs. Furthermore, we obtain upper and lower bounds for the number of linear regions of multi-layer ReLU CNNs. Our results suggest that deeper CNNs have more powerful expressivity than their shallow counterparts, while CNNs have more expressivity than fully-connected NNs per parameter.