On the Number of Regions of Piecewise Linear Neural Networks

TL;DR

This paper provides bounds and estimates on the number of linear regions in piecewise linear neural networks, highlighting the exponential impact of depth and the combined influence of network architecture and activation functions on expressiveness.

Contribution

It generalizes bounds on linear regions to arbitrary CPWL activations and introduces a stochastic framework to estimate average regions, emphasizing depth's exponential role.

Findings

Depth exponentially increases the number of regions.

Bounds depend on network depth, width, and activation complexity.

Average number of regions scales with depth, width, and activation complexity.

Abstract

Many feedforward neural networks (NNs) generate continuous and piecewise-linear (CPWL) mappings. Specifically, they partition the input domain into regions on which the mapping is affine. The number of these so-called linear regions offers a natural metric to characterize the expressiveness of CPWL NNs. The precise determination of this quantity is often out of reach in practice, and bounds have been proposed for specific architectures, including for ReLU and Maxout NNs. In this work, we generalize these bounds to NNs with arbitrary and possibly multivariate CPWL activation functions. We first provide upper and lower bounds on the maximal number of linear regions of a CPWL NN given its depth, width, and the number of linear regions of its activation functions. Our results rely on the combinatorial structure of convex partitions and confirm the distinctive role of depth which, on its own, is able to exponentially increase the number of regions. We then introduce a complementary stochastic framework to estimate the average number of linear regions produced by a CPWL NN. Under reasonable assumptions, the expected density of linear regions along any 1D path is bounded by the product of depth, width, and a measure of activation complexity (up to a scaling factor). This yields an identical role to the three sources of expressiveness: no exponential growth with depth is observed anymore.