The Computational Complexity of Counting Linear Regions in ReLU Neural Networks

TL;DR

This paper analyzes the computational difficulty of counting the linear regions in ReLU neural networks, showing it is generally intractable but with some cases solvable in polynomial space.

Contribution

It systematically compares definitions of linear regions and proves NP- and #P-hardness results for counting them in neural networks.

Findings

Counting linear regions is NP- and #P-hard.

Hardness of approximation increases with network depth.

Some definitions allow polynomial space algorithms.

Abstract

An established measure of the expressive power of a given ReLU neural network is the number of linear regions into which it partitions the input space. There exist many different, non-equivalent definitions of what a linear region actually is. We systematically assess which papers use which definitions and discuss how they relate to each other. We then analyze the computational complexity of counting the number of such regions for the various definitions. Generally, this turns out to be an intractable problem. We prove NP- and #P-hardness results already for networks with one hidden layer and strong hardness of approximation results for two or more hidden layers. Finally, on the algorithmic side, we demonstrate that counting linear regions can at least be achieved in polynomial space for some common definitions.