Hyperplane Arrangements and Fixed Points in Iterated PWL Neural Networks

TL;DR

This paper uses hyperplane arrangements to analyze fixed points in multi-layer PWL neural networks, providing bounds on their number and stability, revealing exponential growth with layers and sharper bounds for specific activations.

Contribution

It introduces a hyperplane arrangement framework to bound fixed points in PWL neural networks, including new bounds for stable fixed points in one-hidden-layer networks.

Findings

Upper bound on fixed points for multi-layer PWL networks

Exponential growth of fixed points with layers demonstrated

Sharper bounds for stable fixed points with hard tanh activation

Abstract

We leverage the framework of hyperplane arrangements to analyze potential regions of (stable) fixed points. We provide an upper bound on the number of fixed points for multi-layer neural networks equipped with piecewise linear (PWL) activation functions with arbitrary many linear pieces. The theoretical optimality of the exponential growth in the number of layers of the latter bound is shown. Specifically, we also derive a sharper upper bound on the number of stable fixed points for one-hidden-layer networks with hard tanh activation.