Hyperplane Arrangements and Fixed Points in Iterated PWL Neural Networks

Hans-Peter Beise

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arXiv · 2405.09878

Hyperplane Arrangements and Fixed Points in Iterated PWL Neural Networks

Hans-Peter Beise

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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.

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Hans-Peter Beise. 2024-05-16. Hyperplane Arrangements and Fixed Points in Iterated PWL Neural Networks. https://arxiv.org/abs/2405.09878

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