Hyperplane Arrangements and Fixed Points in Iterated PWL Neural Networks
This work addresses theoretical limitations in understanding neural network dynamics for researchers in machine learning and mathematics, but it is incremental as it builds on existing hyperplane arrangement frameworks.
The paper tackles the problem of analyzing fixed points in multi-layer neural networks with piecewise linear activations, providing an exponential upper bound on the number of fixed points and a sharper bound for stable fixed points in one-hidden-layer networks with hard tanh activation.
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.