  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.
