  We present results on the number of linear regions of the functions that can
be represented by artificial feedforward neural networks with maxout units. A
rank-k maxout unit is a function computing the maximum of $k$ linear functions.
For networks with a single layer of maxout units, the linear regions correspond
to the upper vertices of a Minkowski sum of polytopes. We obtain face counting
formulas in terms of the intersection posets of tropical hypersurfaces or the
number of upper faces of partial Minkowski sums, along with explicit sharp
upper bounds for the number of regions for any input dimension, any number of
units, and any ranks, in the cases with and without biases. Based on these
results we also obtain asymptotically sharp upper bounds for networks with
multiple layers.
