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We present results on the number of linear regions of the functions that canbe represented by artificial feedforward neural networks with maxout units. Arank-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 correspondto the upper vertices of a Minkowski sum of polytopes. We obtain face countingformulas in terms of the intersection posets of tropical hypersurfaces or thenumber of upper faces of partial Minkowski sums, along with explicit sharpupper bounds for the number of regions for any input dimension, any number ofunits, and any ranks, in the cases with and without biases. Based on theseresults we also obtain asymptotically sharp upper bounds for networks withmultiple layers.

Sharp bounds for the number of regions of maxout networks and vertices of Minkowski sums