Sharp Bounds for the Number of Regions of Maxout Networks and Vertices of Minkowski Sums
Guido Montúfar, Yue Ying Ren, Leon Zhang · SIAM Journal on Applied Algebra and Geometry · 2022
Abstract. 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-[Formula: see text] maxout unit is a function computing the maximum of [Formula: see text] 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 cases with and without biases. Based on these results we also obtain asymptotically sharp upper bounds for networks with multiple layers.