Sharp bounds for the number of regions of maxout networks and vertices of Minkowski sums
This provides foundational theoretical insights into the expressivity of maxout networks, which is important for researchers in machine learning theory and neural network design.
The paper tackles the problem of counting the number of linear regions in feedforward neural networks with maxout units, deriving explicit sharp upper bounds for any input dimension, number of units, and ranks, with and without biases, and extends these to asymptotically sharp bounds for multi-layer networks.
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.