MiS Preprint Repository
We have decided to discontinue the publication of preprints on our preprint server end of 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.
Sharp bounds for the number of regions of maxout networks and vertices of Minkowski sums
Guido Montúfar, Yue Ren and Leon ZhangAbstract
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.
- Received:
- 16.04.21
- Published:
- 16.04.21
- MSC Codes:
- 68T07, 52B05, 14T15, 06A07
- Keywords:
- linear regions of neural networks, upper bound theorem for Minkowski sums, hyperplane arrangement, tropical hypersurface arrangement, Newton polytope