# Maxout Polytopes

arXiv: https://arxiv.org/abs/2509.21286 · HTML: https://mlsanity.com/paper/2509.21286/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 36 references that are in the map.
- First posted on arXiv 25 Sep 2025.

## Builds on (most cited first, 20 of 20)

- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [On the Number of Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1402.1869/) (2014)
- [Understanding Deep Neural Networks with Rectified Linear Units](https://mlsanity.com/paper/1611.01491/) (2016)
- [Bounding and Counting Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1711.02114/) (2017)
- [Deep ReLU Networks Have Surprisingly Few Activation Patterns](https://mlsanity.com/paper/1906.00904/) (2019)
- [Tropical Geometry of Deep Neural Networks](https://mlsanity.com/paper/1805.07091/) (2018)
- [Towards Lower Bounds on the Depth of ReLU Neural Networks](https://mlsanity.com/paper/2105.14835/) (2021)
- [When Deep Learning Meets Polyhedral Theory: A Survey](https://mlsanity.com/paper/2305.00241/) (2023)
- [Sharp bounds for the number of regions of maxout networks and vertices of Minkowski sums](https://mlsanity.com/paper/2104.08135/) (2021)
- [Lower Bounds on the Depth of Integral ReLU Neural Networks via Lattice Polytopes](https://mlsanity.com/paper/2302.12553/) (2023)
- [The Real Tropical Geometry of Neural Networks](https://mlsanity.com/paper/2403.11871/) (2024)
- [Algorithmic Determination of the Combinatorial Structure of the Linear Regions of ReLU Neural Networks](https://mlsanity.com/paper/2207.07696/) (2022)
- [On the Expressiveness of Rational ReLU Neural Networks With Bounded Depth](https://mlsanity.com/paper/2502.06283/) (2025)
- [Better Neural Network Expressivity: Subdividing the Simplex](https://mlsanity.com/paper/2505.14338/) (2025)
- [On Minimal Depth in Neural Networks](https://mlsanity.com/paper/2402.15315/) (2024)
- [Depth-Bounds for Neural Networks via the Braid Arrangement](https://mlsanity.com/paper/2502.09324/) (2025)
- [On the Depth of Monotone ReLU Neural Networks and ICNNs](https://mlsanity.com/paper/2505.06169/) (2025)
- [Neural Networks and (Virtual) Extended Formulations](https://mlsanity.com/paper/2411.03006/) (2024)
- [A Combinatorial Perspective on the Optimization of Shallow ReLU Networks](https://mlsanity.com/paper/2210.00176/) (2022)
- [Revisiting Tropical Polynomial Division: Theory, Algorithms and Application to Neural Networks](https://mlsanity.com/paper/2306.15157/) (2023)

## Abstract (arXiv)

Maxout polytopes are defined by feedforward neural networks with maxout activation function and non-negative weights after the first layer. We characterize the parameter spaces and extremal f-vectors of maxout polytopes for shallow networks, and we study the separating hypersurfaces which arise when a layer is added to the network. We also show that maxout polytopes are cubical for generic networks without bottlenecks.