# Computing Linear Regions in Neural Networks with Skip Connections

arXiv: https://arxiv.org/abs/2509.15441 · HTML: https://mlsanity.com/paper/2509.15441/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 15 references that are in the map.
- First posted on arXiv 18 Sep 2025.

## Builds on (most cited first, 6 of 6)

- [On the Number of Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1402.1869/) (2014)
- [Bounding and Counting Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1711.02114/) (2017)
- [Tropical Geometry of Deep Neural Networks](https://mlsanity.com/paper/1805.07091/) (2018)
- [Mad Max: Affine Spline Insights into Deep Learning](https://mlsanity.com/paper/1805.06576/) (2018)
- [Empirical Studies on the Properties of Linear Regions in Deep Neural Networks](https://mlsanity.com/paper/2001.01072/) (2020)
- [Training Dynamics of Deep Network Linear Regions](https://mlsanity.com/paper/2310.12977/) (2023)

## Built on by (most cited first, 1 of 1)

- [Revisiting the Shape Convention of Transformer Language Models](https://mlsanity.com/paper/2602.06471/) (2026)

## Abstract (arXiv)

Neural networks are important tools in machine learning. Representing piecewise linear activation functions with tropical arithmetic enables the application of tropical geometry. Algorithms are presented to compute regions where the neural networks are linear maps. Through computational experiments, we provide insights on the difficulty to train neural networks, in particular on the problems of overfitting and on the benefits of skip connections.