# Toric geometry of ReLU neural networks

arXiv: https://arxiv.org/abs/2509.05894 · HTML: https://mlsanity.com/paper/2509.05894/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 13 references that are in the map.
- First posted on arXiv 7 Sep 2025.

## Builds on (most cited first, 9 of 9)

- [Understanding Deep Neural Networks with Rectified Linear Units](https://mlsanity.com/paper/1611.01491/) (2016)
- [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)
- [Lower Bounds on the Depth of Integral ReLU Neural Networks via Lattice Polytopes](https://mlsanity.com/paper/2302.12553/) (2023)
- [On transversality of bent hyperplane arrangements and the topological expressiveness of ReLU neural networks](https://mlsanity.com/paper/2008.09052/) (2020)
- [Geometry of Polynomial Neural Networks](https://mlsanity.com/paper/2402.00949/) (2024)
- [Algebra Unveils Deep Learning -- An Invitation to Neuroalgebraic Geometry](https://mlsanity.com/paper/2501.18915/) (2025)
- [Activation degree thresholds and expressiveness of polynomial neural networks](https://mlsanity.com/paper/2408.04569/) (2024)
- [Better Neural Network Expressivity: Subdividing the Simplex](https://mlsanity.com/paper/2505.14338/) (2025)

## Built on by (most cited first, 2 of 2)

- [A Complete Symmetry Classification of Shallow ReLU Networks](https://mlsanity.com/paper/2604.14037/) (2026)
- [Message Tuning Outshines Graph Prompt Tuning: A Prismatic Space Perspective](https://mlsanity.com/paper/2606.03290/) (2026)

## Abstract (arXiv)

Given a continuous finitely piecewise linear function f:ℝn0 → ℝ and a fixed architecture (n0,ldots,nk;1) of feedforward ReLU neural networks, the exact function realization problem is to determine when some network with the given architecture realizes f. To develop a systematic way to answer these questions, we establish a connection between toric geometry and ReLU neural networks. This approach enables us to utilize numerous structures and tools from algebraic geometry to study ReLU neural networks. Starting with an unbiased ReLU neural network with rational weights, we define the ReLU fan, the ReLU toric variety, and the ReLU Cartier divisor associated with the network. This work also reveals the connection between the tropical geometry and the toric geometry of ReLU neural networks. As an application of the toric geometry framework, we prove a necessary and sufficient criterion of functions realizable by unbiased shallow ReLU neural networks by computing intersection numbers of the ReLU Cartier divisor and torus-invariant curves.