# On transversality of bent hyperplane arrangements and the topological expressiveness of ReLU neural networks

arXiv: https://arxiv.org/abs/2008.09052 · HTML: https://mlsanity.com/paper/2008.09052/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Aug 2020.

## Built on by (most cited first, 30 of 35)

- [When Deep Learning Meets Polyhedral Theory: A Survey](https://mlsanity.com/paper/2305.00241/) (2023)
- [Hidden symmetries of ReLU networks](https://mlsanity.com/paper/2306.06179/) (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)
- [Functional dimension of feedforward ReLU neural networks](https://mlsanity.com/paper/2209.04036/) (2022)
- [Polyhedral Complex Extraction from ReLU Networks using Edge Subdivision](https://mlsanity.com/paper/2306.07212/) (2023)
- [Topological Expressivity of ReLU Neural Networks](https://mlsanity.com/paper/2310.11130/) (2023)
- [On the Local Complexity of Linear Regions in Deep ReLU Networks](https://mlsanity.com/paper/2412.18283/) (2024)
- [Local and global topological complexity measures OF ReLU neural network functions](https://mlsanity.com/paper/2204.06062/) (2022)
- [Labeled sample compression schemes for complexes of oriented matroids](https://mlsanity.com/paper/2110.15168/) (2021)
- [Convexity in ReLU Neural Networks: beyond ICNNs?](https://mlsanity.com/paper/2501.03017/) (2025)
- [Most ReLU Networks Admit Identifiable Parameters](https://mlsanity.com/paper/2605.03601/) (2026)
- [Fast Jacobian-Vector Product for Deep Networks](https://mlsanity.com/paper/2104.00219/) (2021)
- [Time to Spike? Understanding the Representational Power of Spiking Neural Networks in Discrete Time](https://mlsanity.com/paper/2505.18023/) (2025)
- [The Symmetries of Three-Layer ReLU Networks](https://mlsanity.com/paper/2605.18319/) (2026)
- [Defining Neural Network Architecture through Polytope Structures of Dataset](https://mlsanity.com/paper/2402.02407/) (2024)
- [Complexity of Injectivity and Verification of ReLU Neural Networks](https://mlsanity.com/paper/2405.19805/) (2024)
- [A Theoretical Study of Neural Network Expressive Power via Manifold Topology](https://mlsanity.com/paper/2410.16542/) (2024)
- [Toric geometry of ReLU neural networks](https://mlsanity.com/paper/2509.05894/) (2025)
- [Polyhedral Complex Derivation from Piecewise Trilinear Networks](https://mlsanity.com/paper/2402.10403/) (2024)
- [A rank decomposition for the topological classification of neural representations](https://mlsanity.com/paper/2404.19710/) (2024)
- [On Functional Dimension and Persistent Pseudodimension](https://mlsanity.com/paper/2410.17191/) (2024)
- [A Quotient Homology Theory of Representation in Neural Networks](https://mlsanity.com/paper/2502.01360/) (2025)
- [Latent Space Topology Evolution in Multilayer Perceptrons](https://mlsanity.com/paper/2506.01569/) (2025)
- [AffineLens: Capturing the Continuous Piecewise Affine Functions of Neural Networks](https://mlsanity.com/paper/2605.06218/) (2026)
- [Region Seeding via Pre-Activation Regularization: A Geometric View of Piecewise Affine Neural Networks](https://mlsanity.com/paper/2605.06300/) (2026)
- [Characterizing the Discrete Geometry of ReLU Networks](https://mlsanity.com/paper/2606.07728/) (2026)
- [Topological Learning in Multi-Class Data Sets](https://mlsanity.com/paper/2301.09734/) (2023)
- [Combinatorial Regularity for Relatively Perfect Discrete Morse Gradient Vector Fields of ReLU Neural Networks](https://mlsanity.com/paper/2412.18005/) (2024)
- [Positivity sets of hinge functions](https://mlsanity.com/paper/2503.13512/) (2025)

## Abstract (arXiv)

Let F:R^n -> R be a feedforward ReLU neural network. It is well-known that for any choice of parameters, F is continuous and piecewise (affine) linear. We lay some foundations for a systematic investigation of how the architecture of F impacts the geometry and topology of its possible decision regions for binary classification tasks. Following the classical progression for smooth functions in differential topology, we first define the notion of a generic, transversal ReLU neural network and show that almost all ReLU networks are generic and transversal. We then define a partially-oriented linear 1-complex in the domain of F and identify properties of this complex that yield an obstruction to the existence of bounded connected components of a decision region. We use this obstruction to prove that a decision region of a generic, transversal ReLU network F: R^n -> R with a single hidden layer of dimension (n + 1) can have no more than one bounded connected component.