# ReLU Neural Networks, Polyhedral Decompositions, and Persistent Homolog

arXiv: https://arxiv.org/abs/2306.17418 · HTML: https://mlsanity.com/paper/2306.17418/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 25 references that are in the map.
- First posted on arXiv 30 Jun 2023.

## Builds on (most cited first, 16 of 16)

- [On the Number of Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1402.1869/) (2014)
- [On the Expressive Power of Deep Neural Networks](https://mlsanity.com/paper/1606.05336/) (2016)
- [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)
- [PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures](https://mlsanity.com/paper/1904.09378/) (2019)
- [Neural Persistence: A Complexity Measure for Deep Neural Networks Using Algebraic Topology](https://mlsanity.com/paper/1812.09764/) (2018)
- [Intrinsic Dimension, Persistent Homology and Generalization in Neural Networks](https://mlsanity.com/paper/2111.13171/) (2021)
- [Reachable Polyhedral Marching (RPM): A Safety Verification Algorithm for Robotic Systems with Deep Neural Network Components](https://mlsanity.com/paper/2011.11609/) (2020)
- [Algorithmic Determination of the Combinatorial Structure of the Linear Regions of ReLU Neural Networks](https://mlsanity.com/paper/2207.07696/) (2022)
- [Reachable Set Estimation and Safety Verification for Piecewise Linear Systems with Neural Network Controllers](https://mlsanity.com/paper/1802.06981/) (2018)
- [Traversing the Local Polytopes of ReLU Neural Networks: A Unified Approach for Network Verification](https://mlsanity.com/paper/2111.08922/) (2021)
- [Reachability Analysis for Feed-Forward Neural Networks using Face Lattices](https://mlsanity.com/paper/2003.01226/) (2020)
- [Locally Linear Attributes of ReLU Neural Networks](https://mlsanity.com/paper/2012.01940/) (2020)
- [Local and global topological complexity measures OF ReLU neural network functions](https://mlsanity.com/paper/2204.06062/) (2022)
- [Hamming Similarity and Graph Laplacians for Class Partitioning and Adversarial Image Detection](https://mlsanity.com/paper/2305.01808/) (2023)

## Built on by (most cited first, 10 of 10)

- [Defining Neural Network Architecture through Polytope Structures of Dataset](https://mlsanity.com/paper/2402.02407/) (2024)
- [Topological Signatures of ReLU Neural Network Activation Patterns](https://mlsanity.com/paper/2510.12700/) (2025)
- [Piecewise Polynomial Regression of Tame Functions via Integer Programming](https://mlsanity.com/paper/2311.13544/) (2023)
- [A rank decomposition for the topological classification of neural representations](https://mlsanity.com/paper/2404.19710/) (2024)
- [A Computational Tropical Geometry Framework for Neural Networks](https://mlsanity.com/paper/2405.20174/) (2024)
- [A Quotient Homology Theory of Representation in Neural Networks](https://mlsanity.com/paper/2502.01360/) (2025)
- [Characterizing the Discrete Geometry of ReLU Networks](https://mlsanity.com/paper/2606.07728/) (2026)
- [Persistent Topological Structures and Cohomological Flows as a Mathematical Framework for Brain-Inspired Representation Learning](https://mlsanity.com/paper/2512.08241/) (2025)
- [Causal Explanations from the Geometric Properties of ReLU Neural Networks](https://mlsanity.com/paper/2605.10396/) (2026)
- [Message Tuning Outshines Graph Prompt Tuning: A Prismatic Space Perspective](https://mlsanity.com/paper/2606.03290/) (2026)

## Abstract (arXiv)

A ReLU neural network leads to a finite polyhedral decomposition of input space and a corresponding finite dual graph. We show that while this dual graph is a coarse quantization of input space, it is sufficiently robust that it can be combined with persistent homology to detect homological signals of manifolds in the input space from samples. This property holds for a variety of networks trained for a wide range of purposes that have nothing to do with this topological application. We found this feature to be surprising and interesting; we hope it will also be useful.