# Hybrid Zonotopes Exactly Represent ReLU Neural Networks

arXiv: https://arxiv.org/abs/2304.02755 · HTML: https://mlsanity.com/paper/2304.02755/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 22 references that are in the map.
- First posted on arXiv 5 Apr 2023.

## Builds on (most cited first, 14 of 14)

- [Provable defenses against adversarial examples via the convex outer adversarial polytope](https://mlsanity.com/paper/1711.00851/) (2017)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Towards Fast Computation of Certified Robustness for ReLU Networks](https://mlsanity.com/paper/1804.09699/) (2018)
- [Exactly Computing the Local Lipschitz Constant of ReLU Networks](https://mlsanity.com/paper/2003.01219/) (2020)
- [Efficiently Computing Local Lipschitz Constants of Neural Networks via Bound Propagation](https://mlsanity.com/paper/2210.07394/) (2022)
- [Reach-SDP: Reachability Analysis of Closed-Loop Systems with Neural Network Controllers via Semidefinite Programming](https://mlsanity.com/paper/2004.07876/) (2020)
- [Reachability Analysis of Neural Feedback Loops](https://mlsanity.com/paper/2108.04140/) (2021)
- [Certifying Incremental Quadratic Constraints for Neural Networks via Convex Optimization](https://mlsanity.com/paper/2012.05981/) (2020)
- [Reachable Set Estimation and Safety Verification for Piecewise Linear Systems with Neural Network Controllers](https://mlsanity.com/paper/1802.06981/) (2018)
- [Hybrid zonotopes: a new set representation for reachability analysis of mixed logical dynamical systems](https://mlsanity.com/paper/2106.14831/) (2021)
- [Safety Verification of Neural Feedback Systems Based on Constrained Zonotopes](https://mlsanity.com/paper/2204.00903/) (2022)
- [Constrained Feedforward Neural Network Training via Reachability Analysis](https://mlsanity.com/paper/2107.07696/) (2021)
- [Reachability Analysis and Safety Verification of Neural Feedback Systems via Hybrid Zonotopes](https://mlsanity.com/paper/2210.03244/) (2022)
- [Utilizing Dependencies to Obtain Subsets of Reachable Sets](https://mlsanity.com/paper/1910.08354/) (2019)

## Built on by (most cited first, 4 of 4)

- [Neural Network Verification with PyRAT](https://mlsanity.com/paper/2410.23903/) (2024)
- [Out of the Shadows: Exploring a Latent Space for Neural Network Verification](https://mlsanity.com/paper/2505.17854/) (2025)
- [Provably-Safe Neural Network Training Using Hybrid Zonotope Reachability Analysis](https://mlsanity.com/paper/2501.13023/) (2025)
- [Language Models That Walk the Talk: A Framework for Formal Fairness Certificates](https://mlsanity.com/paper/2505.12767/) (2025)

## Abstract (arXiv)

We show that hybrid zonotopes offer an equivalent representation of feed-forward fully connected neural networks with ReLU activation functions. Our approach demonstrates that the complexity of binary variables is equal to the total number of neurons in the network and hence grows linearly in the size of the network. We demonstrate the utility of the hybrid zonotope formulation through three case studies including nonlinear function approximation, MPC closed-loop reachability and verification, and robustness of classification on the MNIST dataset.