# Data-Driven Reachability Analysis from Noisy Data

arXiv: https://arxiv.org/abs/2105.07229 · HTML: https://mlsanity.com/paper/2105.07229/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 57 references that are in the map.
- First posted on arXiv 15 May 2021.

## Builds on (most cited first, 21 of 21)

- [A General Safety Framework for Learning-Based Control in Uncertain Robotic Systems](https://mlsanity.com/paper/1705.01292/) (2017)
- [Formulas for Data-driven Control: Stabilization, Optimality and Robustness](https://mlsanity.com/paper/1903.06842/) (2019)
- [A Tutorial on Concentration Bounds for System Identification](https://mlsanity.com/paper/1906.11395/) (2019)
- [Data informativity: a new perspective on data-driven analysis and control](https://mlsanity.com/paper/1908.00468/) (2019)
- [Sparse Polynomial Zonotopes: A Novel Set Representation for Reachability Analysis](https://mlsanity.com/paper/1901.01780/) (2019)
- [Data-Driven Reachable Set Computation using Adaptive Gaussian Process Classification and Monte Carlo Methods](https://mlsanity.com/paper/1910.02500/) (2019)
- [Data-Driven Reachability Analysis with Christoffel Functions](https://mlsanity.com/paper/2104.13902/) (2021)
- [From noisy data to feedback controllers: non-conservative design via a matrix S-lemma](https://mlsanity.com/paper/2006.00870/) (2020)
- [Sampling-based Reachability Analysis: A Random Set Theory Approach with Adversarial Sampling](https://mlsanity.com/paper/2008.10180/) (2020)
- [Data-Driven Reachability Analysis Using Matrix Zonotopes](https://mlsanity.com/paper/2011.08472/) (2020)
- [Combining Prior Knowledge and Data for Robust Controller Design](https://mlsanity.com/paper/2009.05253/) (2020)
- [Linear robust adaptive model predictive control: Computational complexity and conservatism -- extended version](https://mlsanity.com/paper/1909.01813/) (2019)
- [Robust data-driven state-feedback design](https://mlsanity.com/paper/1909.04314/) (2019)
- [Robust Adaptive Model Predictive Control: Performance and Parameter Estimation](https://mlsanity.com/paper/1911.00865/) (2019)
- [On-The-Fly Control of Unknown Smooth Systems from Limited Data](https://mlsanity.com/paper/2009.12733/) (2020)
- [Data-Driven Set-Based Estimation using Matrix Zonotopes with Set Containment Guarantees](https://mlsanity.com/paper/2101.10784/) (2021)
- [Verifying dissipativity properties from noise-corrupted input-state data](https://mlsanity.com/paper/2004.07270/) (2020)
- [Provably robust verification of dissipativity properties from data](https://mlsanity.com/paper/2006.05974/) (2020)
- [Robust Data-Driven Predictive Control using Reachability Analysis](https://mlsanity.com/paper/2103.14110/) (2021)
- [Dissipativity verification with guarantees for polynomial systems from noisy input-state data](https://mlsanity.com/paper/2008.03231/) (2020)
- [Simulation-based reachability analysis for nonlinear systems using componentwise contraction properties](https://mlsanity.com/paper/1709.06661/) (2017)

## Built on by (most cited first, 14 of 14)

- [Data-Driven Reachability Analysis of Stochastic Dynamical Systems with Conformal Inference](https://mlsanity.com/paper/2309.09187/) (2023)
- [Safe LLM-Controlled Robots with Formal Guarantees via Reachability Analysis](https://mlsanity.com/paper/2503.03911/) (2025)
- [Data-driven Reachability using Christoffel Functions and Conformal Prediction](https://mlsanity.com/paper/2309.08976/) (2023)
- [Formal Verification of Unknown Dynamical Systems via Gaussian Process Regression](https://mlsanity.com/paper/2201.00655/) (2021)
- [Safe Reinforcement Learning using Data-Driven Predictive Control](https://mlsanity.com/paper/2211.11027/) (2022)
- [Pick-to-Learn for Systems and Control: Data-driven Synthesis with State-of-the-art Safety Guarantees](https://mlsanity.com/paper/2512.04781/) (2025)
- [Statistical Reachability Analysis of Stochastic Cyber-Physical Systems under Distribution Shift](https://mlsanity.com/paper/2407.11609/) (2024)
- [Pedestrian Motion Prediction Using Transformer-based Behavior Clustering and Data-Driven Reachability Analysis](https://mlsanity.com/paper/2408.15250/) (2024)
- [Data-Driven Safety Verification using Barrier Certificates and Matrix Zonotopes](https://mlsanity.com/paper/2504.01007/) (2025)
- [Data-driven Set-based Estimation of Polynomial Systems with Application to SIR Epidemics](https://mlsanity.com/paper/2111.04704/) (2021)
- [Data-Driven Reachability Analysis via Diffusion Models with PAC Guarantees](https://mlsanity.com/paper/2604.00283/) (2026)
- [Data-driven Reachable Set Estimation with Tunable Adversarial and Wasserstein Distributional Guarantees](https://mlsanity.com/paper/2604.12654/) (2026)
- [Matrix Zonotopic Attention: A Context-Adaptive Value Projection for Set Transformers](https://mlsanity.com/paper/2608.05472/) (2026)
- [Safe Multi-Robot Coordination via VLM-LLM Reasoning and Reachability Analysis](https://mlsanity.com/paper/2609.27816/) (2026)

## Abstract (arXiv)

We consider the problem of computing reachable sets directly from noisy data without a given system model. Several reachability algorithms are presented for different types of systems generating the data. First, an algorithm for computing over-approximated reachable sets based on matrix zonotopes is proposed for linear systems. Constrained matrix zonotopes are introduced to provide less conservative reachable sets at the cost of increased computational expenses and utilized to incorporate prior knowledge about the unknown system model. Then we extend the approach to polynomial systems and, under the assumption of Lipschitz continuity, to nonlinear systems. Theoretical guarantees are given for these algorithms in that they give a proper over-approximate reachable set containing the true reachable set. Multiple numerical examples and real experiments show the applicability of the introduced algorithms, and comparisons are made between algorithms.