# Data-driven Reachability using Christoffel Functions and Conformal Prediction

arXiv: https://arxiv.org/abs/2309.08976 · HTML: https://mlsanity.com/paper/2309.08976/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 19 references that are in the map.
- First posted on arXiv 16 Sep 2023.

## Builds on (most cited first, 5 of 5)

- [A Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantification](https://mlsanity.com/paper/2107.07511/) (2021)
- [Testing for Outliers with Conformal p-values](https://mlsanity.com/paper/2104.08279/) (2021)
- [Data-Driven Reachability Analysis from Noisy Data](https://mlsanity.com/paper/2105.07229/) (2021)
- [Data-Driven Reachability Analysis with Christoffel Functions](https://mlsanity.com/paper/2104.13902/) (2021)
- [On the Christoffel function and classification in data analysis](https://mlsanity.com/paper/2203.14571/) (2022)

## Built on by (most cited first, 6 of 6)

- [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)
- [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)
- [Vision-Based Safe Human-Robot Collaboration with Uncertainty Guarantees](https://mlsanity.com/paper/2604.15221/) (2026)
- [Learning Goal-Reaching Quasimetric Geometry From Finite-Time Reachability](https://mlsanity.com/paper/2610.00778/) (2026)

## Abstract (arXiv)

An important mathematical tool in the analysis of dynamical systems is the approximation of the reach set, i.e., the set of states reachable after a given time from a given initial state. This set is difficult to compute for complex systems even if the system dynamics are known and given by a system of ordinary differential equations with known coefficients. In practice, parameters are often unknown and mathematical models difficult to obtain. Data-based approaches are promised to avoid these difficulties by estimating the reach set based on a sample of states. If a model is available, this training set can be obtained through numerical simulation. In the absence of a model, real-life observations can be used instead. A recently proposed approach for data-based reach set approximation uses Christoffel functions to approximate the reach set. Under certain assumptions, the approximation is guaranteed to converge to the true solution. In this paper, we improve upon these results by notably improving the sample efficiency and relaxing some of the assumptions by exploiting statistical guarantees from conformal prediction with training and calibration sets. In addition, we exploit an incremental way to compute the Christoffel function to avoid the calibration set while maintaining the statistical convergence guarantees. Furthermore, our approach is robust to outliers in the training and calibration set.