On the Limits of Sampling-Based Reachability: Geometry, Dynamics, and Sample Complexity
This paper investigates how the geometry of the initial set, dynamics, and sampling distribution affect the accuracy of sampling-based reachability analysis. By formulating the problem as geometric support estimation, the authors identify two regularity conditions—positive reach of the initial set's complement and Lipschitz continuity of the dynamics—that allow a probability-mass coverage guarantee to be upgraded to Hausdorff distance accuracy. The sample complexity scales exponentially with state dimension and time horizon, and this exponential dependence is intrinsic, not an artifact of the method. Experiments on nonlinear systems confirm that adversarial sampling improves constants but not the scaling.
-->
[Submitted on 21 Jul 2026]
Title:On the Limits of Sampling-Based Reachability: Geometry, Dynamics, and Sample Complexity
View a PDF of the paper titled On the Limits of Sampling-Based Reachability: Geometry, Dynamics, and Sample Complexity, by Jixian Liu and 3 other authors
View PDF HTML (experimental)
Abstract:Reachability analysis is central to safety-critical control, robotics, and neural network verification, but classical computational methods, such as Hamilton--Jacobi reachability and set propagation, scale poorly with state dimension. Sampling-based methods have emerged as a promising alternative, often providing finite-sample guarantees that bound the probability-mass left uncovered. However, an explicit account of how the geometry of the initial set, the dynamics, and the sampling law affect the accuracy of the estimator is not fully available in the literature. We study this by casting sampling-based reachable-set recovery as geometric support estimation over a family of problems specified by an initial set, its dynamics, and a sampling law. First, we identify two regularity properties, positive reach of the initial set's complement and Lipschitz continuity of the dynamics, that together make recovery well-posed: a probability-mass coverage guarantee can be upgraded to accuracy $r$ in Hausdorff distance. Second, we bound the resulting sample complexity: recovery is achievable with $\tilde{\mathcal{O}}\big((e^{3LT}/r)^n\big)$ samples, exponential in both the state dimension and the time horizon. Third, we show that neither can be removed: an minimax lower bound of $\Omega\big((e^{LT}/r)^n\big)$ holds for every estimator, so the exponential dependence on dimension and the degradation over the horizon are both intrinsic, not artifacts of a particular method. Experiments on nonlinear systems confirm that adversarial sampling improves constants but not the scaling.
Subjects:
Robotics (cs.RO)
Cite as: arXiv:2607.18606 [cs.RO]
(or arXiv:2607.18606v1 [cs.RO] for this version)
https://doi.org/10.48550/arXiv.2607.18606
arXiv-issued DOI via DataCite (pending registration)
Submission history
From: Jixian Liu [view email] [v1] Tue, 21 Jul 2026 00:54:27 UTC (5,890 KB)
Full-text links:
Access Paper:
View a PDF of the paper titled On the Limits of Sampling-Based Reachability: Geometry, Dynamics, and Sample Complexity, by Jixian Liu and 3 other authors
View PDF
HTML (experimental)
TeX Source
view license
Current browse context:
cs.RO
new | recent | 2026-07
Change to browse by:
cs
References & Citations
NASA ADS
Google Scholar
Semantic Scholar
Loading...
Data provided by:
Bibliographic Tools
Bibliographic and Citation Tools
Bibliographic Explorer Toggle
Bibliographic Explorer (What is the Explorer?)
Connected Papers Toggle
Connected Papers (What is Connected Papers?)
Litmaps Toggle
Litmaps (What is Litmaps?)
scite.ai Toggle
scite Smart Citations (What are Smart Citations?)
Code, Data, Media
Code, Data and Media Associated with this Article
alphaXiv Toggle
alphaXiv (What is alphaXiv?)
Links to Code Toggle
CatalyzeX Code Finder for Papers (What is CatalyzeX?)
DagsHub Toggle
DagsHub (What is DagsHub?)
GotitPub Toggle
Gotit.pub (What is GotitPub?)
Huggingface Toggle
Hugging Face (What is Huggingface?)
ScienceCast Toggle
ScienceCast (What is ScienceCast?)
Demos
Demos
Replicate Toggle
Replicate (What is Replicate?)
Spaces Toggle
Hugging Face Spaces (What is Spaces?)
Spaces Toggle
TXYZ.AI (What is TXYZ.AI?)
Related Papers
Recommenders and Search Tools
Link to Influence Flower
Influence Flower (What are Influence Flowers?)
Core recommender toggle
CORE Recommender (What is CORE?)
Author
Venue
Institution
Topic
About arXivLabs
arXivLabs: experimental projects with community collaborators
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.
Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)