# Kernel-Based Learning of Safety Barriers

arXiv: https://arxiv.org/abs/2601.12002 · HTML: https://mlsanity.com/paper/2601.12002/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 52 of its 95 references that are in the map.
- First posted on arXiv 17 Jan 2026.

## Builds on (most cited first, 30 of 52)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Reluplex: An Efficient SMT Solver for Verifying Deep Neural Networks](https://mlsanity.com/paper/1702.01135/) (2017)
- [Safety Verification of Deep Neural Networks](https://mlsanity.com/paper/1610.06940/) (2016)
- [Gaussian Processes and Kernel Methods: A Review on Connections and Equivalences](https://mlsanity.com/paper/1807.02582/) (2018)
- [A Measure-Theoretic Approach to Kernel Conditional Mean Embeddings](https://mlsanity.com/paper/2002.03689/) (2020)
- [Optimal Rates for Regularized Conditional Mean Embedding Learning](https://mlsanity.com/paper/2208.01711/) (2022)
- [How to Train Your Neural Control Barrier Function: Learning Safety Filters for Complex Input-Constrained Systems](https://mlsanity.com/paper/2310.15478/) (2023)
- [Non-parametric Models for Non-negative Functions](https://mlsanity.com/paper/2007.03926/) (2020)
- [Safe Learning of Quadrotor Dynamics Using Barrier Certificates](https://mlsanity.com/paper/1710.05472/) (2017)
- [Learning with little mixing](https://mlsanity.com/paper/2206.08269/) (2022)
- [A Rigorous Theory of Conditional Mean Embeddings](https://mlsanity.com/paper/1912.00671/) (2019)
- [Learning linear operators: Infinite-dimensional regression as a well-behaved non-compact inverse problem](https://mlsanity.com/paper/2211.08875/) (2022)
- [Formal Synthesis of Stochastic Systems via Control Barrier Certificates](https://mlsanity.com/paper/1905.04585/) (2019)
- [Control Barrier Functions for Unknown Nonlinear Systems using Gaussian Processes](https://mlsanity.com/paper/2010.05818/) (2020)
- [Automated Verification and Synthesis of Stochastic Hybrid Systems: A Survey](https://mlsanity.com/paper/2101.07491/) (2021)
- [Fossil 2.0: Formal Certificate Synthesis for the Verification and Control of Dynamical Models](https://mlsanity.com/paper/2311.09793/) (2023)
- [A Barrier Function Approach to Finite-Time Stochastic System Verification and Control](https://mlsanity.com/paper/1909.05109/) (2019)
- [Global Optimization of Gaussian processes](https://mlsanity.com/paper/2005.10902/) (2020)
- [Hard Shape-Constrained Kernel Machines](https://mlsanity.com/paper/2005.12636/) (2020)
- [Data-driven verification and synthesis of stochastic systems via barrier certificates](https://mlsanity.com/paper/2111.10330/) (2021)
- [Formal Synthesis of Controllers for Safety-Critical Autonomous Systems: Developments and Challenges](https://mlsanity.com/paper/2402.13075/) (2024)
- [Stochastic Omega-Regular Verification and Control with Supermartingales](https://mlsanity.com/paper/2405.17304/) (2024)
- [Data-Driven Distributionally Robust Safety Verification Using Barrier Certificates and Conditional Mean Embeddings](https://mlsanity.com/paper/2403.10497/) (2024)
- [Formal Policy Synthesis for Continuous-Space Systems via Reinforcement Learning](https://mlsanity.com/paper/2005.01319/) (2020)
- [Data-Driven Abstraction-Based Control Synthesis](https://mlsanity.com/paper/2206.08069/) (2022)
- [Maximum Mean Discrepancy Distributionally Robust Nonlinear Chance-Constrained Optimization with Finite-Sample Guarantee](https://mlsanity.com/paper/2204.11564/) (2022)
- [Robust Control Barrier Functions for Nonlinear Control Systems with Uncertainty: A Duality-based Approach](https://mlsanity.com/paper/2208.05955/) (2022)
- [Distributionally Robust Optimal and Safe Control of Stochastic Systems via Kernel Conditional Mean Embedding](https://mlsanity.com/paper/2304.00644/) (2023)
- [Simultaneous Synthesis and Verification of Neural Control Barrier Functions through Branch-and-Bound Verification-in-the-loop Training](https://mlsanity.com/paper/2311.10438/) (2023)
- [Bounding Multivariate Trigonometric Polynomials with Applications to Filter Bank Design](https://mlsanity.com/paper/1802.09588/) (2018)

## Abstract (arXiv)

The rapid integration of AI algorithms in safety-critical applications such as autonomous driving and healthcare is raising significant concerns about the ability to meet stringent safety standards. Traditional tools for formal safety verification struggle with the black-box nature of AI-driven systems and lack the flexibility needed to scale to the complexity of real-world applications. In this paper, we present a data-driven approach for safety verification and synthesis of black-box systems with discrete-time stochastic dynamics. We employ the concept of control barrier certificates, which can guarantee safety of the system, and learn the certificate directly from a set of system trajectories. We use conditional mean embeddings to embed data from the system into a reproducing kernel Hilbert space (RKHS) and construct an RKHS ambiguity set that can be inflated to robustify the result to out-of-distribution behavior. We provide the theoretical results on how to apply the approach to general classes of temporal logic specifications beyond safety. For the data-driven computation of safety barriers, we leverage a finite Fourier expansion to cast a typically intractable semi-infinite optimization problem as a linear program. The resulting spectral barrier allows us to leverage the fast Fourier transform to generate the relaxed problem efficiently, offering a scalable yet distributionally robust framework for verifying safety. Our work moves beyond restrictive assumptions on system dynamics and uncertainty, as demonstrated on two case studies including a black-box system with a neural network controller.