Data-Driven Reachability analysis and Support set Estimation with Christoffel Functions
This work addresses safety-critical applications in control and data science by providing finite-data guarantees for support estimation, though it is incremental as it builds on existing Christoffel function methods.
The paper tackles the problem of estimating forward reachable sets of dynamical systems and support sets of random variables using finite data samples, achieving provable accuracy guarantees under the Probably Approximately Correct (PAC) framework with improved sample efficiency via PAC-Bayes bounds.
We present algorithms for estimating the forward reachable set of a dynamical system using only a finite collection of independent and identically distributed samples. The produced estimate is the sublevel set of a function called an empirical inverse Christoffel function: empirical inverse Christoffel functions are known to provide good approximations to the support of probability distributions. In addition to reachability analysis, the same approach can be applied to general problems of estimating the support of a random variable, which has applications in data science towards detection of novelties and outliers in data sets. In applications where safety is a concern, having a guarantee of accuracy that holds on finite data sets is critical. In this paper, we prove such bounds for our algorithms under the Probably Approximately Correct (PAC) framework. In addition to applying classical Vapnik-Chervonenkis (VC) dimension bound arguments, we apply the PAC-Bayes theorem by leveraging a formal connection between kernelized empirical inverse Christoffel functions and Gaussian process regression models. The bound based on PAC-Bayes applies to a more general class of Christoffel functions than the VC dimension argument, and achieves greater sample efficiency in experiments.