Data-Driven Reachability analysis and Support set Estimation with Christoffel Functions

TL;DR

This paper introduces algorithms for estimating the forward reachable set of a dynamical system using empirical inverse Christoffel functions, providing finite-sample guarantees and applications in data science and safety verification.

Contribution

It develops novel algorithms for reachability and support estimation using Christoffel functions, with PAC-Bayes bounds offering improved sample efficiency and finite-data guarantees.

Findings

PAC-Bayes bounds outperform VC bounds in sample efficiency

Algorithms accurately estimate support and reachable sets from finite samples

Applications include outlier detection and safety verification

Abstract

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.