Convex and Nonconvex Sublinear Regression with Application to Data-driven Learning of Reach Sets

TL;DR

This paper introduces methods for estimating compact sets from data by learning their support functions through sublinear regression, using convex quadratic programming and nonconvex neural networks, with applications to reach set estimation in control systems.

Contribution

It presents two novel algorithms for sublinear regression—one convex and one nonconvex—that enable data-driven learning of support functions for compact sets.

Findings

The convex approach solves a quadratic program for support function estimation.

The nonconvex approach employs neural networks to learn support functions.

Numerical examples demonstrate effective learning of reach sets from trajectory data.

Abstract

We consider estimating a compact set from finite data by approximating the support function of that set via sublinear regression. Support functions uniquely characterize a compact set up to closure of convexification, and are sublinear (convex as well as positive homogeneous of degree one). Conversely, any sublinear function is the support function of a compact set. We leverage this property to transcribe the task of learning a compact set to that of learning its support function. We propose two algorithms to perform the sublinear regression, one via convex and another via nonconvex programming. The convex programming approach involves solving a quadratic program (QP). The nonconvex programming approach involves training a input sublinear neural network. We illustrate the proposed methods via numerical examples on learning the reach sets of controlled dynamics subject to set-valued input uncertainties from trajectory data.