  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.
