Parametrizing Convex Sets Using Sublinear Neural Networks
It provides a new way to represent convex sets for shape optimization and inverse design, though the impact is limited to domain-specific applications.
The paper introduces a neural parameterization of convex sets using sublinear functions, proving a universal approximation theorem and demonstrating accurate shape reconstruction in optimization and inverse design tasks.
We propose a neural parameterization of convex sets by learning sublinear (positively homogeneous and convex) functions. Our networks implicitly represent both the support and gauge functions of a convex body. We prove a universal approximation theorem for convex sets under this parametrization. Empirically, we demonstrate the method on shape optimization and inverse design tasks, achieving accurate reconstruction of target shapes.