Parametrizing Convex Sets Using Sublinear Neural Networks

TL;DR

This paper introduces a neural network approach to parameterize convex sets using sublinear functions, enabling effective shape reconstruction and optimization.

Contribution

It presents a novel neural parametrization of convex sets via sublinear functions, with a universal approximation theorem and practical applications.

Findings

Achieved accurate shape reconstruction in inverse design tasks

Proved a universal approximation theorem for convex sets

Demonstrated effectiveness in shape optimization

Abstract

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.