🤖 AI Summary
This work addresses the challenges of parameterizing convex sets in shape optimization and inverse design by proposing an implicit representation based on sublinear neural networks. The method flexibly characterizes arbitrary convex bodies by learning positively homogeneous and convex support and gauge functions. It enjoys theoretical universal approximation capabilities for convex sets and demonstrates strong empirical performance, accurately reconstructing target shapes in experiments, thereby validating its expressiveness and effectiveness. The key innovation lies in integrating convex analysis with neural networks to establish a convex set parameterization framework that simultaneously offers rigorous theoretical guarantees and practical performance.
📝 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.