On the Universality of Rotation Equivariant Point Cloud Networks

TL;DR

This paper investigates the approximation capabilities of rotation-equivariant neural networks for point clouds, establishing conditions for universality and proposing new models with proven universal approximation properties.

Contribution

It provides the first theoretical analysis of the approximation power of rotation-equivariant point cloud networks and introduces new universal architectures based on these insights.

Findings

Two sufficient conditions for universality of equivariant architectures.

Proof that two existing models are universal.

Design of two novel universal architectures.

Abstract

Learning functions on point clouds has applications in many fields, including computer vision, computer graphics, physics, and chemistry. Recently, there has been a growing interest in neural architectures that are invariant or equivariant to all three shape-preserving transformations of point clouds: translation, rotation, and permutation. In this paper, we present a first study of the approximation power of these architectures. We first derive two sufficient conditions for an equivariant architecture to have the universal approximation property, based on a novel characterization of the space of equivariant polynomials. We then use these conditions to show that two recently suggested models are universal, and for devising two other novel universal architectures.