On the Universality of Rotation Equivariant Point Cloud Networks

Nadav Dym, Haggai Maron

Keywords: 3D deep learning, Rotation invariance, Invariant and equivariant deep networks, Universal approximation, Point clouds

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, Tensor field Networks and SE3-Transformers, are universal, and for devising two other novel universal architectures.

One-sentence Summary: We provide sufficient conditions for universality of rotation equivariant point cloud networks and use these conditions to show that current models are universal as well as for devising new universal architectures.

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