ON THE UNIVERSALITY OF ROTATION EQUIVARIANT POINT CLOUD NETWORKS

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 (Thomas et al., 2018; Fuchs et al., 2020) are universal, and for devising two other novel universal architectures.

1. INTRODUCTION

Designing neural networks that respect data symmetry is a powerful approach for obtaining efficient deep models. Prominent examples being convolutional networks which respect the translational invariance of images, graph neural networks which respect the permutation invariance of graphs (Gilmer et al., 2017; Maron et al., 2019b) , networks such as (Zaheer et al., 2017; Qi et al., 2017a) which respect the permutation invariance of sets, and networks which respect 3D rotational symmetries (Cohen et al., 2018; Weiler et al., 2018; Esteves et al., 2018; Worrall & Brostow, 2018; Kondor et al., 2018a) . While the expressive power of equivariant models is reduced by design to include only equivariant functions, a desirable property of equivariant networks is universality: the ability to approximate any continuous equivariant function. This is not always the case: while convolutional networks and networks for sets are universal (Yarotsky, 2018; Segol & Lipman, 2019) , popular graph neural networks are not (Xu et al., 2019; Morris et al., 2018) . In this paper, we consider the universality of networks that respect the symmetries of 3D point clouds: translations, rotations, and permutations. Designing such networks is a popular paradigm in recent years (Thomas et al., 2018; Fuchs et al., 2020; Poulenard et al., 2019; Zhao et al., 2019) . While there have been many works on the universality of permutation invariant networks (Zaheer et al., 2017; Maron et al., 2019c; Keriven & Peyré, 2019) , and a recent work discussing the universality of rotation equivariant networks (Bogatskiy et al., 2020) , this is a first paper which discusses the universality of networks which combine rotations, permutations and translations. We start the paper with a general, architecture-agnostic, discussion, and derive two sufficient conditions for universality. These conditions are a result of a novel characterization of equivariant polynomials for the symmetry group of interest. We use these conditions in order to prove universality of the prominent Tensor Field Networks (TFN) architecture (Thomas et al., 2018; Fuchs et al., 2020) . The following is a weakened and simplified statement of Theorem 2 stated later on in the paper: Theorem (Simplification of Theorem 2). Any continuous equivariant function on point clouds can be approximated uniformly on compact sets by a composition of TFN layers. We use our general discussion to prove the universality of two additional equivariant models: the first is a simple modification of the TFN architecture which allows for universality using only low dimensional filters. The second is a minimal architecture which is based on tensor product representations, rather than the more commonly used irreducible representations of SO(3). We discuss the advantages and disadvantages of both approaches. To summarize, the contributions of this paper are: (1) A general approach for proving the universality of rotation equivariant models for point clouds; (2) A proof that two recent equivariant models (Thomas et al., 2018; Fuchs et al., 2020) are universal; (3) Two additional simple and novel universal architectures.

2. PREVIOUS WORK

Deep learning on point clouds. (Qi et al., 2017a; Zaheer et al., 2017) were the first to apply neural networks directly to the raw point cloud data, by using pointwise functions and pooling operations. Many subsequent works used local neighborhood information (Qi et al., 2017b; Wang et al., 2019b; Atzmon et al., 2018) . We refer the reader to a recent survey for more details (Guo et al., 2020) . In contrast with the aforementioned works which focused solely on permutation invariance, more related to this paper are works that additionally incorporated invariance to rigid motions. (Thomas et al., 2018) proposed Tensor Field Networks (TFN) and showed their efficacy on physics and chemistry tasks. (Kondor et al., 2018b) also suggested an equivariant model for continuous rotations. (Li et al., 2019) suggested models that are equivariant to discrete subgroups of SO(3). (Poulenard et al., 2019) suggested an invariant model based on spherical harmonics. (Fuchs et al., 2020) followed TFN and added an attention mechanism. Recently, (Zhao et al., 2019) proposed a quaternion equivariant point capsule network that also achieves rotation and translation invariance. Universal approximation for invariant networks. Understanding the approximation power of invariant models is a popular research goal. Most of the current results assume that the symmetry group is a permutation group. (Zaheer et al., 2017; Qi et al., 2017a; Segol & Lipman, 2019; Maron et al., 2020; Serviansky et al., 2020) proved universality for several S n -invariant and equivariant models. (Maron et al., 2019b; a; Keriven & Peyré, 2019; Maehara & NT, 2019) studied the approximation power of high-order graph neural networks. (Maron et al., 2019c; Ravanbakhsh, 2020) targeted universality of networks that use high-order representations for permutation groups(Yarotsky, 2018) provided several theoretical constructions of universal equivariant neural network models based on polynomial invariants, including an SE(2) equivariant model. In a recent work (Bogatskiy et al., 2020) presented a universal approximation theorem for networks that are equivariant to several Lie groups including SO(3). The main difference from our paper is that we prove a universality theorem for a more complex group that besides rotations also includes translations and permutations.

3. A FRAMEWORK FOR PROVING UNIVERSALITY

In this section, we describe a framework for proving the universality of equivariant networks. We begin with some mathematical preliminaries: 3.1 MATHEMATICAL SETUP An action of a group G on a real vector space W is a collection of maps ρ(g) : W → W defined for any g ∈ G, such that ρ(g 1 ) • ρ(g 2 ) = ρ(g 1 g 2 ) for all g 1 , g 2 ∈ G, and the identity element of G is mapped to the identity mapping on W . We say ρ is a representation of G if ρ(g) is a linear map for every g ∈ G. As is customary, when it does not cause confusion we often say that W itself is a representation of G . In this paper, we are interested in functions on point clouds. Point clouds are sets of vectors in R 3 arranged as matrices: X = (x 1 , . . . , x n ) ∈ R 3×n . Many machine learning tasks on point clouds, such as classification, aim to learn a function which is invariant to rigid motions and relabeling of the points. Put differently, such functions are required

