On Universal Equivariant Set Networks
Authors: Nimrod Segol, Yaron Lipman
Keywords: deep learning, universality, set functions, equivariance
TL;DR: Settling permutation equivariance universality for popular deep models. 
Abstract: Using deep neural networks that are either invariant or equivariant to permutations in order to learn functions on unordered sets has become prevalent. The most popular, basic models are DeepSets (Zaheer et al. 2017) and PointNet (Qi et al. 2017). While known to be universal for approximating invariant functions, DeepSets and PointNet are not known to be universal when approximating equivariant set functions. On the other hand, several recent equivariant set architectures have been proven equivariant universal (Sannai et al. 2019, Keriven and Peyre 2019), however these models either use layers that are not permutation equivariant (in the standard sense) and/or use higher order tensor variables which are less practical. There is, therefore, a gap in understanding the universality of popular equivariant set models versus theoretical ones. 
			
In this paper we close this gap by proving that: (i) PointNet is not equivariant universal; and (ii) adding a single linear transmission  layer makes PointNet universal. We call this architecture PointNetST and argue it is the simplest permutation equivariant universal model known to date. Another consequence is that DeepSets is universal, and also PointNetSeg, a popular point cloud segmentation network (used e.g., in Qi et al. 2017) is universal.
		
The key theoretical tool used to prove the above results is an explicit characterization of all permutation equivariant polynomial layers. Lastly, we provide numerical experiments validating the theoretical results and comparing different permutation equivariant models.