In this paper we prove new universal approximation theorems for deep learning on point clouds that do not assume fixed cardinality. We do this by first generalizing the classical universal approximation theorem to general compact Hausdorff spaces and then applying this to the permutation-invariant architectures presented in PointNet (Qi et al) and Deep Sets (Zaheer et al). Moreover, though both architectures operate on the same domain, we show that the constant functions are the only functions they can mutually uniformly approximate. In particular, DeepSets architectures cannot uniformly approximate the diameter function but can uniformly approximate the center-of-mass function but it is the other way around for PointNet. Additionally, even when the point clouds are limited to at most k points, PointNet cannot uniformly approximate center-of-mass and we obtain explicit error bounds and a method to produce geometrically derived adversarial examples.
This work examines the fundamental properties of two popular architectures -- PointNet and DeepSets -- for processing point clouds (and other unordered sets). The authors provide a new universal approximation theorem on real-valued functions that doesn't require the assumption of a fixed cardinality of the input set. They further provide examples of functions that can't be mutually approximated by PointNets and DeepSets.
- It is important in order to know the representational power and fundamental limitations of basic algorithmic building blocks. However, for the two base architectures that are examined in this work, UATs where already provided in the original manuscripts. The presentation in this paper does remove the assumption of a fixed cardinality, but since this seems to be a mild assumption, it is not clear what is gained by this (beyond mathematical elegance). The paper doesn't give any hints here.
- The paper shows two specific functions, where one can be approximated by PointNet but not by DeepSets and vice-versa. The authors again note that this might vanish when a fixed cardinality of the input point cloud is assumed. Again, it seems that this is a mild restriction and I'd like the authors to elaborate on the importance of removing this restriction.
- The paper is clearly targeted at a specialist audience and invokes dense and advanced mathematical concepts. It might find a better audience at a venue that is more specialized in this type of work (e.g. applied mathematics or more theory-focused machine learning venues).
Summary: UATs are important and interesting, however, they do exist for the architectures that are targeted in this paper. It is not clear what is gained by the main difference, i.e. removing the cardinality. I'd be grateful if the authors could comment on this. 
Disclaimer: While I believe to have a reasonable mathematical background, I'm not an expert in this field and my assessment is primarily based on the bottom line of these proofs.
=== Post rebuttal update ===
I'd like to thank the authors for their efforts and additional insights. The additional illustration improves the accessibility of the paper. However, the rebuttal does not alleviate my concerns about additional impact beyond the UATs in the original paper. I thus maintain my recommendation of weak reject