Christian Bueno, Alan Hylton
Point clouds and sets are input data-types which pose unique problems to deep learning. Since sets can have variable cardinality and are unchanged by permutation, the input space for these problems naturally form infinite-dimensional non-Euclidean spaces. Despite these mathematical difficulties, PointNet (Qi et al. 2017) and Deep Sets (Zaheer et al. 2017) introduced foundational neural network architectures to address these problems. In this paper we present a unified framework to study the expressive power of such networks as well as their extensions beyond point clouds (partially addressing a conjecture on the extendibility of DeepSets along the way). To this end, we demonstrate the crucial role that the Hausdorff and Wasserstein metrics play and prove new cardinality-agnostic universality results to characterize exactly which functions can be approximated by these models. In particular, these results imply that PointNet generally cannot approximate averages of continuous functions over sets (e.g. center-of-mass or higher moments) implying that DeepSets is strictly more expressive than PointNet in the constant cardinality setting. Moreover, we obtain explicit lower-bounds on the approximation error and present a simple method to produce arbitrarily many examples of this failure-mode. Counterintuitively, we also prove that in the unbounded cardinality setting that any function which can be uniformly approximated by both PointNet and normalized-DeepSets must be constant. Finally, we also prove theorems on the Lipschitz properties of PointNet and normalized-DeepSets which shed insight into exploitable inductive bias in these networks.