  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 are universal, and for
devising two other novel universal architectures.
