  Based on the theory of homogeneous spaces we derive geometrically optimal
edge attributes to be used within the flexible message-passing framework. We
formalize the notion of weight sharing in convolutional networks as the sharing
of message functions over point-pairs that should be treated equally. We define
equivalence classes of point-pairs that are identical up to a transformation in
the group and derive attributes that uniquely identify these classes. Weight
sharing is then obtained by conditioning message functions on these attributes.
As an application of the theory, we develop an efficient equivariant group
convolutional network for processing 3D point clouds. The theory of homogeneous
spaces tells us how to do group convolutions with feature maps over the
homogeneous space of positions $\mathbb{R}^3$, position and orientations
$\mathbb{R}^3 {\times} S^2$, and the group $SE(3)$ itself. Among these,
$\mathbb{R}^3 {\times} S^2$ is an optimal choice due to the ability to
represent directional information, which $\mathbb{R}^3$ methods cannot, and it
significantly enhances computational efficiency compared to indexing features
on the full $SE(3)$ group. We support this claim with state-of-the-art results
-- in accuracy and speed -- on five different benchmarks in 2D and 3D,
including interatomic potential energy prediction, trajectory forecasting in
N-body systems, and generating molecules via equivariant diffusion models.
