Based on the theory of homogeneous spaces we derive \textit{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 empirically support this claim
by reaching state-of-the-art results -- in accuracy and speed -- on three
different benchmarks: interatomic potential energy prediction, trajectory
forecasting in N-body systems, and generating molecules via equivariant
diffusion models.

基于均匀空间理论，我们得出了在灵活的消息传递框架中使用的几何最优边属性，将卷积网络中的权重共享定义为在应该等同处理的点对上共享消息函数的共享点对的等价类，并导出能够唯一标识这些等价类的属性。我们通过在处理 3D 点云时开发了一种高效的等变群卷积网络来应用这一理论。在三个不同的基准测试中，即原子间势能预测、N 体系统轨迹预测和通过等变扩散模型生成分子，我们以准确性和速度方面达到了最先进的结果。