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.

翻译：基于齐次空间理论，我们推导出可在柔性消息传递框架中使用的\emph{几何最优边属性}。我们形式化卷积网络中权值共享的概念，即对等对待的点对之间共享消息函数。我们定义在群变换下等价的点对等价类，并推导出唯一标识这些类的属性。通过将消息函数条件化于这些属性，实现权值共享。作为该理论的应用，我们开发了一种处理三维点云的高效等变群卷积网络。齐次空间理论指导我们如何对位置齐次空间$\mathbb{R}^3$、位置与取向齐次空间$\mathbb{R}^3 {\times} S^2$以及SE$(3)$群本身上的特征图执行群卷积。其中$\mathbb{R}^3 {\times} S^2$是最优选择，因其能表示方向信息（$\mathbb{R}^3$方法无法做到），同时相较在完整SE$(3)$群上索引特征显著提升计算效率。我们在三个不同基准测试中分别达到最先进结果（准确性与速度），实证支撑了这一论断：原子间势能预测、N体系统轨迹预测，以及通过等变扩散模型生成分子。