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，我们推导了可在灵活消息传递框架中使用的\emph{几何最优边属性}。我们将卷积网络中的权值共享形式化为对应等处理点对的消息函数共享机制。我们定义了在群变换下等价且仅差一个变换的点对等价类，并推导了唯一标识这些类别的属性。通过使消息函数依赖于这些属性，实现了权值共享。作为该理论的应用，我们开发了用于处理3D点云的高效等变群卷积网络。齐次空间理论指导我们如何在位置齐次空间$\mathbb{R}^3$、位置与方向$\mathbb{R}^3 {\times} S^2$以及SE$(3)$群本身上，利用特征图执行群卷积。其中，$\mathbb{R}^3 {\times} S^2$是最优选择：它既能表示方向信息（$\mathbb{R}^3$方法无法实现），又比在全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上——包括原子间势能预测、N体系统轨迹预测及通过等变扩散模型生成分子——均以精度和速度达到当前最优结果，实证支持了上述论断。