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.

翻译：基于齐次空间理论，我们推导了适用于灵活消息传递框架的几何最优边属性。我们形式化了卷积网络中权重共享的概念，即对应被平等处理的点对共享消息函数。我们定义了在群变换下等价的点对等价类，并推导了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，实现了权重共享。作为该理论的应用，我们开发了一种高效处理3D点云的等变群卷积网络。齐次空间理论指导我们如何在位置齐次空间$\mathbb{R}^3$、位置与取向齐次空间$\mathbb{R}^3 {\times} S^2$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，因为它能够表征方向信息（这是$\mathbb{R}^3$方法无法做到的），同时相比在全$SE(3)$群上索引特征，显著提升了计算效率。我们在二维和三维的五个不同基准测试上取得了最先进的结果（在准确率和速度上），包括原子间势能预测、N体系统轨迹预测以及通过等变扩散模型生成分子，支持了这一论断。