In numerous applications of geometric deep learning, the studied systems exhibit spatial symmetries and it is desirable to enforce these. For the symmetry of global rotations and reflections, this means that the model should be equivariant with respect to the transformations that form the group of $\mathrm O(d)$. While many approaches for equivariant message passing require specialized architectures, including non-standard normalization layers or non-linearities, we here present a framework based on local reference frames ("local canonicalization") which can be integrated with any architecture without restrictions. We enhance equivariant message passing based on local canonicalization by introducing tensorial messages to communicate geometric information consistently between different local coordinate frames. Our framework applies to message passing on geometric data in Euclidean spaces of arbitrary dimension. We explicitly show how our approach can be adapted to make a popular existing point cloud architecture equivariant. We demonstrate the superiority of tensorial messages and achieve state-of-the-art results on normal vector regression and competitive results on other standard 3D point cloud tasks.

翻译：在几何深度学习的众多应用中，所研究的系统展现出空间对称性，因此需要强制实施这些对称性。对于全局旋转和反射的对称性而言，这意味着模型应对构成 $\mathrm O(d)$ 群的变换具有等变性。尽管许多等变消息传递方法需要专门的架构，包括非标准的归一化层或非线性激活函数，本文提出了一种基于局部参考系（"局部规范化"）的框架，该框架可与任何架构无缝集成，且不受限制。我们通过引入张量消息来增强基于局部规范化的等变消息传递，以在不同局部坐标系之间一致地传递几何信息。我们的框架适用于任意维度欧几里得空间中几何数据的消息传递。我们明确展示了如何调整我们的方法，使一个流行的现有点云架构具有等变性。我们证明了张量消息的优越性，在法向量回归任务上取得了最先进的结果，并在其他标准3D点云任务上获得了有竞争力的结果。