We introduce Clifford Group Equivariant Simplicial Message Passing Networks, a method for steerable E(n)-equivariant message passing on simplicial complexes. Our method integrates the expressivity of Clifford group-equivariant layers with simplicial message passing, which is topologically more intricate than regular graph message passing. Clifford algebras include higher-order objects such as bivectors and trivectors, which express geometric features (e.g., areas, volumes) derived from vectors. Using this knowledge, we represent simplex features through geometric products of their vertices. To achieve efficient simplicial message passing, we share the parameters of the message network across different dimensions. Additionally, we restrict the final message to an aggregation of the incoming messages from different dimensions, leading to what we term shared simplicial message passing. Experimental results show that our method is able to outperform both equivariant and simplicial graph neural networks on a variety of geometric tasks.

翻译：我们提出Clifford群等变单纯消息传递网络，一种用于单纯复形上可操控E(n)-等变消息传递的方法。该方法将Clifford群等变层的表达力与单纯消息传递相结合，后者在拓扑结构上比常规图消息传递更为精细。Clifford代数包含诸如双向量和三向量等高阶对象，可表达由向量导出的几何特征（如面积、体积）。利用这一特性，我们通过顶点间的几何积来表示单纯形特征。为实现高效的单纯消息传递，我们在不同维度间共享消息网络的参数。此外，我们将最终消息限制为来自不同维度的传入消息的聚合，由此提出共享单纯消息传递机制。实验结果表明，该方法在多种几何任务上均优于现有的等变图神经网络和单纯图神经网络。