This paper presents $\mathrm{E}(n)$ Equivariant Message Passing Simplicial Networks (EMPSNs), a novel approach to learning on geometric graphs and point clouds that is equivariant to rotations, translations, and reflections. EMPSNs can learn high-dimensional simplex features in graphs (e.g. triangles), and use the increase of geometric information of higher-dimensional simplices in an $\mathrm{E}(n)$ equivariant fashion. EMPSNs simultaneously generalize $\mathrm{E}(n)$ Equivariant Graph Neural Networks to a topologically more elaborate counterpart and provide an approach for including geometric information in Message Passing Simplicial Networks. The results indicate that EMPSNs can leverage the benefits of both approaches, leading to a general increase in performance when compared to either method. Furthermore, the results suggest that incorporating geometric information serves as an effective measure against over-smoothing in message passing networks, especially when operating on high-dimensional simplicial structures. Last, we show that EMPSNs are on par with state-of-the-art approaches for learning on geometric graphs.

翻译：本文提出$\mathrm{E}(n)$等变消息传递单纯复形网络（EMPSNs），这是一种作用于几何图与点云数据的新方法，其对于旋转、平移和反射具有等变性。EMPSNs能够学习图中高维单纯形特征（例如三角形），并以$\mathrm{E}(n)$等变方式利用高维单纯形的几何信息增强。该方法同时将$\mathrm{E}(n)$等变图神经网络推广为拓扑上更复杂的对应结构，并为消息传递单纯复形网络提供了引入几何信息的途径。实验结果表明，EMPSNs能够融合两种方法的优势，相较于单一方法整体性能显著提升。此外，研究发现几何信息的引入可有效缓解消息传递网络中的过度平滑问题，尤其在高维单纯形结构处理中效果显著。最后，我们证明EMPSNs在几何图学习任务上达到与当前最优方法相当的性能水平。