$\mathrm{E}(n)$ Equivariant Message Passing Simplicial Networks

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.