  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.
