Clifford Group Equivariant Simplicial Message Passing Networks

Cong Liu, David Ruhe, Floor Eijkelboom, Patrick Forré

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Keywords: Clifford Algebra, Geometric Algebra, Graph Neural Networks, Simplicial Message Passing, Topological Deep Learning, Geometric Deep Learning, Equivariance

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Abstract: We introduce Clifford Group Equivariant Simplicial Message Passing Networks, a method for steerable $\mathrm{E}(n)$-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.

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Primary Area: learning on graphs and other geometries & topologies

Submission Number: 5751