2024 | Cong Liu, David Ruhe, Floor Eijkelboom, Patrick Forré
Clifford Group Equivariant Simplicial Message Passing Networks (CSMPNs) are introduced as a method for steerable $ \mathrm{E}(n) $-equivariant message passing on simplicial complexes. The 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 derived from vectors. Using this knowledge, simplex features are represented through geometric products of their vertices. To achieve efficient simplicial message passing, parameters of the message network are shared across different dimensions. Additionally, the final message is restricted to an aggregation of incoming messages from different dimensions, leading to shared simplicial message passing. Experimental results show that the method outperforms both equivariant and simplicial graph neural networks on various geometric tasks. The implementation is available on GitHub. The method combines the merits of simplicial message passing and geometric equivariance, resulting in a method based on EGNNs called $ \mathrm{E}(n) $ Equivariant Message Passing Simplicial Networks (EMPSNs). However, the method has limitations, such as initializing higher-dimensional simplices with manually calculated geometric information and relying on scalarization methods. The proposed CSMPNs use Clifford algebra-based simplicial message passing, enabling efficient message passing across different simplex orders. The method is equivariant to the Clifford group's orthogonal action, ensuring respect for geometric symmetries. Experimental results show that the method outperforms existing models on tasks such as convex hulls volume prediction, human walking motion prediction, molecular motion prediction, and NBA player trajectory prediction. The method is based on Clifford group equivariant neural networks and simplicial complexes. The method embeds scalar and vector features of nodes in the Clifford algebra and creates simplex features through geometric products. The method uses shared parameters for message passing across different simplex orders and ensures equivariance through Clifford group equivariant neural networks. The method is tested on various geometric tasks, including 5D convex hulls, CMU motion capture, MD17 atomic motion prediction, and NBA player trajectory prediction. The results show that the method outperforms existing models on these tasks. The method is also compared with other models, showing that it achieves outstanding scores across the board. The method is efficient and can be applied to various geometric tasks. The method is a significant advancement in geometric deep learning, combining the expressivity of Clifford group-equivariant layers with the topological intricacy of simplicial message passing.Clifford Group Equivariant Simplicial Message Passing Networks (CSMPNs) are introduced as a method for steerable $ \mathrm{E}(n) $-equivariant message passing on simplicial complexes. The 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 derived from vectors. Using this knowledge, simplex features are represented through geometric products of their vertices. To achieve efficient simplicial message passing, parameters of the message network are shared across different dimensions. Additionally, the final message is restricted to an aggregation of incoming messages from different dimensions, leading to shared simplicial message passing. Experimental results show that the method outperforms both equivariant and simplicial graph neural networks on various geometric tasks. The implementation is available on GitHub. The method combines the merits of simplicial message passing and geometric equivariance, resulting in a method based on EGNNs called $ \mathrm{E}(n) $ Equivariant Message Passing Simplicial Networks (EMPSNs). However, the method has limitations, such as initializing higher-dimensional simplices with manually calculated geometric information and relying on scalarization methods. The proposed CSMPNs use Clifford algebra-based simplicial message passing, enabling efficient message passing across different simplex orders. The method is equivariant to the Clifford group's orthogonal action, ensuring respect for geometric symmetries. Experimental results show that the method outperforms existing models on tasks such as convex hulls volume prediction, human walking motion prediction, molecular motion prediction, and NBA player trajectory prediction. The method is based on Clifford group equivariant neural networks and simplicial complexes. The method embeds scalar and vector features of nodes in the Clifford algebra and creates simplex features through geometric products. The method uses shared parameters for message passing across different simplex orders and ensures equivariance through Clifford group equivariant neural networks. The method is tested on various geometric tasks, including 5D convex hulls, CMU motion capture, MD17 atomic motion prediction, and NBA player trajectory prediction. The results show that the method outperforms existing models on these tasks. The method is also compared with other models, showing that it achieves outstanding scores across the board. The method is efficient and can be applied to various geometric tasks. The method is a significant advancement in geometric deep learning, combining the expressivity of Clifford group-equivariant layers with the topological intricacy of simplicial message passing.