Hierarchy-based Clifford Group Equivariant Message Passing Neural Networks

We introduce Hierarchy-based Clifford Group Equivariant Message Passing Neural Network (HCGE-MPNN), a Clifford group equivariant U-Net with skip-connection. Our method integrates the expressivity of Clifford group-equivariant layers with hierarchical pooling/unpooling in an encoder-decoder fashion. Our architecture admits major classes of pooling methods, sparse and dense pooling methods. Additionally, we introduce a Clifford group invariant projection operator, a generalized projection operator defined on the Clifford space, to make our end-to-end architecture equivariant to Clifford group action. Our method outperforms state-of-the-art (Clifford-)Equivariant MPNNs by up to 7\% in prediction MSE for Multi-Nbody datasets and 22\% for motion capture dataset.