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Clifford Group Equivariant Simplicial Message Passing Networks
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Proposes Clifford Group Equivariant Simplicial Message Passing Networks (CSMPNs), which embed simplices via geometric products of vertex features and perform parameter-shared, hypergraph-style simplicial message passing that is E(n)-equivariant.
Problem
Equivariant geometric graph networks are limited to standard message passing expressivity, while existing simplicial equivariant methods (EMPSN) manually initialize higher-order simplex features and rely on scalarization that updates vectors only by scaling.
Approach
Initialize simplex features as geometric products (and learned linear combinations) of vertex Clifford features, share one message network across simplex dimensions conditioned on source/target dimensions, and aggregate messages from all adjacency types so the update reduces to standard hypergraph message passing built from Clifford group-equivariant layers.
Key results
- On MD17 atomic motion, CSMPN (300K) achieves the best ADE/FDE on ethanol (4.44/6.30) and malonaldehyde (3.88/5.94), slightly ahead of CGENN, while CGENN is marginally better on aspirin (3.70 vs 3.82 ADE).
- On NBA trajectory prediction, CSMPN (200K) obtains ADE/FDE of 8.88/14.06 on attack and 6.44/9.22 on defense, outperforming DAG-Net and CGENN.
- Shared simplicial message passing reduces per-step time versus separate-parameter CSMPN (e.g., 0.09 vs 0.29 s/step on ethanol), though still slower than EMPSN and CEGNN.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use CSMPN instead of CGenn | MD17 atomic motion prediction · MD17 Aspirin | ADE: 3.70 -> 3.82 | Hurt |
| Use CSMPN instead of CGenn | MD17 atomic motion prediction · MD17 Aspirin | FDE: 5.63 -> 5.75 | Hurt |
| Use CSMPN instead of EMPSN | MD17 atomic motion prediction · MD17 Benzene | FDE: 1.12 -> 1.60 | Hurt |
| Use CSMPN instead of CGenn | MD17 atomic motion prediction · MD17 Ethanol | ADE: 4.53 -> 4.44 | Helped |
| Use CSMPN instead of CGenn | MD17 atomic motion prediction · MD17 Ethanol | FDE: 6.35 -> 6.30 | Helped |
| Use CSMPN instead of CGenn | MD17 atomic motion prediction · MD17 Malonaldehyde | ADE: 4.20 -> 3.88 | Helped |
| Use CSMPN instead of CGenn | MD17 atomic motion prediction · MD17 Malonaldehyde | FDE: 6.55 -> 5.94 | Helped |
| Use CSMPN instead of the strongest listed baseline | NBA player trajectory prediction · NBA Attack | ADE: 8.98 -> 8.88 | Helped |
| Use CSMPN instead of the strongest listed baseline | NBA player trajectory prediction · NBA Attack | FDE: 14.08 -> 14.06 | Helped |
| Use CSMPN instead of CGenn | NBA player trajectory prediction · NBA Defense | ADE: 6.64 -> 6.44 | Helped |
| Use CSMPN instead of CGenn | NBA player trajectory prediction · NBA Defense | FDE: 9.42 -> 9.22 | Helped |
| Share simplicial message-passing networks instead of using separate networks (ablation) | CSMPN · MD17 Aspirin | Seconds/Step: 0.37 -> 0.22 | Helped |
Surprising findings
- On MD17 benzene, the less expressive EMPSN outperforms CSMPN and CGENN, which the authors hypothesize is a good inductive bias for a rigid planar molecule.
- Shared CSMPN is still roughly 3-5x slower per step than CEGNN in the reported timing table.
Limitations
- Simplicial message passing increases computational cost relative to standard message passing.
- CSMPN performance is comparable to other methods in some tasks, with no clear winner across all benchmarks.
- Manual lifts require domain knowledge for simplex construction; experiments use manually chosen or clique/Vietoris-Rips complexes capped at dimension 2.
Experimental setup
Datasets: CMU Human Motion Capture MD17 STATS SportVU NBA dataset synthetic 5D convex hulls
Benchmarks: 5D convex hull volume CMU motion capture MD17 atomic motion (ADE/FDE) NBA player trajectory (ADE/FDE)
Metrics: ADE FDE RMSE inference time per step
Baselines: EMPSN CGENN EGNN TFN SE(3)-Transformer EqMotion DAG-Net GroupNet
Builds on: Clifford group equivariant neural networks EMPSN simplicial message passing networks EGNN
Closest papers by meaning
- Schur Nets: exploiting local structure for equivariance in higher order graph neural networks (NeurIPS 2024 · similarity 0.86)
- Are High-Degree Representations Really Unnecessary in Equivariant Graph Neural Networks? (NeurIPS 2024 · similarity 0.86)
- GLGENN: A Novel Parameter-Light Equivariant Neural Networks Architecture Based on Clifford Geometric Algebras (ICML 2025 · similarity 0.85)
- On the Expressive Power of Sparse Geometric MPNNs (ICLR 2025 · similarity 0.85)
- Demystifying Topological Message-Passing with Relational Structures: A Case Study on Oversquashing in Simplicial Message-Passing (ICLR 2025 · similarity 0.84)
- Simplicial Representation Learning with Neural $k$-Forms (ICLR 2024 · similarity 0.84)
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