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Equivariance via Minimal Frame Averaging for More Symmetries and Efficiency
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Proposes Minimal Frame Averaging (MFA), which builds provably minimal, exactly equivariant frames via canonicalization and stabilizers, extending frame averaging to Lorentz, unitary, linear and permutation groups.
Problem
Frame averaging for equivariance is costly when frames are large, and sampling-based variants sacrifice exact equivariance; existing frames are also limited to few groups.
Approach
Define minimal frames as coset of the stabilizer of a canonical form; derive canonicalization via a generalized QR decomposition on induced G-sets (using a pseudo-inner product) so minimal frames exist for O(1,d-1), U(d), GL, SL, and S_n, giving exact equivariance with one backbone forward pass.
Key results
- MFA-GNN achieves MSE 0.0036 on the n-body task versus 0.0054 for FA-GNN, with inference time 0.0023 s vs 0.0041 s.
- MFA-MLP on GRAPH8C achieves 0.00 invariance error at 1.10 ms inference, versus 0.09 x10^-2 error for FA-MLP-256 at 28.09 ms.
- On OC20 IS2RE, MFAENet has invariance error 8.809e-6 versus 6.199e-2 for FAENet, while FAENet has lower average MAE (0.5679 vs 0.5691).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| apply MFA frame averaging to the FA-GNN backbone | FA-GNN · n-body simulation | MSE: 0.0054 -> 0.0036 | Helped |
| apply MFA frame averaging to the FA-GNN backbone | FA-GNN · n-body simulation | Inference Time (s): 0.0041 -> 0.0023 | Helped |
| apply MFA frame averaging | MinkGNN · top tagging | Accuracy: 94.2136 -> 94.2178 | Helped |
| apply MFA frame averaging | MinkGNN · top tagging | AUC: 98.68 -> 98.69 | Helped |
| use MFA instead of FA | FAENet · OC20 validation split, IS2RE Direct All | ID-MAE (eV): 0.5446 -> 0.5437 | Helped |
| use MFA instead of FA | FAENet · OC20 validation split, IS2RE Direct All | Average-MAE (eV): 0.5679 -> 0.5691 | Hurt |
| use MFA instead of FA | FAENet · OC20 validation split, IS2RE Direct All | Invariance Error (eV): 6.199$\times$10\textsuperscript{--2} -> 8.809$\mathbf{\times}10$\textsuperscript{--6} | Helped |
| use MFA instead of FA | FAENet · OC20 validation split, IS2RE Direct All | Throughput (samples per second): 3863.3 -> 3919.8 | Helped |
| apply MFA frame averaging | MLP · GRAPH8c | Pair separation failures: 4755 -> 0 | Helped |
| apply MFA frame averaging | MLP · EXP | Pair separation failures: 600 -> 0 | Helped |
| use exact MFA instead of sampling-based FA with 256 samples | MLP · GRAPH8C | Invariance Error ($\times 10^{-2}$): 0.09 -> 0.00 | Helped |
| use MFA instead of sampling-based FA with 1 sample | MLP · GRAPH8C | Inference Time (ms): 2.20 -> 1.10 | Helped |
Surprising findings
- On OC20, FAENet achieves a lower out-of-domain MAE than MFAENet, which the authors attribute to the large data volume making exact equivariance less vital.
- On top tagging, MFA gives only a marginal gain (accuracy 94.2178 vs 94.2136), since the large training set lets the non-invariant backbone learn symmetries.
Limitations
- Canonicalization via the generalized QR decomposition may be discontinuous.
- Frame size is lower-bounded by the stabilizer, which can be large for highly symmetric objects such as fully-connected graphs.
- Graph canonical labeling can have factorial worst-case time complexity for highly symmetric graphs.
Experimental setup
Datasets: n-body (E(3) problem) top tagging dataset OC20 GRAPH8c EXP convex hull 5-dimensional dataset synthetic point clouds
Benchmarks: n-body top tagging OC20 IS2RE GRAPH8c EXP convex hull
Metrics: MSE accuracy AUC MAE invariance/equivariance error inference time throughput
Baselines: FA (frame averaging) SFA (stochastic frame averaging) SE(3)-Transformer TFN EGNN SEGNN CN-GNN CGENN FAENet PointNet GCN GIN
Builds on: frame averaging group averaging Canonicalization (Kaba et al.) FAENet SE(3)-Transformer
Closest papers by meaning
- Equivariant Frames and the Impossibility of Continuous Canonicalization (ICML 2024 · similarity 0.90)
- A Canonicalization Perspective on Invariant and Equivariant Learning (NeurIPS 2024 · similarity 0.90)
- Learning symmetries via weight-sharing with doubly stochastic tensors (NeurIPS 2024 · similarity 0.87)
- Probing Equivariance and Symmetry Breaking in Convolutional Networks (NeurIPS 2025 · similarity 0.86)
- Improving Equivariant Networks with Probabilistic Symmetry Breaking (ICLR 2025 · similarity 0.86)
- A Tale of Two Symmetries: Exploring the Loss Landscape of Equivariant Models (NeurIPS 2025 · similarity 0.85)
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