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MatrixNet: Learning over symmetry groups using learned group representations
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Proposes MatrixNet, a sequence model that learns matrix representations of group generators (matrix exponential blocks over signed one-hot encodings) plus a relation loss, to predict order in finite groups and Jordan–Hölder multiplicities over the braid group B3.
Problem
Choosing feature representations for learning functions over symmetry groups; fixed predefined group representations may be suboptimal, and learned models often fail to respect group axioms and relations or generalize to longer words.
Approach
Learn a matrix representation of free-group generators via matrix-exponential blocks (so inverses and identity are built in), multiply generator matrices along the word to get a group-element matrix, and enforce group relations with an auxiliary loss rather than data augmentation.
Key results
- On S10 order prediction, MatrixNet-LN and MatrixNet-MC reach 99.4% and 100% accuracy, versus 77.3% for LSTM and 87% for the fixed-representation ablation.
- On B3 Jordan-Hölder multiplicities, MatrixNet-NL reaches 99.9% average accuracy and MSE 6.4e-4 at 100 epochs, versus 93.0% for LSTM and 89.1% for MLP.
- MatrixNet-LN and MatrixNet-NL keep near-zero MSE on braid words of length 8-10 beyond the training length, while MatrixNet and MatrixNet-MC show explosive MSE growth.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| learn the group representation rather than use a fixed permutation-matrix representation (ablation) | MatrixNet on S10 order prediction · S10 order prediction | accuracy: 87 ± 2 -> 100 ± 0 | Helped |
| replace the single matrix-block parameter matrix with a two-layer linear network (ablation) | MatrixNet · S10 order prediction | accuracy: 100 ± 0 -> 99.4 ± 0.4 | No clear effect |
| use a block-diagonal matrix block (ablation) | MatrixNet · S10 order prediction | accuracy: 100 ± 0 -> 100 ± 0 | No clear effect |
| add a nonlinearity to the matrix block (ablation) | MatrixNet · S10 order prediction | CE Loss (10^-2): 0.02± 0.01 -> 0.0003± 0 | Helped |
| replace the single matrix-block parameter matrix with a two-layer linear network (ablation) | MatrixNet · B3 categorical braid action | MSE Epoch 100: 0.082 ± 0.034 -> 0.001 ± 0.001 | Helped |
| use a block-diagonal matrix block (ablation) | MatrixNet · B3 categorical braid action | MSE Epoch 100: 0.082 ± 0.034 -> 0.014 ± 0.006 | Helped |
| add a nonlinearity to the matrix block (ablation) | MatrixNet · B3 categorical braid action | MSE Epoch 100: 0.082 ± 0.034 -> \mathbf{6.4}\textbf{e--}\mathbf{4} \mathbf{\pm 3.6}\textbf{e--}4 | Helped |
| replace the single matrix-block parameter matrix with a two-layer linear network (ablation) | MatrixNet · B3 length extrapolation | Relational Error: 13.63 ± 6.83 -> 0.071 ± 0.018 | Helped |
| use a block-diagonal matrix block (ablation) | MatrixNet · B3 length extrapolation | Relational Error: 13.63 ± 6.83 -> 2.58 ± 2.10 | Helped |
| add a nonlinearity to the matrix block (ablation) | MatrixNet · B3 length extrapolation | Relational Error: 13.63 ± 6.83 -> 0.066 ± 0.009 | Helped |
| increase braid-word length beyond the training range | MatrixNet and MatrixNet-MC · B3 length extrapolation | MSE | Hurt |
| increase braid-word length beyond the training range | MatrixNet-LN and MatrixNet-NL · B3 length extrapolation | MSE | No clear effect |
Surprising findings
- The fixed permutation-matrix representation ablation performs worse than the plain MLP on S10 order prediction, suggesting learned representations help.
- The simplest MatrixNet variant has high relational error (13.63) and poor length extrapolation, showing that relation-respecting representations matter.
Limitations
- Restricted to finitely presented discrete groups; Lie groups would need Lie algebra generators.
- Relations are enforced via auxiliary loss, so the homomorphism property is only approximate.
- Some variants show training instability and high MSE variance during length extrapolation.
Experimental setup
Datasets: S10 order prediction dataset (500k sampled words) B3 Jordan-Hölder multiplicity dataset (47,831 examples up to length 8)
Benchmarks: S10 order prediction S12, C11x...xC15, S5^4 order prediction B3 Jordan-Hölder multiplicity prediction length extrapolation on braid words of length 8-10
Metrics: cross-entropy loss accuracy MSE relational error
Baselines: MLP LSTM Transformer Fixed representation (permutation matrices)
Builds on: group equivariant neural networks Seidel-Thomas spherical twists Rouquier-Zimmermann theorem
Closest papers by meaning
- Learning symmetries via weight-sharing with doubly stochastic tensors (NeurIPS 2024 · similarity 0.89)
- Lie Group Decompositions for Equivariant Neural Networks (ICLR 2024 · similarity 0.88)
- Graph Automorphism Group Equivariant Neural Networks (ICML 2024 · similarity 0.88)
- Improving Equivariant Networks with Probabilistic Symmetry Breaking (ICLR 2025 · similarity 0.87)
- A Tale of Two Symmetries: Exploring the Loss Landscape of Equivariant Models (NeurIPS 2025 · similarity 0.87)
- Compact Matrix Quantum Group Equivariant Neural Networks (ICML 2025 · similarity 0.87)
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