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REVISITING MULTI-PERMUTATION EQUIVARIANCE THROUGH THE LENS OF IRREDUCIBLE REPRESENTATIONS
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Revisits equivariant linear layer characterization for permutation groups and wreath products using irreducible representations and Schur's lemma, deriving DeepSets, 2-IGN, and DWS layers and a full characterization of non-Siamese wreath-equivariant layers.
Problem
Characterizing all linear equivariant layers for permutation groups, multi-permutation weight-space symmetries, and wreath products G wr S_k (unaligned sets), where prior work relied on parameter-sharing and assumed transitive actions.
Approach
Decompose the representation into irreducibles and apply Schur's lemma to get equivariant maps directly, instead of parameter-sharing bookkeeping; extend to wreath products without transitivity, showing s^2 extra non-Siamese layers where s is the number of trivial summands.
Key results
- On synthetic graph anomaly detection, SchurNet reaches 100.0% accuracy without noise and 97.0% with noise (eta=0.1), versus 97.5%/92.0% for DSS and 48%/28% for Siamese.
- On Wasserstein distance learning, SchurNet beats NProductNet in most settings (e.g., noisy-sphere-3 [300,500]: 0.1026 vs 0.1580), but loses in several noisy-sphere-6 and RNA-seq settings.
- On MLP weight-space alignment, SchurNet obtains lower reconstruction loss than Siamese on MNIST (0.251346 vs 0.262913) and CIFAR10 (1.7822 vs 1.7876).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add the full set of four non-Siamese layers instead of using Siamese layers only | Graph anomaly detection model · Graph anomaly detection, η = 0.0 | accuracy: 48% -> 100.0% | Helped |
| Add the full set of four non-Siamese layers instead of using Siamese layers only | Graph anomaly detection model · Graph anomaly detection, η = 0.1 | accuracy: 28% -> 97.0% | Helped |
| Add the full set of four non-Siamese layers instead of the single DSS layer | Graph anomaly detection model · Graph anomaly detection, η = 0.0 and η = 0.1 | accuracy: 97.5% -> 100.0%; 92.0% -> 97.0% | Helped |
| Add non-Siamese layers to NProductNet | Wasserstein distance model · noisy-sphere-3, both input-size settings; uniform [200, 300]; ModelNet-small; ModelNet-large; RNA-seq [300, 500] | relative absolute mean error: 0.0460 -> 0.0389; 0.1580 -> 0.1026; 0.1089 -> 0.1043; 0.0840 -> 0.0623; 0.1110 -> 0.0738; 0.1400 -> 0.0468; 0.1620 -> 0. | Helped |
| Add non-Siamese layers to NProductNet | Wasserstein distance model · noisy-sphere-6, both input-size settings; uniform 256; RNA-seq [20, 200] | relative absolute mean error: 0.0150 -> 0.0217; 0.0490 -> 0.0795; 0.0970 -> 0.0974; 0.0120 -> 0.0123 | Hurt |
| Add non-Siamese weight/bias self-mappings | Siamese weight-space alignment model · MNIST MLP weight alignment | Acc(↓): 1.5e-5 -> 1.25e-5 | Helped |
| Add non-Siamese weight/bias self-mappings | Siamese weight-space alignment model · MNIST MLP weight alignment | Loss (↓): 0.262913 -> 0.251346 | Helped |
| Add non-Siamese weight/bias self-mappings | Siamese weight-space alignment model · CIFAR10 MLP weight alignment | Acc(↓): 1.0e-4 -> 0.0 | Helped |
| Add non-Siamese weight/bias self-mappings | Siamese weight-space alignment model · CIFAR10 MLP weight alignment | Loss (↓): 1.7876 -> 1.7822 | Helped |
Surprising findings
- Siamese-only layers achieve only 48% accuracy on the graph anomaly task without noise, near chance for the task.
- NProductNet outperforms SchurNet in several settings, e.g., noisy-sphere-6 and RNA-seq at small set sizes.
Limitations
- The irreducible decomposition must be derived case by case, which is the main challenge of the approach.
- For Wasserstein experiments only the d^2 extra layers were added to an existing model and SchurNet does not win everywhere.
- Experiments use small synthetic tasks and few hyperparameter configurations.
Experimental setup
Datasets: MNIST CIFAR10 ModelNet RNA-seq synthetic graphs for anomaly detection synthetic Wasserstein distance datasets
Benchmarks: synthetic graph anomaly detection Wasserstein distance learning datasets (noisy-sphere, uniform, ModelNet, RNA-seq) MLP weight-space alignment on MNIST and CIFAR10
Metrics: accuracy relative absolute mean error reconstruction loss
Baselines: Siamese networks DSS NProductNet DWS-based weight alignment (Siamese)
Builds on: DeepSets 2-IGN DWSNets DSS
Closest papers by meaning
- Equivariance Everywhere All At Once: A Recipe for Graph Foundation Models (NeurIPS 2025 · similarity 0.89)
- Permutation Equivariant Neural Networks for Symmetric Tensors (ICML 2025 · similarity 0.89)
- A Characterization Theorem for Equivariant Networks with Point-wise Activations (ICLR 2024 · similarity 0.88)
- Learning symmetries via weight-sharing with doubly stochastic tensors (NeurIPS 2024 · similarity 0.88)
- A Tale of Two Symmetries: Exploring the Loss Landscape of Equivariant Models (NeurIPS 2025 · similarity 0.87)
- Graph Automorphism Group Equivariant Neural Networks (ICML 2024 · similarity 0.87)
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