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Incorporating Arbitrary Matrix Group Equivariance into KANs
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Proposes EKAN, a KAN variant equivariant to arbitrary matrix groups via gated spline basis functions, equivariant linear weights, and a lift layer, outperforming EMLP, KAN, and CGENN on particle scattering, three-body, and top quark tagging with small data or few parameters.
Problem
Standard KANs use spline activations that do not respect symmetries of the task, so they fail to exploit known group equivariance; KANs also underperform on non-symbolic tasks.
Approach
Split each KAN layer into basis functions and linear weights; make the spline basis equivariant by gating non-scalar features with scalar gates passed through the basis functions, and solve for equivariant linear weights via the EMLP nullspace method, with a lift layer aligning the input to the dataset feature space.
Key results
- On particle scattering, EKAN-O(1,3) reaches about 3.8e-6 test MSE at 10^4 samples vs EMLP ~2.0e-4, often orders of magnitude better; EKAN with 10^3 samples achieves ~10% of EMLP's 10^4-sample MSE.
- On three-body problem, EKAN-SO(2) with 10^4.5 parameters achieves 1.12e-3 test MSE and matches baselines with 10^5.5 parameters, saving about 90% of parameters.
- On top quark tagging (n_comp=3), EKAN uses 34K parameters vs 133K for EMLP and reaches comparable accuracy (76.93% vs 77.12% at 10^4 samples), improving accuracy by 0.23%-6.44% on small datasets.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add Lorentz-group equivariance with EKAN instead of using EMLP | Particle scattering · Particle scattering, training set size 10^3 | test MSE: (3.30 \pm 0.86) \times 10^{-4} -> (2.01 \pm 1.93) \times 10^{-5} | Helped |
| Use EKAN-SO(2) instead of EMLP-SO(2) | Three-body problem · Three-body problem, 10^5 parameters | test MSE: (3.55 \pm 1.59) \times 10^{-3} -> (6.09 \pm 0.27) \times 10^{-4} | Helped |
| Use EKAN-SO+(1,3) instead of EMLP-SO+(1,3) | Top quark tagging, n_comp=3 · Top quark tagging, training set size 10^2 | test accuracy (%): 65.48 \pm 1.21 -> 71.92 \pm 0.88 | Helped |
| Use EKAN-SO+(1,3) instead of EMLP-SO+(1,3) | Top quark tagging, n_comp=3 · Top quark tagging, training set size 10^4 | test accuracy (%): 77.10 \pm 0.02 -> 76.93 \pm 0.02 | Hurt |
| Increase the number of observed jet constituents | Top quark tagging with EKAN-SO+(1,3) · Top quark tagging, training set size 10^4 | test accuracy (%): 76.93 \pm 0.02 -> 87.14 \pm 0.03 -> 89.30 \pm 0.12 | Helped |
| Use EKAN instead of EMLP with comparable equivariant performance and fewer parameters | Top quark tagging, n_comp=3 · Top quark tagging | number of parameters: 133K -> 34K | Helped |
Surprising findings
- KAN does not outperform MLP on the non-symbolic top quark tagging task, as expected by the authors' framing.
- EKAN's advantage diminishes with abundant data and when more jet constituents are observed (n_comp=10,20 show EKAN comparable to baselines, not superior, at larger n_comp).
Limitations
- Gating mechanisms can reduce expressive power of EKAN.
- Advantages diminish when data is abundant, so EKAN improves generalization on small data but not the expressive ceiling.
- Equivariant advantage shrinks with more observed jet constituents (n_comp=10, 20).
Experimental setup
Datasets: particle scattering (synthetic, EMLP generation process) three-body problem (HNN dataset) top quark tagging dataset
Benchmarks: particle scattering three-body problem top quark tagging
Metrics: test MSE test accuracy equivariant error number of parameters
Baselines: MLP MLP + augmentation EMLP KAN KAN + augmentation CGENN
Builds on: Kolmogorov-Arnold Networks EMLP gated equivariant networks Adan optimizer
Closest papers by meaning
- KAN: Kolmogorov–Arnold Networks (ICLR 2025 · similarity 0.89)
- On the expressiveness and spectral bias of KANs (ICLR 2025 · similarity 0.88)
- Equivariant Neural Tangent Kernels (ICML 2025 · similarity 0.87)
- Kolmogorov-Arnold Transformer (ICLR 2025 · similarity 0.87)
- Lie Group Decompositions for Equivariant Neural Networks (ICLR 2024 · similarity 0.87)
- Affine Equivariant Networks Based on Differential Invariants (CVPR 2024 · similarity 0.87)
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