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Lie Neurons: Adjoint-Equivariant Neural Networks for Semisimple Lie Algebras
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Proposes Lie Neurons, an adjoint-equivariant MLP-style network that takes semisimple Lie algebra elements (as R^K vectors) as input, generalizing Vector Neurons via the Killing form and adding Lie bracket and geometric channel mixing layers.
Problem
Building neural networks that are equivariant to the adjoint action of semisimple Lie groups on data living in Lie algebras, which existing equivariant networks (designed for vectors under left group action) do not handle.
Approach
Represent Lie algebra elements as coefficient vectors in R^K and use the negative Killing form (adjoint-invariant) for a ReLU-style nonlinearity and invariant layer, plus a Lie bracket nonlinearity (commutator with skip connection) and a geometric channel mixing layer built from outer products; with so(3) this reduces to Vector Neurons.
Key results
- On BCH formula regression, Lie Neurons reaches Frobenius error 6.9e-4 versus 2.6e-3 for EMLP and about 0.6 for Vector Neurons and e3nn.
- On Platonic solid classification, LN-LR + LN-LB achieves 99.62% accuracy and 99.61% on rotated test sets versus 95.76% / 36.54% for an MLP.
- On point cloud registration, Lie Neurons (Mixing) reaches 1.879 degrees average error versus 2.227 for Vector Neurons and 1.934 for Vector Neurons with mixing.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use Lie Neurons with the bracket nonlinearity instead of Vector Neurons | BCH regression on so(3) · BCH regression, Id and SO(3)-augmented test | Frobenius Error: 0.617 -> 6.9\times 10^{-4} (Id); 0.617 -> 6.9\times 10^{-4} (SO(3)) | Helped |
| Add geometric channel mixing (ablation) | Lie Neurons for free-rotating ISS dynamics · Multiple trajectories, 5 sec, Id | trajectory norm distance: 0.739 -> 0.005 | Helped |
| Add geometric channel mixing | Vector Neurons for ModelNet40 registration · ModelNet40 point cloud registration | Average Registration Error (deg): 2.227 -> 1.934 | Helped |
| Use Lie Neurons instead of MLP | Free-rotating ISS dynamics · Multiple trajectories, 5 sec, Id | trajectory norm distance: 0.428 -> 0.005 | Helped |
| Use adjoint-equivariant Lie Neurons instead of MLP | Platonic solid classification · Rotated pose | accuracy: 36.54\% -> 99.61\% | Helped |
| Train MLP with augmentation | Platonic solid classification · Original pose | accuracy: 95.76\% -> 81.47\% | Hurt |
| Use LN-LB instead of LN-LR | sl(3) invariant regression · Id test | MSE: 1.30 $\times$ 10$^{-3}$ -> 0.557 | Hurt |
| Use 2 LN-LB layers instead of 2 LN-LR layers | sl(3) equivariant regression · Id test | MSE: 0.213 -> $\mathbf{9.83\times10^{-10}}$ | Helped |
| Add the residual connection to LN-Bracket (ablation) | LN-LB · sl(3) invariant regression | MSE: 4.838 -> 0.558 | Helped |
| Add the residual connection to LN-Bracket (ablation) | LN-LB · sl(3) equivariant regression | MSE: 0.276 -> 9.6 \times 10^{-10} | Helped |
| Add the residual connection to LN-Bracket (ablation) | LN-LB · Platonic solid classification | accuracy: 0.967 -> 0.986 (original); 0.959 -> 0.979 (rotated) | Helped |
| Use Lie Neurons instead of MLP | sp(4) invariant regression · SP(4) test augmentation | MSE: 0.446 -> $\mathbf{2.70\times 10^{-4}}$ | Helped |
Surprising findings
- Vector Neurons and e3nn fail to converge on the BCH regression task, so the bracket nonlinearity is essential there.
- Without geometric mixing layers, the Lie Neurons dynamics model fails to converge on the ISS task.
Limitations
- LN-ReLU relies on a non-degenerate Killing form, so it does not apply to non-semisimple algebras and cannot use normalization as in Vector Neurons.
- The geometric channel mixing layer is only defined for so(n); extension to general semisimple Lie algebras is conjectured but not shown.
- Assumes a basis of the target Lie algebra is available; mapping standard sensor measurements into the Lie algebra is left as future work.
Experimental setup
Datasets: ModelNet40 synthetic sp(4) invariant function data synthetic BCH data synthetic ISS free-rotation trajectories synthetic Platonic solid homography data
Benchmarks: sp(4) invariant function regression BCH formula regression ISS dynamics learning ModelNet40 point cloud registration sl(3) Platonic solid classification sl(3) invariant/equivariant function regression
Metrics: mean squared error invariance error Frobenius error log error trajectory norm error (rad/s) registration error (deg) classification accuracy equivariance error
Baselines: MLP MLP Augmentation EMLP e3nn Vector Neurons Vector Neurons (Mixing) Neural ODE MLP
Builds on: Vector Neurons Killing form Neural ODE EMLP
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