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Lie Group Decompositions for Equivariant Neural Networks
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Proposes a framework for building equivariant convolutional networks over non-compact, non-abelian matrix Lie groups (GL+(n,R), SL(n,R), and affine groups R^n ⋊ G) via Haar-measure factorization and a Cartan/polar decomposition that avoids the limits of the group exponential.
Problem
Equivariant networks for non-compact, non-abelian Lie groups are hard because the group exponential is not surjective and Haar-measure sampling is difficult, so standard harmonic/exponential-map methods do not apply to affine groups.
Approach
Decompose GL+(n,R) and SL(n,R) via the polar/Cartan decomposition A = S^{1/2}R into an SPD (or unit-det SPD) factor and an SO(n) factor, so the Haar measure factorizes as a product of invariant measures that can be sampled independently; a global parametrization map Phi^{-1}(A) = (1/2 logm(AA^T), expm(-1/2 logm(AA^T))A) is used to build Lie-algebra kernels for affine-equivariant convolutions.
Key results
- R^2 ⋊ SL(2) affine-equivariant model reaches 98.5% affNIST accuracy after training on MNIST, outperforming affConv (95.08%) and capsule networks (VB CapsNet 98.1%)
- R^2 ⋊ GL+(2) reaches 97.4% on affNIST and 97.71% on homNIST; R^2 ⋊ SL(2) reaches 98.3% on homNIST vs affConv 95.71%
- The SL(2) model needs 10 Monte Carlo samples versus 100 for affConv
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use volume-preserving affine group R^2 ⋊ SL(2) instead of R^2 ⋊ GL+(2) | affine-equivariant CNN · affNIST | classification accuracy (%): 98.5 vs 97.4 | Helped |
| Model with R^2 ⋊ SL(2) group equivariance outperforms prior equivariant models and Capsule Networks | affine-equivariant CNN · affNIST | classification accuracy (%): 98.5 vs 98.1 (best CapsNet) | Helped |
| Monte Carlo sampling reduced to 10 samples instead of 100 used by affConv | affine-equivariant CNN · affNIST | classification accuracy (%): 98.5 vs 95.08 (affConv, 100 samples) | Helped |
| Model with R^2 ⋊ SL(2) equivariance outperforms affConv on homography transformations | affine-equivariant CNN · homNIST | classification accuracy (%): 98.3 vs 95.71 | Helped |
Surprising findings
- The GL+(2) model performs slightly worse than the volume-preserving SL(2) model, attributed to affNIST containing little scaling
Limitations
- Equivariance holds only in expectation due to Monte Carlo approximation of the convolution integral
- Experiments are preliminary and limited to small MNIST-based affine/homography benchmarks
- Framework is primarily theoretical and currently limited to matrix Lie groups with finitely many connected components
Experimental setup
Datasets: MNIST affNIST homNIST
Benchmarks: affNIST homNIST
Metrics: classification accuracy
Baselines: affConv VB CapsNet RU CapsNet affine CapsNet Equivariant CapsNet
Builds on: group equivariant CNNs Lie group convolutional networks (affine convolution) Capsule Networks
Closest papers by meaning
- Lie Algebra Canonicalization: Equivariant Neural Operators under arbitrary Lie Groups (ICLR 2025 · similarity 0.91)
- Affine Equivariant Networks Based on Differential Invariants (CVPR 2024 · similarity 0.90)
- A Tale of Two Symmetries: Exploring the Loss Landscape of Equivariant Models (NeurIPS 2025 · similarity 0.90)
- Equivariance by Contrast: Identifiable Equivariant Embeddings from Unlabeled Finite Group Actions (NeurIPS 2025 · similarity 0.90)
- Learning symmetries via weight-sharing with doubly stochastic tensors (NeurIPS 2024 · similarity 0.89)
- A Characterization Theorem for Equivariant Networks with Point-wise Activations (ICLR 2024 · similarity 0.89)
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