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AtlasD: Automatic Local Symmetry Discovery
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Introduces atlas equivariance as a formal notion of local symmetry and AtlasD, a pipeline that trains local chart predictors and learns Lie algebra generators and discrete cosets to which they are equivariant.
Problem
Existing symmetry discovery methods only find global symmetries, which can misrepresent the true symmetry of systems whose symmetries are local (e.g., on arbitrary manifolds or with heat sources breaking global symmetry).
Approach
Define local symmetry as atlas equivariance: localized predictors on each chart must be globally equivariant to a Lie group. AtlasD learns the Lie algebra basis and component-group coset representatives by optimizing equivariance error of chart predictors, with standard basis regularization and coset normalization, and uses the discovered group as a gauge group in gauge equivariant CNNs.
Key results
- On top-quark tagging, the discovered invariant metric has cosine correlation 0.9996 with the Minkowski tensor (LieGAN: 0.9975), and the discovered O+(1,3) group includes a parity coset.
- AtlasGNN (LorentzNet with discovered metric) reaches 0.939 accuracy and 0.9852 AUROC, close to LorentzNet with ground-truth symmetry (0.941 / 0.9862).
- On projected MNIST with 180-degree test rotations, the gauge equivariant CNN with discovered SO(2) gets 0.9381 accuracy versus 0.6975 for a vanilla CNN.
- On ClimateNet, the GL+(2) gauge model uses 111k parameters versus 766k for SO(2), with mean IoU 0.487 vs 0.490.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use the discovered SO(2) symmetry in a gauge equivariant CNN rather than a vanilla CNN | MNIST on Sphere pixel classification · MNIST on Sphere | Accuracy: 0.6975 -> 0.9381 | Helped |
| Use the AtlasD-discovered GL+(2) gauge group with a flat kernel instead of the SO(2) structure group | icoCNN ClimateNet segmentation · ClimateNet | Parameters: 766k -> 111k | Helped |
| Use the AtlasD-discovered GL+(2) gauge group with a flat kernel instead of the SO(2) structure group | icoCNN ClimateNet segmentation · ClimateNet | Mean IoU: 0.4896 -> 0.4865 | No clear effect |
| Use a 3-chart atlas instead of a 19-chart atlas (ablation) | AtlasD heat-equation symmetry discovery · Heat equation PDE | Symmetry recovery | Hurt |
| Replace standard basis regularization with cosine similarity (ablation) | AtlasD top-tagging generator discovery · Top Quark Tagging | Generator overlap: single such pair -> 21 pairs | Hurt |
| Remove coset normalization (ablation) | AtlasD top-tagging coset discovery · Top Quark Tagging | Coset discovery quality | Hurt |
| Remove coset normalization (ablation) | AtlasD PDE coset discovery · Heat equation PDE | Coset discovery quality | Hurt |
| Use a partially sheared chart (ablation) | AtlasD heat-equation symmetry discovery · Heat equation PDE | Generator quality | Hurt |
| Add the discovered GL+(2) gauge group with a flat kernel instead of using the SO(2) structure group | icoCNN ClimateNet segmentation · ClimateNet | Mean IoU: 0.4896 -> 0.4865 | Hurt |
| Use the discovered GL+(2) gauge group with a flat kernel instead of the SO(2) structure group | icoCNN ClimateNet segmentation · ClimateNet | Parameters: 766k -> 111k | Helped |
Surprising findings
- On ClimateNet, the discovered symmetry group is 4-dimensional (GL+(2)) rather than SO(2), and the flat-kernel model matches baseline performance with about 7x fewer parameters.
Limitations
- Requires a priori knowledge of a suitable atlas, which may not be available.
- Atlas equivariances describe only a subset of all manifold diffeomorphisms, so larger groups are not covered.
- Basis dimension k is chosen by heuristic and the standard basis regularization can yield duplicate generators.
Experimental setup
Datasets: Top Quark Tagging Reference Dataset synthetic heat equation dataset projected MNIST ClimateNet
Benchmarks: Top Quark Tagging Reference Dataset ClimateNet projected MNIST
Metrics: accuracy AUROC IoU cosine correlation
Baselines: LieGAN LieGG LorentzNet (w/o) EGNN LieGNN vanilla CNN
Builds on: LieGAN LieGG LorentzNet gauge equivariant CNN icoCNN
Closest papers by meaning
- Symmetry Discovery Beyond Affine Transformations (NeurIPS 2024 · similarity 0.87)
- Learning Infinitesimal Generators of Continuous Symmetries from Data (NeurIPS 2024 · similarity 0.87)
- Discovering Symmetry Breaking in Physical Systems with Relaxed Group Convolution (ICML 2024 · similarity 0.86)
- Explicit Discovery of Nonlinear Symmetries from Dynamic Data (ICML 2025 · similarity 0.86)
- Symmetry-Informed Governing Equation Discovery (NeurIPS 2024 · similarity 0.86)
- Improving Equivariant Networks with Probabilistic Symmetry Breaking (ICLR 2025 · similarity 0.85)
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