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O$n$ Learning Deep O($n$)-Equivariant Hyperspheres
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Proposes Deep Equivariant Hyperspheres, O(n)-equivariant neurons built from spherical decision surfaces copied to regular n-simplex vertices, stackable across dimensions, with a two-point-sphere Gram-matrix invariant operator.
Problem
Learning deep features equivariant (and invariant) to O(n) reflections and rotations in arbitrary dimension n, where prior steerable spherical neurons were limited to 3D and it was unclear how to stack them or extend to other dimensions.
Approach
Generalize steerable 3D spherical neurons to n-D by replicating a learnable hypersphere across the n+1 vertices of a regular n-simplex (with a change-of-basis matrix M_n in O(n+1)), yielding O(n)-equivariant outputs that can be cascaded nD->(n+1)D; invariants come from a Gram matrix of equivariant features, interpreted as a two-point-sphere relation.
Key results
- On O(3) action recognition, DEH with the two-point invariant Delta reaches 87.36% accuracy vs. 82.92% with the edge variant, trained only on non-transformed skeletons.
- On O(5) regression, DEH achieves MSE 0.0007 with the Delta invariant (vs. 0.0033 for the Delta_E variant and 0.0084 for l2-norm), with 275 parameters.
- On O(5) convex hull volume prediction, DEH reports MSE 1.3166 with 49.8K parameters, outperforming the listed equi- and invariant baselines across training sizes.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use the invariant operator Δ instead of sum aggregation (ablation) | DEH · O(3) Action recognition | accuracy: 69.86 -> 87.36 | Helped |
| use the invariant operator Δ instead of the edge-weighted operator Δ_E (ablation) | DEH · O(3) Action recognition | accuracy: 82.92 -> 87.36 | Helped |
| use the invariant operator Δ instead of l2-norm (ablation) | DEH · O(5) Regression | MSE: 0.0084 -> 0.0007 | Helped |
| use the invariant operator Δ instead of the edge-weighted operator Δ_E (ablation) | DEH · O(5) Regression | MSE: 0.0033 -> 0.0007 | Helped |
| use the invariant operator Δ instead of l2-norm (ablation) | DEH · O(5) Convex hulls | MSE: 7.5398 -> 1.3166 | Helped |
| use the invariant operator Δ instead of the edge-weighted operator Δ_E (ablation) | DEH · O(5) Convex hulls | MSE: 1.3843 -> 1.3166 | Helped |
| use two equivariant layers instead of one (ablation) | DEH · O(3) Action recognition | accuracy: 70.05 -> 70.84 | Helped |
| use two equivariant layers instead of one (ablation) | DEH · O(5) Convex hulls | MSE: 2.1633 -> 2.1024 | Helped |
| remove learnable parameters from normalization while keeping bias (ablation) | DEH · O(5) Convex hulls | MSE: 2.1024 -> 2.0936 | Helped |
| remove the equivariant bias while keeping learnable normalization parameters (ablation) | DEH · O(5) Convex hulls | MSE: 2.1024 -> 2.0623 | Helped |
| remove both the equivariant bias and learnable normalization parameters (ablation) | DEH · O(5) Convex hulls | MSE: 2.1024 -> 2.1468 | Hurt |
| remove the bias (ablation) | DEH · O(5) Regression | MSE: 0.0011 -> 0.0007 | Helped |
Surprising findings
- Removing the equivariant bias improves average MSE on the convex hull task (2.0623 vs. 2.1024 with bias), and a single-layer DEH is already close to the two-layer model.
Limitations
- Experiments are small-scale (n=3 and n=5 tasks) and the architecture is not scaled to large point clouds; scalability would need a GNN wrapper such as DGCNN.
- Only invariant outputs are considered; equivariant-output tasks are not addressed.
- Translation equivariance (E(n)) is not covered directly and requires centering the input.
Experimental setup
Datasets: UTKinect-Action3D synthetic O(5) regression data (Gaussian samples) synthetic O(5) convex hull volume data
Benchmarks: UTKinect-Action3D (O(3)-transformed test) O(5) regression synthetic task O(5) convex hull volume prediction
Metrics: accuracy MSE inference time
Baselines: MLP MLP Aug VN GVP CGENN SGM Scalars EMLP
Builds on: TetraSphere steerable 3D spherical neurons VN-DGCNN Scalars conformal geometric algebra
Closest papers by meaning
- Efficient Generalized Spherical CNNs (ICLR 2021 · similarity 0.88)
- Reducing SO(3) Convolutions to SO(2) for Efficient Equivariant GNNs (ICML 2023 · similarity 0.88)
- Equivariance versus Augmentation for Spherical Images (ICML 2022 · similarity 0.87)
- Enabling Efficient Equivariant Operations in the Fourier Basis via Gaunt Tensor Products (ICLR 2024 · similarity 0.86)
- A Characterization Theorem for Equivariant Networks with Point-wise Activations (ICLR 2024 · similarity 0.85)
- Equivariant Neural Tangent Kernels (ICML 2025 · similarity 0.85)
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