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Equivariant Frames and the Impossibility of Continuous Canonicalization
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Shows continuous canonicalization and unweighted frames are impossible for S_n, SO(d), O(d) on point clouds, and introduces weighted "robust frames" that always preserve continuity, with constructions of moderate size.
Problem
Canonicalization and frame-averaging for enforcing group invariance/equivariance often destroy continuity of the input function, causing lack of robustness; existing frames are large (e.g., n! for permutations).
Approach
Proves topological impossibility results (fundamental group, homology, Borsuk-Ulam) for continuous canonicalization and continuity-preserving unweighted frames, then defines weighted, weakly equivariant, continuous "robust frames" whose invariant projections always preserve continuity, and constructs polynomial-size robust frames for S_n, SO(d), O(d).
Key results
- Robust frames for S_n reach 75.5-88.4% test accuracy with 1-25 samples, versus 85.6% for discontinuous sorting canonicalization.
- Sampled Reynolds operator reaches only 21.0-22.6% accuracy, close to no-invariance (25.5%).
- Permutation robust frame on the distinct-point set uses cardinality n(d-1) and is shown to be optimal via a matching lower bound.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Enforce permutation invariance with robust frames instead of discontinuous x-axis sorting | MLP on MNIST point clouds · MNIST point-cloud classification, test | accuracy (%): 85.6 -> 75.5 / 85.6 / 87.1 / 88.4; 85.6 -> 74.2 / 85.9 / 87.6 / 88.7 | Mixed |
| Enforce permutation invariance with sampled Reynolds averaging rather than no invariance | MLP on MNIST point clouds · MNIST point-cloud classification, test | accuracy (%): 25.5 -> 21.0 / 22.4 / 22.6 / 22.6 | Hurt |
| Increase inference samples for robust frames | Robust Frames (Sec. 4.1 and Sec. 4.2) on the MLP · MNIST point-cloud classification, test | accuracy (%): 75.5 / 85.6 / 87.1 / 88.4; 74.2 / 85.9 / 87.6 / 88.7 | Helped |
Surprising findings
- Sorting-based canonicalization (discontinuous) is unavoidable in some cases: no continuous canonicalization exists for S_n, SO(d) with n>=d, O(d) with n>d.
- Unweighted frames for SO(2) admit no finite-cardinality continuity-preserving version.
Limitations
- Robust frames for all of R^{d x n} with S_n are large (O(n^{2(d-1)}) cardinality) and need sampling in practice.
- Experiments are a single small toy MNIST point-cloud task; only S_n studied empirically.
- Optimality of SO(d) and O(d) frame constructions is not established.
Experimental setup
Datasets: MNIST
Benchmarks: MNIST point-cloud classification (2D, 100 points)
Metrics: test accuracy (%)
Baselines: discontinuous canonicalization (sorting along x-axis) Reynolds operator no invariance
Builds on: frame averaging canonicalization group averaging (Reynolds operator)
Closest papers by meaning
- A Canonicalization Perspective on Invariant and Equivariant Learning (NeurIPS 2024 · similarity 0.91)
- Equivariance via Minimal Frame Averaging for More Symmetries and Efficiency (ICML 2024 · similarity 0.90)
- Generalization Bounds for Canonicalization: A Comparative Study with Group Averaging (ICLR 2025 · similarity 0.89)
- Equivariance by Contrast: Identifiable Equivariant Embeddings from Unlabeled Finite Group Actions (NeurIPS 2025 · similarity 0.85)
- Learning symmetries via weight-sharing with doubly stochastic tensors (NeurIPS 2024 · similarity 0.84)
- Improving Equivariant Networks with Probabilistic Symmetry Breaking (ICLR 2025 · similarity 0.83)
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