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REVIEW 3 major objections 5 minor 5 cited by

Probing Equivariance and Symmetry Breaking in Convolutional Networks

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Symmetry, not scale, wins on geometry-heavy point-cloud tasks

desk verdict A useful controlled testbed and a mostly sound empirical comparison, but the headline capacity-scaling claim is not yet backed by the scaling experiment that matters. read the letter →

arxiv 2501.01999 v3 pith:BFLDLF65 submitted 2025-01-01 cs.CV cs.AIcs.LG

Sharvaree Vadgama , Mohammad Mohaiminul Islam , Domas Buracas , Christian Shewmake , Artem Moskalev , Erik Bekkers This is my paper · ORCID
classification cs.CVcs.AIcs.LG
keywords groupequivariancesymmetrybreakingregularconvolutionposition-orientationspacepointcloudsmolecularpropertyprediction3Dshapesegmentationhumanmotion
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper intervenes in the debate over whether explicit group equivariance still matters as models grow. It claims that when an equivariant model's symmetry matches the geometry of a 3D point-cloud task, the more constrained model outperforms less constrained alternatives, and that simply scaling channel capacity of the looser models does not eliminate the gap. It also claims that explicit symmetry breaking—conditioning on a global reference frame or pose—consistently improves performance, while feeding geometric inputs as scalars can help but only when aligned with task geometry. If right, the paper gives concrete model-selection guidance for molecular property prediction and generation, shape segmentation, and motion prediction.

What carries the argument

The load-bearing object is the regular group convolution over position-orientation space $\mathbb{R}^3 \times S^2$, the quotient space of SE(3) by the stabilizer SO(2). Convolution kernels are conditioned on three bijective geometric invariants—the projection of the neighbour displacement onto the orientation axis, the length of its orthogonal component, and the inner product of the two orientation vectors—which automatically satisfy the SO(2) axial-symmetry constraint. Around this backbone, Rapidash adds input/output flexibility to treat geometric quantities as scalars or vectors, and a global-frame input for symmetry breaking. The theoretical support is the spherical-Fourier equivalence to steerable tensor field networks, the universal approximation result built on it, and the pose-entropy argument that turns symmetry breaking into an expressivity gain.

What would settle it

Take a T(3)-equivariant $\mathbb{R}^3$ model and inflate its channels until its total parameter count and FLOPs match an $\mathbb{R}^3 \times S^2$ SE(3) model on the same task; if the $\mathbb{R}^3$ model then reaches or beats the $\mathbb{R}^3 \times S^2$ accuracy on QM9 or rotated ShapeNet, the claim that capacity cannot close the gap would fail. A simpler check is to ablate the orientation axis of the $\mathbb{R}^3 \times S^2$ model while keeping its compute budget fixed and compare.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that equivariance constraints should be chosen by task geometry, not minimized or maximized: within the Rapidash framework, SE(3)-equivariant regular group convolutions on position-orientation space $\mathbb{R}^3 \times S^2$ consistently beat translation-equivariant $\mathbb{R}^3$ convolutions on geometry-critical tasks such as QM9 property prediction and generation and CMU motion prediction, and inflated $\mathbb{R}^3$ models with matched total feature count and up to 2048 channels do not close the gap. The paper further establishes that pose-conditioned symmetry breaking, implemented by supplying a global frame as input, improves segmentation, generation, and motion results, and that this has a formal basis: standard invariant models behave as if pose has maximum entropy and therefore cannot represent orientation-dependent optimal functions, while pose-conditioned jointly invariant models can, with provable generalization gains. In addition, the paper shows that regular $\mathbb{R}^3 \times S^2$ group convolutions are Fourier-equivalent to steerable tensor field networks and inherit universal approximation.

Load-bearing premise

The empirical comparison assumes that the advantage of the $\mathbb{R}^3 \times S^2$ models over the inflated $\mathbb{R}^3$ models is explained by the equivariance constraint itself, not by the orientation axis adding effective parameters or compute that the $\mathbb{R}^3$ models were not matched on; the paper itself notes in Section 3.1 that the O-axis is a correlated, structured domain rather than independent channels.

Editorial extensions

If this is right

  • For geometry-critical point-cloud tasks, a default starting point is an SE(3)-equivariant $\mathbb{R}^3 \times S^2$ convolutional backbone rather than a translation-equivariant or plain-$\mathbb{R}^3$ one.
  • Channel inflation of weaker models is a limited substitute for the right inductive bias; observed saturation suggests architecture choice dominates capacity after a point.
  • Adding a global reference frame as symmetry-breaking input is a cheap, consistent accuracy lift across segmentation, generation, and motion tasks.
  • Equivariance pays off most in low-data regimes, so pose-aware equivariant models are especially attractive for small datasets.
  • Molecule generation on QM9 with stability metrics achieves strong results comparable to or better than existing specialized diffusion models.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • My extension: because the paper matches total features rather than parameter count and FLOPs, a stricter capacity-matched comparison would settle whether the orientation axis is just extra capacity; I would predict some, but not all, of the gap closes.
  • My extension: the pose-entropy argument suggests any task on near-symmetric inputs with a canonical reference, such as protein structure prediction or rigid-body docking, should benefit from explicit frame conditioning, which is directly testable.
  • My extension: the Fourier equivalence to steerable networks implies the conclusions should carry over to tensor-field architectures, provided their nonlinearities are matched to scalar-field activations.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript introduces Rapidash, a unified regular group convolutional architecture defined over R3 and R3×S2, and uses it to compare SE(3)-equivariant, T(3)-equivariant, equivariance-broken, and symmetry-broken variants on QM9 molecule prediction/generation, ShapeNet part segmentation/generation, CMU motion prediction, and ModelNet40 classification. The central empirical claims are that SE(3)-equivariant R3×S2 models outperform less constrained alternatives when aligned with task geometry, that explicit symmetry breaking via geometric reference frames improves performance, and that increasing channel capacity of R3 models does not fully close the performance gap. The theoretical sections formalize relations to steerable networks, universal approximation, and generalization of pose-conditioned models, but most formal statements are imported from prior work rather than newly proven.

Significance. If accepted, the manuscript would provide a valuable controlled comparison from a single architecture family, with error bars, data-efficiency curves, capacity inflation, and several diverse tasks. The explicit limitation statements, including the note that total-feature matching does not equate to capacity matching and the modest computational budget, are honest and useful. The main unresolved issue is that the paper's headline capacity-scaling claim is not tested on the task where the SE(3)-vs-T(3) gap is largest, and several 'state-of-the-art' statements rest on a narrow and outdated baseline set. The absence of released code also weakens the reproducibility of an otherwise extensive empirical study.

major comments (3)
  1. [Section 3.1, Table 1, Table 3, App. Table 5] The central claim that increasing representation capacity does not fully eliminate performance gaps is not tested on the task where the gap is largest. Table 3 and App. Table 5 report inflated C=2048 R3 variants only for ShapeNet part segmentation and CMU motion prediction; no analogous inflated T(3) R3 run is reported for QM9 property prediction or generation, even though Table 1 shows the largest advertised SE(3)-vs-T(3) gap there (MAE_mu 10.39 vs 22.11). On aligned ShapeNet the inflated T(3) R3 result (85.38) is effectively tied with the SE(3) R3×S2 result (85.44), and on aligned CMU the inflated T(3) R3 result (5.3) is close to the best R3×S2 value (4.88). The remaining gaps on rotated test inputs are out-of-distribution failures of a translation-equivariant model without rotation augmentation, not direct evidence about capacity saturation. Furthermore, Section 3.1 itself concedes that matching total features O×C against C' is not a true capacity match because the O-axis is a structured correlated domain, and the normalized epoch times in Table 3 differ by roughly an order of magnitude. Without a QM9 scaling experiment, the RQ2 conclusion in Section 6 is not settled; the manuscript should either provide such runs or explicitly restrict the claim to the domains where it is tested.
  2. [Section 5, Table 1] The 'state-of-the-art' claims are not supported by the comparisons as presented. The property-prediction baselines in Table 1 are only EGNN, DimeNet++, and SE(3)-Transformer, which are older methods, and no recent equivariant or non-equivariant baselines on QM9 are included; no code is released. The text also states that the best QM9 generation results are 'decisively achieved' by SE(3)-equivariant models, but Table 1 shows Discover % for the T3 R3 model (91.83±0.45) above the best R3×S2 SE(3) value (90.78±0.11), so the claim is not decisive on all reported generation metrics. Please either update the baseline set and statistical comparison or soften and qualify these claims to reflect what the table actually shows.
  3. [Section 3.1, Tables 1 and 2] The main SE(3)-versus-T(3) comparison conflates the equivariance constraint with the feature domain: the headline contrast is mostly R3×S2 against R3, so both the group constraint and the presence of the orientation axis change simultaneously. Table 2 includes a T(3)-equivariant R3×S2 variant (row 13) that is competitive on aligned ShapeNet (IoU 85.46) and CMU (MSE 4.73), but Table 1 has no T(3) R3×S2 baseline on QM9. Adding a T(3) R3×S2 model on QM9 and a capacity-matched T(3) R3 model would help isolate whether the observed benefits come from SE(3) weight sharing or from the extra structured orientation dimension; without this control, the equivariance-based interpretation in Section 6 is partly confounded.
minor comments (5)
  1. [Appendix F] The implementation details contain duplicated phrases 'we trained we trained' in two places; please fix these grammar errors.
  2. [Figure 1] The caption contains the typo 'Molecule Stabiilty' and the meaning of '(inflated)' is not defined in the caption; please clarify the axis label and the inflation protocol.
  3. [Tables 3 and 4] The table rows contain four IoU values per model without an in-table explanation; the caption should state explicitly which columns correspond to C=256 versus C=2048 and to aligned versus rotated test inputs.
  4. [References] The reference list contains two entries for Lawrence et al. with the same title, one marked 2025a (ICLR) and one marked 2025b (arXiv); if they are the same paper, they should be cited once with the appropriate venue.
  5. [Appendices B and Section 3.1] Proposition B.1 duplicates Proposition 3.1 almost verbatim; the paper should state the result once and refer to it from the other location.

Circularity Check

1 steps flagged · score 2.0 of 10

Empirical core is self-contained; only minor circularity is the universal-approximation claim inherited from the authors' own PΘNITA paper.

  1. self citation load bearing [Section 3.1, Proposition 3.2 (see also App. B.3, Corollary B.1)]
    "Proposition 3.2 (Universal Approximation for Rapidash). Rapidash, as an instance of message passing networks over R3 × S2 with message functions conditioned on the bijective invariant attributes (derived in [Bekkers et al., 2024, Thm. 1]), is an SE(3)-equivariant universal approximator. This specific universality follows from [Bekkers et al., 2024, Cor. 1.1], leveraging the sufficient expressivity of feature maps over R3 × S2."

    The paper presents a 'theoretical analysis' contribution, but the key expressivity theorem for Rapidash is not proved here. It is directly inherited from Bekkers et al. 2024, Cor. 1.1, a paper co-authored by two of the present authors (Vadgama and Bekkers). Thus the claim 'Rapidash is an SE(3)-equivariant universal approximator' reduces to a same-group citation rather than an independent derivation in this manuscript. This citation is load-bearing for the theoretical framing, including the assumption in Corollary 3.1 that pose-conditioned models can represent the optimal mapping. It is not load-bearing for the paper's central empirical conclusions, which rest on benchmark comparisons, so the circularity is minor.

full rationale

The paper's headline claims—that more constrained equivariant models outperform less constrained alternatives when aligned with task geometry, and that increasing capacity does not fully eliminate gaps—are supported by direct empirical comparisons on QM9, ShapeNet3D, CMU, and ModelNet40. These results are generated by training and evaluating model variants, not derived from the paper's own propositions. The generalization theory is imported from external work by Lawrence et al. (2025a/b) and Elesedy & Zaidi, and the regular-versus-steerable equivalence is attributed to external sources (Brandstetter et al., Cesa et al.) as well as the authors' prior work. The one notable self-citation is Proposition 3.2, where Rapidash's universal approximation is inherited from the same authors' PΘNITA paper; because that prior work is not machine-checked or otherwise independently verified in this manuscript, the theoretical expressivity claim is partly circular. However, that claim is a supporting motivation rather than the source of the empirical numbers. The capacity-scaling comparison (Sec. 3.1, Tab. 3) has a possible confound because inflated R3 models are not matched in FLOPs or parameter count, but the paper explicitly acknowledges that the O-axis is a structured, correlated domain, and this is a methodological limitation rather than a case of fitting a parameter and then predicting the same quantity. Overall, the central empirical derivation is self-contained and the circularity burden is low.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The paper's empirical claims rest on standard group-theoretic foundations and prior architectural results (mostly PΘNITA, Bekkers et al. 2024) and prior generalization theory (Lawrence et al. 2025). The main free choices are the spherical fiber discretization and hidden dimensions, neither swept comprehensively.

free parameters (2)
  • Fiber dimension on S2 = 8 orientations (R3×S2 models)
    Choice of spherical grid resolution for discretizing the orientation fiber; not swept; coarser grids would approximate SO(3) equivariance less accurately.
  • Hidden dimension C = 256, with inflated 2048 for R3 models
    Architectural capacity setting; the inflation experiments use these values but do not FLOP-match across base types.
assumptions (6)
  • domain assumption The invariant pair-wise attributes from Bekkers et al. 2024, Thm. 1 parametrize the full set of SE(3)-equivariant kernels on R3×S2
    Used in Eq. 3-4 to implement equivariant message passing; if this characterization is incomplete, the architecture's equivariance guarantee fails.
  • standard math Universal approximation of SE(3)-equivariant message passing over R3×S2 (Bekkers et al. 2024, Cor. 1.1; Dym and Maron 2020)
    Invoked for Proposition 3.2; not proved in this paper.
  • standard math Lawrence et al. 2025a, Thm. 6.1 generalization bound for jointly invariant pose-conditioned models
    Used for Proposition D.2; the paper applies the theorem rather than proving it.
  • domain assumption Data distribution P(X) is G-invariant and pose condition P(Z|X)=δ(Z−R) is equivariant
    Required for the generalization advantage of pose-conditioned models; not empirically verified on the datasets.
  • domain assumption Curie's principle: an equivariant map cannot produce a less symmetric output distribution from a more symmetric input
    Used in App. D.3-D.4 to motivate symmetry breaking; assumes deterministic equivariant maps and exact group actions.
  • domain assumption Task geometry alignment: QM9 and CMU motion are SE(3)-equivariant tasks, ShapeNet segmentation is invariant/equivariant with canonical frame, generation target is aligned
    Underlies the 'aligned with task geometry' interpretation of results; misclassification would weaken the conclusions.

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Cite this review

Pith. "Pith review of Probing Equivariance and Symmetry Breaking in Convolutional Networks." pith.science (2026). https://pith.science/paper/BFLDLF65

@misc{pith2026250101999,
  author       = {Pith},
  title        = {Pith review of: Probing Equivariance and Symmetry Breaking in Convolutional Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BFLDLF65}},
  note         = {Machine review of arXiv:2501.01999}
}
read the original abstract

In this work, we explore the trade-offs of explicit structural priors, particularly group equivariance. We address this through theoretical analysis and a comprehensive empirical study. To enable controlled and fair comparisons, we introduce \texttt{Rapidash}, a unified group convolutional architecture that allows for different variants of equivariant and non-equivariant models. Our results suggest that more constrained equivariant models outperform less constrained alternatives when aligned with the geometry of the task, and increasing representation capacity does not fully eliminate performance gaps. We see improved performance of models with equivariance and symmetry-breaking through tasks like segmentation, regression, and generation across diverse datasets. Explicit \textit{symmetry breaking} via geometric reference frames consistently improves performance, while \textit{breaking equivariance} through geometric input features can be helpful when aligned with task geometry. Our results provide task-specific performance trends that offer a more nuanced way for model selection.

Figures

Figures reproduced from arXiv: 2501.01999 by the authors.

Figure 2
Figure 2. Comparison of Rapidash model variants with different equivariance-breaking on Shapenet part segmentation and genera￾tion. (X%) indicates what portion of the training dataset the model was trained on. We also evaluate on aligned and rotated ver￾sions of the test set. To address this debate, we systematically investigate the impact of equivariant and non-equivariant models across a range of learning tasks involving ge… view at source ↗
Figure 3
Figure 3. Input variations to Rapidash with base space [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Smoothed IoU (Instance average) perfor￾mance curves over time for ShapeNet part segmen￾tation. The legend codes correspond to the effective equivariance and the percentage of the training dataset trained on [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figures from the paper (3 more)
Figure 6
Figure 6. Figure 6: Block design with base space R 3 × S 2 . Regular group convolutions over the full spacew SE(3) can be efficiently computed when the kernel is factorized via kc ′c(x, R) = k R 3 c (x)k SO(3) c (R)k (channel) c ′c , with c, c′ the row and column indices of the "channel m…
Figure 7
Figure 7. Figure 7: Depiction of an instance from CMU motion capture dataset. [PITH_FULL_IMAGE:figures/full_fig_p027_7.png]
Figure 8
Figure 8. Figure 8: Progression of tasks based on geometric complexities of each task. [PITH_FULL_IMAGE:figures/full_fig_p027_8.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. CP$^2$: Leveraging Geometry for Conformal Prediction via Canonicalization

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    Canonicalizing inputs before conformal prediction preserves coverage and shrinks prediction sets under rotation shifts, without retraining the underlying model.

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    Point clouds are lifted to Anti-de Sitter space and message passing uses the AdS proper distance, producing a network approximately equivariant under the full conformal group.

  3. Platonic Transformers: A Solid Choice For Equivariance

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    Platonic Transformers achieve exact equivariance to translations plus discrete Platonic-solid rotations by lifting features into multiple reference frames and sharing one RoPE attention across them, with a linear-time...

  4. On Equivariant Model Selection through the Lens of Uncertainty

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    Conformal and calibration uncertainty scores align with test error when choosing among pretrained equivariant models, while last-layer Laplace marginal likelihood does not.

  5. Quick ViTs: Speeding up Vision Transformers through Equivariance

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    D8-equivariant linear layers give ViTs a 5.33x FLOP reduction and 8x parameter reduction per layer, and hybrid octic ViTs match or slightly exceed ImageNet-1K accuracy while using about 40% fewer FLOPs.

Reference graph

Works this paper leans on

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    This is because the action of g ∈ SE(3) permutes these values on the fiber or spatially, but the activation acts on each scalar value independently

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    By the joint SO(3)-invariance of f, we have f (X, Z) = f (Z −1X, Z−1Z) = f (Z −1X, Id)

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