{"id":"28b322ce-bc97-412a-9b9d-7765b3abe2aa","arxiv_id":"2501.01999","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A controlled empirical study with a unified architecture finds SE(3)-equivariant position-orientation convolutions outperform less constrained models on geometry-aligned tasks, and pose-based symmetry breaking gives consistent gains.","lead":"This paper introduces Rapidash, a unified group-convolutional architecture, and uses it to compare SE(3)-equivariant, translation-equivariant, and non-equivariant models on molecular, shape, and motion tasks. It reports that equivariance aligned with task geometry beats less constrained models, that more capacity does not close the gap, and that pose-conditioned symmetry breaking consistently helps.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The abstract's capacity-scaling claim is not tested on QM9, the task where the SE(3)-vs-T(3) gap is largest; Table 3/App. Tab. 5 inflate only ShapeNet and CMU R3 models, where inflated R3 T3 already nearly matches on aligned data.","rationale":"The paper's central empirical thesis is that stronger geometric priors win and that scaling a less-constrained model cannot erase the gap. The reader's weakest assumption points at capacity matching; I partially agree. In the existing scaling tables the inflated R3 model is not obviously under-powered: with C=2048 it has as many or more independent channels and roughly comparable epoch cost to the O=8, C=256 R3xS2 model. The real problem is coverage. The scaling study is run on ShapeNet and CMU, where on aligned data the inflated T3 R3 is already close to or tied with R3xS2, and where the residual gap is dominated by rotated test sets. QM9, where the gap is largest and the SOTA claim is made, has no capacity-scaling experiment. Since Section 3.1 concedes that O*C vs C' is not a clean capacity match, one cannot assume the QM9 gap is unscalable. This does not mean the paper's conclusion is false; it means the headline capacity claim currently rests on tasks that do not display the advertised unscalable gap. A targeted QM9 inflation experiment would settle it. If the gap persists there, the paper's main claims are substantially supported; if it closes, the abstract overclaims. Hence I recommend keeping the conditional verdict, with the QM9 scaling run as the condition.","tokens_in":27662,"tokens_out":11565,"duration_ms":115466,"concrete_test":"Run Table 1's QM9 property (mu, alpha, HOMO) and generation ablations with the internal-T3 R3 model at C=2048 (and, if budget allows, C=4096), keeping R3xS2 at C=256/O=8 and using the same training schedule, epochs, and diffusion sampler as the other Rapidash runs. If inflated R3 T3 MAE_mu approaches ~10.4 or generation molecule stability approaches ~93%, the capacity-scaling conclusion fails; if the gap persists at C=2048 or larger, the central claim is supported.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim that 'increasing representation capacity does not fully eliminate performance gaps' depends on showing that a well-scaled less-constrained model still trails. The paper's scaling evidence is Table 3 and App. Tab. 5: R3 models inflated from C=256 to C=2048 on ShapeNet part segmentation and CMU motion prediction. On aligned ShapeNet, inflated R3 T3 (85.38) is essentially tied with R3xS2 SE(3) (85.44); on aligned CMU, inflated R3 T3 (5.3) is close to R3xS2 (4.88). The remaining large gaps appear on rotated test inputs, where a T3 model without rotation augmentation is expected to fail; those are out-of-distribution robustness results, not evidence about capacity saturation. The largest advertised gap is QM9 property prediction (R3 T3 MAE_mu 22.11 vs R3xS2 10.39, Table 1), yet no C=2048 R3 T3 result is reported for QM9 prediction or generation; Fig. 5 only varies training time. Section 3.1 itself concedes that total-feature matching O*C vs C' is not a true capacity match because the O-axis is structured. Until an inflated R3 T3 is run on QM9, the possibility that matched/independent capacity closes that gap remains live, and the headline conclusion is not settled.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":28021,"tokens_out":6475,"duration_ms":62247,"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":[{"comment":"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.","section":"Section 3.1, Table 1, Table 3, App. Table 5"},{"comment":"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.","section":"Section 5, Table 1"},{"comment":"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.","section":"Section 3.1, Tables 1 and 2"}],"minor_comments":[{"comment":"The implementation details contain duplicated phrases 'we trained we trained' in two places; please fix these grammar errors.","section":"Appendix F"},{"comment":"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.","section":"Figure 1"},{"comment":"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.","section":"Tables 3 and 4"},{"comment":"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.","section":"References"},{"comment":"Proposition B.1 duplicates Proposition 3.1 almost verbatim; the paper should state the result once and refer to it from the other location.","section":"Appendices B and Section 3.1"}],"recommendation":"major_revision","confidential_remarks":"This is a useful empirical study with a sensible unified architecture, but the editorial bar for the headline claims requires either additional experiments (QM9 inflated baselines, T(3) R3×S2 on QM9, proper capacity/FLOP matching) or a careful narrowing of the conclusions. The 'state-of-the-art' wording should also be moderated. I would be willing to review a revision that addresses these points; the requested experiments are within the scope of the manuscript, so this is not a reject."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The real contribution here is Rapidash and the systematic comparison it enables. The theory is mostly imported—restatements of Bekkers et al., Dym and Maron, and Lawrence et al.—but the empirical study is the substance, and it is generally well done. The architecture cleanly varies SE(3) vs T(3) constraints, symmetry breaking via global frames, and equivariance breaking via scalar/vector inputs. The breadth across QM9, ShapeNet, CMU, and ModelNet40, with error bars, data-efficiency curves, and rotated test sets, is a solid package. The data-efficiency results and the consistent gains from explicit reference frames are genuinely useful findings.\n\nThe main soft spot is the capacity-scaling claim. The paper says increasing representation capacity does not fully eliminate performance gaps, but the scaling experiments only inflate R3 models on ShapeNet and CMU. On aligned data, the inflated R3 T3 models nearly tie the R3xS2 SE(3) models—the remaining gap is mostly on rotated inputs, which is an out-of-distribution robustness result, not evidence about capacity saturation. The largest advertised gap is on QM9 (R3 T3 MAE 22.11 vs R3xS2 10.39), but no inflated R3 T3 result is reported there. Section 3.1 itself concedes that matching O*C channels against C' is not a true capacity match because the orientation axis is structured. Until an inflated R3 T3 is run on QM9, the abstract's headline conclusion is not settled.\n\nA few secondary issues: the QM9 'state-of-the-art' claim rests on only three older baselines, and no code or data is released, which makes the SOTA framing hard to verify. The theoretical propositions are honest about being restatements, which is fine, but they add little beyond context.\n\nOverall, this is a serious empirical paper that will be useful to the equivariant deep learning community as a testbed and a reference point. It deserves a careful referee, but the referee should push for a QM9 scaling experiment, a clearer distinction between in-distribution and OOD gains, and a code release before the capacity claim is accepted. I would send it to review rather than desk reject, and I'd tell the authors to temper the abstract until the missing experiment is done.","headline":"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.","tokens_in":28530,"tokens_out":2146,"would_cite":true,"duration_ms":21522,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Symmetry, not scale, wins on geometry-heavy point-cloud tasks","keywords":["group equivariance","symmetry breaking","regular group convolution","position-orientation space","point clouds","molecular property prediction","3D shape segmentation","human motion prediction"],"falsifier":"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.","tokens_in":27530,"feed_emoji":"⚛️","tokens_out":5223,"duration_ms":49341,"temperature":0.7,"pith_summary":"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.","feed_headline":"Symmetry, not scale, wins on geometry-heavy point-cloud tasks","feed_subtitle":"A unified architecture shows pose-aware equivariant networks beat looser high-capacity models on molecules, shapes, and motion.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the $\\mathbb{R}^3 \\times S^2$ regular group convolution with bijective invariant attributes and the universality corollary that grounds Rapidash's expressivity.","marker":"[Bekkers et al., 2024]"},{"why":"Establishes universal approximation for rotation-equivariant point cloud networks, used to support Rapidash's universal approximation proposition.","marker":"[Dym and Maron, 2020]"},{"why":"Provides the probabilistic symmetry-breaking framework and the joint-equivariance generalization theorem that the paper's symmetry-breaking analysis builds on.","marker":"[Lawrence et al., 2025b]"},{"why":"Gives the provable generalization benefit of equivariant models that the paper extends to pose-conditioned jointly invariant models.","marker":"[Elesedy and Zaidi, 2021]"},{"why":"Supplies the QM9 generation metrics and the EDM diffusion baseline that Rapidash must match or beat on molecule generation.","marker":"[Hoogeboom et al., 2022]"},{"why":"Provides the EGNN baseline reference on QM9 property prediction and CMU motion prediction used in the comparisons.","marker":"[Satorras et al., 2021]"},{"why":"Frames the regular-versus-steerable distinction and the nonlinearity issue that motivates scalar signals on $\\mathbb{R}^3 \\times S^2$.","marker":"[Weiler and Cesa, 2019]"}],"fun_headline_variants":["Equivariance choice beats model size on geometric tasks","Task geometry dictates when equivariance helps most","Symmetry breaking boosts point-cloud nets beyond scale","Pose-aware models win where geometry rules","Design equivariance by task, not by maximization"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Equivariance choice beats model size on geometric tasks","Task geometry dictates when equivariance helps most","Symmetry breaking boosts point-cloud nets beyond scale","Pose-aware models win where geometry rules","Design equivariance by task, not by maximization"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000229,"raw_usage":{"total_tokens":1465,"prompt_tokens":921,"completion_tokens":544,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":537,"completion_tokens_details":{"reasoning_tokens":473}},"tokens_in":537,"tokens_out":544,"duration_ms":4829,"temperature":1.0,"reasoning_tokens":473,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T22:43:06.353592+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}