{"id":"8828d24b-1c17-4230-a270-1988805d3e52","arxiv_id":"2507.09753","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Rotation-augmented CNNs learn equivariance easily for denoising and prediction, but only large models keep generation outputs invariant to seed rotations, and their latent codes do not identify rotated molecules as the same object.","lead":"This study tests whether molecule-generating neural networks need built-in rotation symmetry, or can learn it from rotated training data. It finds that augmented CNNs match equivariant models for denoising and property prediction, but need much larger models to keep outputs stable when the seed molecule is rotated during generation.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The equivariance metric in Eq. (4) is applied to voxel grids, but the paper never specifies the rotation operator, and interpolation-based rotations are not orthogonal—so the measured 'equivariance error' may be dominated by discretization artifacts rather than model behavior.","rationale":"The reader's weakest_assumption is that voxel rotation discretization artifacts could confound all equivariance measurements. I agree with that assessment and consider it the single most load-bearing concern, because it attacks the internal validity of every quantitative claim in Sections 4.1 and 4.2. The paper's headline contribution is an empirical study of learned equivariance; if the rotation operator is not faithfully orthogonal, then both the theoretical decomposition and the empirical metric lose their meaning. This is a correctness risk, not a disagreement with consensus. The concern is concrete: one needs to measure the discretization floor on an oracle and compare it to the reported model errors. If the oracle error is tiny, the paper's conclusions survive; if not, the central claim is not currently supported. I have not raised the capacity mismatch or lack of significance tests as the primary issue because, while real, those would only weaken the comparison between specific models; an uncharacterized rotation operator undermines the very definition of equivariance used throughout. The reader's verdict is CONDITIONAL, and this concern is precisely a condition that must be checked; hence no change to the verdict is needed.","tokens_in":14115,"tokens_out":6626,"duration_ms":80604,"concrete_test":"On a sample of 100 GEOM-Drugs molecules, compute the Eq. (4) equivariance error for the perfect oracle denoiser f(x) = x, using the paper's exact rotation pipeline (the same interpolation method and grid size). Also compute R_θ R_{−θ} − I in operator norm on the voxel space for the angles used in the paper. If the oracle's median equivariance error is comparable to the smallest model errors reported in Figures 2–5, or if the operator norm of R_θ R_{−θ} − I exceeds the model-to-model differences in those figures, then the equivariance metric is dominated by discretization rather than by model equivariance.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central quantitative tool is Theorem 3.2, Eq. (3), which is exact only when R is an orthogonal transformation acting on the function's input vector space. For voxelized molecules, an arbitrary rotation is realized by resampling a 64^3 Cartesian grid; the paper never states how this is done. Such a resampling operator is typically not orthogonal, and it need not satisfy the group law R(θ1)R(θ2) = R(θ1+θ2). If R is not orthogonal, the step ||R(g_R − y)||² = ||g_R − y||² in the MSE derivation fails, so Eq. (3) is not exact for the actual transformation used. Consequently, the 'equivariance error' in Eq. (4) conflates the model's failure to commute with R with the interpolation error introduced by R itself. The same confound enters the generation experiments of Section 4.2, where a rotated seed grid may be a distorted version of the original molecule before any model is applied. This does not merely shift numbers; it makes the reported errors—and the claim that only the 111M model achieves generative equivariance—uninterpretable without a characterization of the rotation operator. Even the E3NN baseline is affected, since it also operates on voxel grids and is evaluated through the same approximate rotation.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper asks whether explicitly equivariant GNNs are necessary for 3D molecular generation by comparing a voxel-based CNN (VoxMol) with an E3NN equivariant baseline. It derives a loss decomposition that separates prediction error from equivariance error, then empirically evaluates reconstruction, seeded generation, and property prediction on the GEOM-Drugs dataset. The main findings are that CNNs trained with rotation augmentation easily learn reconstruction equivariance, but that generative equivariance under rotated seed molecules is robust only for the full-size VoxMol model, while smaller, un-augmented, or data-limited models diverge. The paper concludes with a hypothesis that latent-alignment auxiliary losses could improve robustness without large models.","tokens_in":14342,"tokens_out":4408,"duration_ms":49545,"significance":"If the empirical claims were fully supported, this would be a valuable systematic study of learned versus built-in equivariance in molecular generative models, with practical implications for model selection and scaling. The loss decomposition in Theorem 3.2 is a useful conceptual tool, and the use of a public dataset together with ablations over model size, data fraction, and training duration is a methodological strength. However, the central quantitative claims are currently weakened by an unspecified and potentially non-orthogonal rotation operator on voxel grids, a mismatched equivariant baseline (0.5M versus 111M parameters), and a lack of statistical testing on the generation differences.","major_comments":[{"comment":"The decomposition in Eq. (3) is exact only when R is an orthogonal transformation of the input vector space, since it uses the step ||R(g_R - y)||^2 = ||g_R - y||^2. The experiments in Sections 4.1 and 4.2 apply rotations to 64^3 voxel grids, but the rotation operator is never specified. Resampling a Cartesian grid by a general rotation is not orthogonal and generally does not satisfy the group law R(θ1)R(θ2)=R(θ1+θ2), so the equality behind Eq. (3) does not hold for the actual transformation used. Consequently, the equivariance error in Eq. (4) conflates model non-equivariance with voxel discretization and interpolation artifacts, and the same confound enters the generation experiments of Section 4.2. Please specify the rotation implementation; if arbitrary angles are used, either rotate continuous atomic coordinates before voxelization, or restrict evaluation to exact voxel-preserving rotations (such as multiples of 90 degrees), and quantify the interpolation error separately.","section":"Section 3.1, Eqs. (3)-(4)"},{"comment":"The abstract claims that CNNs can 'match the performance of equivariant models,' but the only equivariant baseline, E3NN, has 0.5M parameters versus 111M for VoxMol and is explicitly reported as not matching VoxMol on denoising or generation metrics (Appendix A.1). Thus the experiments support at most that a small equivariant model underperforms a much larger augmented CNN, not that equivariant models are unnecessary at comparable capacity. Please either train an equivariant model of comparable capacity, or restrict the claim to the baselines actually evaluated.","section":"Section 4.1, Abstract, Appendix A.1"},{"comment":"The assertion that smaller, un-augmented, or data-limited models 'diverge, losing generative equivariance' rests on differences that appear comparable to the reported standard deviations. For example, Table 1 shows VoxMol (No Aug.) with stable 66±25 unrotated versus 61±23 rotated, and uniqueness 75±19 versus 83±17; the KL divergences in Table 2 (0.01-0.07) are reported without confidence intervals. No statistical test is provided. Please report paired tests (e.g., bootstrap or permutation) over the 100 seed molecules, or otherwise show that the rotated/unrotated differences exceed within-model variability.","section":"Section 4.2, Tables 1, 2"}],"minor_comments":[{"comment":"There are several typos and formatting inconsistencies: 'processnig' in Section 1, 'demostrating' in the caption of Figure 6, and inconsistent spacing in 'V oxMol' throughout the text and tables.","section":"Throughout"},{"comment":"The proof sketch says 'This follows by a second-order Taylor expansion,' but the regularity conditions on the loss l are not stated. Please specify the differentiability assumptions or note that the approximation is informal.","section":"Section 3.1, Theorem 3.2"},{"comment":"The derivation of p(Ry)=p(y) uses invariance of the Lebesgue measure under rotations; it would be clearer to state explicitly that det R = 1 implies the Jacobian factor is one before performing the substitution x = R x~.","section":"Section 3.2, Eq. (7)"},{"comment":"The metrics 'stable mol' and 'stable atom' are used in Tables 1 and 4 but are not defined in the main text; please define them or cite the original definitions.","section":"Section 4.2"},{"comment":"The description of the second generation experiment says the seed molecule is 'noised with the same noise' as the rotated experiment, but the procedure for fixing the noise is not fully specified. Please describe how the random noise is controlled across runs.","section":"Section 4.2"}],"recommendation":"major_revision","confidential_remarks":"This is a well-motivated empirical study, but the primary comparison between an augmented CNN and a much smaller equivariant baseline, plus the uncharacterized rotation discretization, affects all stated equivariance measurements. I do not think the paper should be rejected—the concerns are addressable—but they are load-bearing for the central claims, so the manuscript needs substantial revision before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, the practical map: rotation-augmented CNNs learn equivariance for denoising and property prediction even at small scale, but for seeded generation only a 111M-parameter VoxMol holds up under input rotation. That's a useful and somewhat surprising cost-benefit result, and the latent embedding analysis (rotated inputs land far apart in latent space) is the most illuminating part. The paper is also the first to study learned equivariance in generative molecular modeling, which fills a real gap.\n\nThe loss decomposition in Theorem 3.2 is a standard Taylor expansion, but the exact MSE equality depends on the rotation being orthogonal. The paper never says how rotations are applied to the 64^3 voxel grids. Any resampling-based rotation is non-orthogonal and may not even satisfy the group law, so Eq. (3) is not exact for the actual transformation, and the equivariance error in Eq. (4) conflates model behavior with interpolation artifacts. This affects every rotation experiment, including generation. The authors need to specify the rotation operator or rotate continuous coordinates before voxelization. This is a genuine measurement flaw, not a nitpick.\n\nThe baseline comparison is also lopsided: E3NN has 0.5M parameters versus 111M for VoxMol and underperforms on stability and uniqueness. Saying 'match the performance of equivariant models' is only true against a model that isn't competitive for generation. The authors mention the constraint, but it undercuts the headline.\n\nAdditionally, the KL divergence table has no error bars, and the generation robustness differences between model sizes are not significance-tested. The qualitative trends look plausible, but they're not as clean as the text implies.\n\nThat said, the paper is honest about its limitations and the empirical sweep (size, data, epochs) is broad. No circularity in the evaluation. For a reader deciding between equivariant GNNs and augmented CNNs, this gives a useful starting point, but the measurement issue means I'd treat the quantitative equivariance errors as provisional.\n\nRecommendation: send to review; the question matters and the study is worth referee time. Expect major revision: specify rotations, add a matched-capacity equivariant baseline, and add error bars. I wouldn't cite it yet for the equivariance numbers.","headline":"A useful empirical map of when augmented CNNs match equivariant models, but the rotation operator is unspecified and the equivariant baseline is too small to support the headline.","tokens_in":14883,"tokens_out":3357,"would_cite":false,"duration_ms":38995,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Rotation-augmented CNNs can learn molecular equivariance, but matching explicit symmetry in generation takes a 111M-parameter model.","keywords":["equivariance","rotation augmentation","molecule generation","3D convolutional networks","voxel grids","walk-jump sampling","learned equivariance","generative equivariance"],"falsifier":"Repeat the reconstruction and seeded-generation experiments with two rotation families: exact grid symmetries (multiples of $90^\\circ$ about grid axes, which a Cartesian grid can represent exactly) versus arbitrary angles that require interpolation and resampling. If the equivariance error is much lower for grid-aligned rotations, part of the claimed learned equivariance is a voxelization artifact rather than model behavior. Alternatively, train the 7M-parameter VoxMol on finer grids at $0.125$ Å resolution with the same rotation augmentation: if generative equivariance then appears without the 111M-parameter budget, the paper's capacity requirement is a resolution artifact, while if it still fails, the capacity claim survives.","tokens_in":13895,"feed_emoji":"🧪","tokens_out":10085,"duration_ms":92005,"temperature":0.7,"pith_summary":"Most 3D molecular generative models build rotation symmetry directly into their architecture with equivariant graph neural networks, on the assumption that explicit equivariance is essential, even though such models are complex, hard to train, and scale poorly. This paper asks whether a plain 3D CNN trained with random rotation augmentation can learn the same behavior instead. The answer depends on the task: for denoising and property prediction, even small augmented models match the equivariance level of an explicitly equivariant network, and a derived loss decomposition separates prediction error from equivariance error to make the result quantitative. For generation, however, only the full 111M-parameter VoxMol model produces stable outputs when the seed molecule is rotated; smaller models and models trained on less data diverge, losing generative equivariance. The paper concludes that learned equivariance is possible in principle but demands redundant capacity that explicit equivariance would save, and suggests auxiliary objectives that align latent embeddings across rotations could close the gap.","feed_headline":"Large augmented CNNs match built-in symmetry for molecule generation","feed_subtitle":"Denoising and property prediction learn rotation robustness with tiny models, but stable generation under rotated seeds needs full scale.","key_machinery":"The load-bearing object is Theorem 3.2, the loss decomposition of Eq. (2), which splits the rotation-augmented training loss into a prediction-error term plus an equivariance-error term involving a Hessian-weighted variance of the rotated outputs, and which becomes exact for MSE loss in Eq. (3). That decomposition justifies the operational criterion in Eq. (4): a model is approximately equivariant when the reconstruction of a rotated input $R(\\hat{x}_0)$ matches the rotated reconstruction of the unrotated input. The experiments run this criterion over VoxMol, a U-Net 3D CNN that denoises $64^3$ voxel grids at $0.25$ Å resolution and generates molecules by walk-jump sampling, compared against E3NN, an SE(3)-equivariant steerable CNN that serves as the explicit-equivariance baseline. A third component is the latent-embedding cosine-similarity analysis, which shows rotated molecules embed differently and thereby diagnoses why the augmented CNN pays a large parameter overhead.","core_discovery":"On its own terms, the paper claims that rotation augmentation alone teaches a non-equivariant 3D CNN approximate rotational equivariance for molecule reconstruction: the equivariance error, measured by comparing the reconstruction of a rotated input with the rotated reconstruction of the unrotated input, stays below the ground-truth reconstruction floor, and this holds with as little as 1% of the training data and within a few hundred epochs of training. The central discovery is that reconstruction equivariance does not transfer to generation. In seeded walk-jump sampling, only the full-size rotation-augmented VoxMol (111M parameters) produces generation metrics and chemical-property distributions that are nearly identical whether the seed lead is rotated or not, matching the explicitly equivariant E3NN baseline; smaller VoxMol variants (28M and 7M parameters), the unaugmented model, and models trained on half the data lose stability and shift property distributions when the seed is rotated. The paper also shows that latent embeddings of the same molecule under different rotations have low cosine similarity even when reconstructions agree, concluding that the CNN learns redundant parallel representations instead of recognizing rotated inputs as equivalent, which it offers as the reason augmented CNNs need so many more parameters.","pith_inferences":["If latent alignment is the bottleneck, a moderate CNN trained with an invariance-penalty auxiliary objective could plausibly beat both the 111M-parameter augmented CNN and the equivariant baseline on generation robustness; the paper floats this idea but does not run it.","The 1%-data success for reconstruction equivariance may partly reflect that isotropic Gaussian noise is rotation-invariant by construction; anisotropic or spatially correlated noise would likely expose the augmented CNN's equivariance as much shallower than an explicitly equivariant model's.","Because the paper never specifies how arbitrary rotations are applied to a Cartesian grid, a natural extension is to compare grid-aligned rotations (multiples of $90^\\circ$) with interpolated arbitrary angles; a large gap in equivariance error would mean voxelization, not the model, carries part of the apparent equivariance.","The redundant-representation finding suggests a general regularization principle: when a task demands invariance, a model trained only on the target loss will spend capacity on duplicate input representations unless the loss penalizes that redundancy, a lesson that may extend beyond molecules to any augmented generative system."],"forward_implications":["Rotation augmentation can replace architectural equivariance for reconstruction and property prediction, so molecule-generation pipelines without built-in symmetry remain viable for those stages.","Generative robustness under rotated seeds is a separate property that emerges only at full model scale; deploying a smaller augmented model in a lead-optimization loop could silently change its outputs, so rotation checks are needed before use.","The loss decomposition gives a practical diagnostic: computing the equivariance-error term on a validation set tells a practitioner whether an augmented model is safe to trust under rotation.","Auxiliary losses that pull the latent embeddings of rotated molecules together, which the paper proposes, are a direct follow-up that could reduce the parameter cost of learned equivariance."],"supporting_citations":[{"why":"VoxMol, the U-Net 3D CNN and walk-jump sampling pipeline that all reconstruction and generation experiments are built on.","marker":"[Pinheiro et al., 2023]"},{"why":"3D steerable CNNs, the SE(3)-equivariant layers used in the E3NN baseline.","marker":"[Weiler et al., 2018]"},{"why":"EDM, the SE(3)-equivariant diffusion model that exemplifies the equivariant-GNN approach the paper questions.","marker":"[Hoogeboom et al., 2022]"},{"why":"The spherical-image comparison claiming equivariant models beat augmented CNNs, which this paper contrasts and partially revises for generative tasks.","marker":"[Gerken et al., 2022]"},{"why":"Neural empirical Bayes, the theory behind the walk-jump sampling used for generation.","marker":"[Saremi & Hyvärinen, 2019]"},{"why":"GEOM-Drugs, the public dataset all models are trained and evaluated on.","marker":"[Axelrod & Gomez-Bombarelli, 2022]"},{"why":"MiDi, the source of the generation quality metrics and the data splits used for evaluation.","marker":"[Vignac et al., 2023]"},{"why":"The invariance and equivariance measurement methodology used for the latent-embedding cosine-similarity analysis.","marker":"[Kvinge et al., 2022]"},{"why":"Evidence that equivariance emerges from augmentation in predictive tasks, which the paper extends to generation.","marker":"[Gerken & Kessel, 2024]"}],"fun_headline_variants":["Augmented CNNs learn equivariance, but generation needs scale","Rotation augmentation teaches reconstruction, not generation","Equivariance from augmentation: reconstruction yes, generation no","Large augmented CNN matches equivariant generation performance","Small augmented models fail at rotation-robust generation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"All the equivariance measurements assume that rotating a $64^3$ voxel grid at $0.25$ Å resolution, followed by peak-finding to recover atoms, preserves molecular geometry faithfully enough that any difference between rotated and unrotated outputs is caused by the model's missing equivariance rather than by voxel discretization or resampling artifacts, yet the paper never states how rotations are applied to the grids.","fun_headline_variants_meta":{"raw":{"variants":["Augmented CNNs learn equivariance, but generation needs scale","Rotation augmentation teaches reconstruction, not generation","Equivariance from augmentation: reconstruction yes, generation no","Large augmented CNN matches equivariant generation performance","Small augmented models fail at rotation-robust generation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000273,"raw_usage":{"total_tokens":1616,"prompt_tokens":904,"completion_tokens":712,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":520,"completion_tokens_details":{"reasoning_tokens":638}},"tokens_in":520,"tokens_out":712,"duration_ms":8064,"temperature":1.0,"reasoning_tokens":638,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:48:54.494622+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Repeat the reconstruction and seeded-generation experiments with two rotation families: exact grid symmetries (multiples of $90^\\circ$ about grid axes, which a Cartesian grid can represent exactly) versus arbitrary angles that require interpolation and resampling. If the equivariance error is much lower for grid-aligned rotations, part of the claimed learned equivariance is a voxelization artifact rather than model behavior. Alternatively, train the 7M-parameter VoxMol on finer grids at $0.125$ Å resolution with the same rotation augmentation: if generative equivariance then appears without the 111M-parameter budget, the paper's capacity requirement is a resolution artifact, while if it still fails, the capacity claim survives.","supporting_citations":[],"review_version":1}