{"id":"88fa3373-9dd9-4f72-a1ee-2bf974a5e806","arxiv_id":"2506.17340","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A maximally stochastic reverse sampler (StoMax) improves molecule stability and validity across DDPM- and BFN-based generators, at a cost in diversity.","lead":"This paper finds that a maximally random sampling procedure, called StoMax, improves the stability and validity of molecules generated by diffusion models. It suggests that sampling design, not just model architecture, is a key lever for molecular generation quality.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"StoMax (Eq. 7 with gamma_t=0) is not shown to be a faithful sampler: it omits the posterior covariance of x0 given x_t and has no well-behaved continuous-time limit, so observed gains may reflect distributional bias rather than superior sampling.","rationale":"The reader's weakest assumption identifies the same general area: StoMax's theoretical validity as a sampler is unproven. I agree that this is the most load-bearing concern. However, the specific mechanism I would emphasize is not only that beta(t) -> infinity violates the bounded-beta condition for Eq. 3, but that the discrete StoMax update is not the exact marginal-preserving transition because it replaces a posterior draw of x0 with its conditional mean. The beta-bound argument is suggestive but not fully decisive, since the discrete update need not be viewed as a discretization of Eq. 3; the omitted posterior covariance is a more direct reason to expect bias that does not vanish with step size. The empirical V*U drops on both BFN models are consistent with this bias, and the paper's own Section 4 admits the lack of a rigorous explanation. The proposed toy-mixture experiment would settle whether the bias is real: if StoMax matches the true distribution in that controlled setting, the theoretical concern is refuted and the empirical finding stands on its own; if not, the headline claim needs substantial qualification. Missing error bars and selective reporting on GEOM-Drugs are secondary issues that would also need addressing, but they are not as central as sampler validity. The reader's CONDITIONAL verdict remains appropriate: the paper should either prove or explicitly qualify the sampler-validity assumption, and should report full metrics with error bars before the claim of consistent improvement is accepted.","tokens_in":8877,"tokens_out":14066,"duration_ms":153810,"concrete_test":"On a 2D Gaussian mixture with an analytically known score, implement the StoMax update (Eq. 7 with gamma_t=0) and the DDPM and DDIM updates, each with the paper's 1000-step discretization. Draw 10^5 initial points from the true terminal marginal p_T, propagate to t=0, and estimate the 2-Wasserstein distance and KL divergence between each sampler's output distribution and the true data mixture. Repeat with 500 and 100 steps. If StoMax's output distribution shows substantially larger divergence than DDPM/DDIM or if the divergence does not shrink with more steps, StoMax is a biased sampler and the empirical gains on QM9/GEOM-Drugs are at least partly an artifact of targeting a different distribution. Optionally, compute the omitted term mu_{t-dt}^2 Var(x0|xt) along the trajectory to confirm it is non-negligible in the relevant noise regime.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that StoMax 'consistently outperforms' default samplers. This requires StoMax to be a valid sampler for the learned distribution, not merely a procedure that improves quality metrics by changing the effective target distribution. Two concrete problems make this insecure. First, the reverse-SDE family in Eq. 3 is guaranteed to share marginals only for bounded beta(t). For gamma_t=0, matching Eq. 7 to Eq. 4 gives beta(t) proportional to 1/Delta t, so beta(t) -> infinity and the shared-marginal guarantee does not apply. Second, even with an exact score, the StoMax update x_{t-dt} = mu_{t-dt} E[x0|xt] + sigma_{t-dt} z is not the marginal-preserving transition. The exact transition that preserves the forward marginals is x_{t-dt} = mu_{t-dt} x0* + sigma_{t-dt} z with x0* ~ p(x0|xt), whose conditional covariance is sigma_{t-dt}^2 I + mu_{t-dt}^2 Var(x0|xt). StoMax drops the posterior-covariance term, and because its noise per step is O(1) rather than O(sqrt(Delta t)), the bias need not vanish as the step size decreases. The paper provides no quantitative argument that this bias is negligible for the learned score. The observed diversity loss in Table 1 (V*U: 92.7 to 91.9 for GeoBFN; 93.0 to 91.3 for UniGEM(BFN)) is exactly the signature of a diversity-reducing bias, so the 'consistently outperforming' claim rests on weighting stability and validity above diversity. Section 4 concedes that a rigorous theoretical explanation is lacking and that the discrete implementation differs from Langevin dynamics; this unverified sampler-validity assumption is the load-bearing weak point.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper revisits the choice of sampling strategy in diffusion-based 3D molecular generation. It works within the general reverse-SDE framework of Ni et al. (2025), derives a family of discrete samplers parameterized by γ_t in Eq. (7), and interprets DDPM/BFN default sampling as the Markov case, DDIM as the deterministic case, and the new 'StoMax' strategy (γ_t = 0) as the maximally stochastic case. The authors evaluate StoMax against default samplers for EDM, GeoBFN, UniGEM(EDM), and UniGEM(BFN) on QM9, and for EDM and UniGEM(EDM) on GEOM-Drugs, reporting improvements in atom stability, molecule stability, and validity, along with a diversity decrease that is described as minor. The paper concludes that StoMax consistently outperforms default sampling and motivates future theoretical work on the diversity-fidelity trade-off.","tokens_in":9305,"tokens_out":7842,"duration_ms":85800,"significance":"If the central claim were established, this would be a practically useful finding: a simple, almost parameter-free change to the sampling procedure that improves the quality of generated molecules across several models and datasets. The paper also provides a useful unifying derivation of DDPM, BFN, DDIM, and StoMax within one update rule, and the QM9 evaluation covers four model/sampler combinations at 1000 sampling steps. However, the significance is currently limited by two issues: StoMax is not shown to be a faithful sampler for the learned distribution, and the empirical record contradicts the word 'consistently' when diversity-aware metrics are included. The paper's own Section 4 concedes that a rigorous theoretical explanation is lacking.","major_comments":[{"comment":"StoMax is defined by γ_t = 0 in Eq. (7). In the continuous reverse SDE of Eq. (3), matching this discrete choice to the SDE requires β(t) ∝ 1/Δt, so β(t) → ∞, which is outside the bounded-β condition under which the reverse-process family shares the forward marginals. The exact discrete transition that preserves forward marginals samples x0* from p(x0|xt) and has conditional covariance σ_{t-Δt}^2 I + μ_{t-Δt}^2 Var(x0|xt); Eq. (7) replaces x0* by E[x0|xt] and drops that posterior-covariance term. Because the StoMax noise term √(2σ_t(σ_{t-Δt} - γ_t)) is O(1) rather than O(√Δt), the bias need not vanish as Δt → 0. Section 4 explicitly concedes that the rigorous theoretical explanation is lacking and that the discrete implementation differs from Langevin dynamics. The paper must either prove that the discrete StoMax update samples from the learned distribution (or is provably close to it), or explicitly reframe StoMax as an alternative generative procedure and remove the claim that it is a theoretically valid sampler within the family of Eq. (3).","section":"§2.3, §4, Eq. (3), Eq. (7)"},{"comment":"The abstract and Section 1 claim that StoMax 'consistently outperforms default sampling methods'. In Table 1, V×U drops for GeoBFN (92.7 to 91.9) and for UniGEM(BFN) (93.0 to 91.3), which are two of the four QM9 model/sampler combinations. Section 3.4 also uses U×V as the headline metric for the favorable trade-off, so these decreases directly contradict the unqualified wording of the central claim. The authors should either qualify the claim to the three quality metrics (atom stability, molecule stability, validity), explicitly acknowledge the V×U decreases in the abstract, or provide an aggregate justification for the word 'consistently'.","section":"Table 1"},{"comment":"The GEOM-Drugs results report only atom stability and validity, omitting molecule stability and all diversity-aware metrics such as V×U. As a result, the GEOM-Drugs experiments cannot support the global conclusions that StoMax consistently outperforms defaults or that the diversity trade-off is minor. The same metric suite as in Table 1 should be reported, or the GEOM-Drugs conclusions should be restricted to the two reported metrics.","section":"Table 2"},{"comment":"The interpolation experiment is performed only with UniGEM(EDM), so the statements that 'StoMax achieves the highest score on the U×V metric' and 'StoMax emerges as the most balanced and effective approach' are not general empirical results; they describe one model's interpolation curve. This is consistent with the V×U decreases for GeoBFN and UniGEM(BFN) in Table 1 and should be presented with that limitation.","section":"§3.4, Figure 1"}],"minor_comments":[{"comment":"The text 'the proportion of generated generated 3D structures' contains a duplicated word and should read 'generated 3D structures'.","section":"§3.1, Metrics"},{"comment":"Section 3.1 refers to UniGEM(GeoBFN) but Table 1 labels the same variant UniGEM(BFN); please align the notation throughout.","section":"§3.1 and Table 1"},{"comment":"The density f(x0) is used in the score derivation but is never defined; it should be written as p(x0) or q(x0) consistently with the rest of the paper.","section":"Appendix A.2, Eq. (12)"},{"comment":"Proposition A.2 verifies only the BFN sampler; the corresponding explicit verification for the DDPM/Markov case is asserted in the text but not shown, so the claim that both default samplers belong to the Markov family would benefit from an explicit derivation or citation.","section":"Proposition A.1 and A.2"},{"comment":"The interpolation coefficients p′ and p″ are used in the formulas but are only described as being in [0,1]; please state which endpoint of each interpolation each value corresponds to.","section":"§3.4, Interpolation"},{"comment":"The statement that the StoMax variance 'can be interpreted as a form of temperature control (Ni et al., 2025)' is vague; a precise relationship between the variance scale and a temperature parameter would be helpful.","section":"§2.3, StoMax variance"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is positioned in an active area and the empirical comparisons are extensive, but the central claim is currently stronger than the evidence. The most important issue is the unproven sampler validity of StoMax: because the discrete update at γ_t = 0 is outside the bounded-β regime of the shared-marginal guarantee and drops the posterior covariance, the observed quality gains may reflect distributional bias rather than superior sampling of the model distribution. The V×U decreases in Table 1 further undermine the 'consistently outperforming' wording. These problems are fixable by a combination of a validity analysis or explicit reframing, a more careful statement of the empirical claims, and complete metric reporting on GEOM-Drugs. The heavy reliance on the authors' own prior work (Ni et al. 2025; Xue et al. 2024b) is not itself a reason for concern."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know this paper is an empirical study of sampling strategies for 3D molecular diffusion models. The main finding: setting gamma_t=0 in the reverse-process family (StoMax) improves atom and molecule stability and validity across EDM, GeoBFN, and UniGEM variants on QM9 and GEOM-Drugs, at some cost in uniqueness. That is genuinely useful, and the paper mostly documents it honestly.\n\nWhat is new is narrow but real. The gamma-parameterized family comes from DDIM and SA-Solver, and gamma=0 is a natural endpoint. The new part is evaluating that endpoint specifically for molecular generation and showing consistent gains on quality metrics. The interpolation experiments give a clear picture of the quality-diversity trade-off and are a good way to present the design space.\n\nThe paper does several things well. The derivations in Appendix A are standard and clearly written. Re-training GeoBFN because the original weights are unavailable is transparent. Section 4 frankly admits that a rigorous theory is lacking and that the discrete update differs from Langevin dynamics. That honesty earns credit.\n\nThe soft spots are real. The headline claim that StoMax \"consistently outperforms\" default sampling is contradicted by Table 1: V*U drops for GeoBFN (92.7 to 91.9) and UniGEM(BFN) (93.0 to 91.3). That is a diversity drop, not noise. Calling that consistent outperformance weights stability and validity above diversity without saying so. There are no error bars, so it is hard to know how reliable the gains are. GEOM-Drugs results omit diversity metrics entirely, which is selective reporting for a claim about overall performance.\n\nThe bigger issue is whether StoMax is a faithful sampler for the learned distribution. The reverse-SDE family in Eq. 3 has a shared-marginal guarantee only for bounded beta(t). For gamma_t=0, matching Eq. 7 to Eq. 4 gives beta(t) ~ 1/Delta_t, which blows up, so the guarantee does not apply. The discrete StoMax update drops the posterior covariance of x0 given x_t; because its noise per step is O(1) rather than O(sqrt(Delta_t)), the bias need not vanish as step size decreases. The paper provides no quantitative argument that this bias is negligible for the learned score. The observed V*U drops are exactly the signature of a diversity-reducing bias, so the empirical gains may partly reflect a change in the effective target distribution rather than better sampling from the intended one.\n\nWho should read this: people working on diffusion or flow samplers for molecules, especially with EDM- or BFN-based models. It is a useful empirical datapoint and may push the field to consider Langevin-like samplers more seriously.\n\nI would engage with it and send it to review. The issues are addressable: add error bars, report full diversity metrics on both datasets, and either prove the sampler-validity assumption or qualify the claim. The authors also need to soften \"consistently outperforms.\" As it stands, the paper is an interesting empirical observation with a load-bearing caveat.","headline":"Useful empirical result, but the 'consistently outperforms' claim is too strong and the sampler-validity question is unanswered.","tokens_in":9833,"tokens_out":1867,"would_cite":true,"duration_ms":21103,"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":"The paper claims that switching to maximally stochastic sampling, StoMax, outperforms default DDPM and BFN samplers across four 3D molecular generation models, improving stability and validity on QM9 and GEOM-Drugs.","keywords":["molecular generation","diffusion models","sampling strategies","StoMax","Bayesian flow networks","3D molecule generation","stochastic differential equations","maximally stochastic sampling"],"falsifier":"A direct check is to run the StoMax update with $\\gamma_t=0$ on a Gaussian target whose score function is known exactly: if the update is unbiased, the sample mean and covariance will match the target's as the step size shrinks, and any systematic offset would show that StoMax samples a different distribution. A molecule-level version of the same test is to increase the number of sampling steps by an order of magnitude and see whether the diversity drop in Table 1 persists; if it persists, the loss is a bias of the StoMax discretization rather than a natural property of the learned density.","tokens_in":8676,"feed_emoji":"🧪","tokens_out":9439,"duration_ms":89474,"temperature":0.7,"pith_summary":"Generating a molecule from a trained diffusion model is usually treated as a fixed recipe, but this paper argues that the sampling strategy itself is a major, underused control on output quality. It works within the family of reverse samplers parameterized by how the next noisy sample depends on the previous one, and identifies the maximally stochastic extreme, StoMax, in which each reverse step is conditionally independent of the last given the clean data. The paper's central claim is that StoMax consistently beats the default DDPM and BFN samplers for four 3D molecular generators, raising atom stability, molecule stability, and validity on QM9 and GEOM-Drugs while trading away a little diversity. If the claim is right, better molecules do not require new architectures or retraining; they can come from a one-line change to the sampling loop, and the optimal sampler may be closer to Langevin dynamics than to the standard DDPM and DDIM recipes.","feed_headline":"Max-stochastic sampling beats defaults in molecular generation","feed_subtitle":"On QM9 and GEOM-Drugs, StoMax lifts validity to near-perfect levels across four diffusion generators.","key_machinery":"The load-bearing design is the one-parameter family of conditional reverse distributions $p_\\lambda(x_{t-\\Delta t}\\mid x_t,x_0)=\\mathcal{N}(\\mu_{t-\\Delta t}x_0+\\gamma_t (x_t-\\mu_t x_0)/\\sigma_t,\\ \\lambda_t^2 I)$ with $\\gamma_t^2+\\lambda_t^2=\\sigma_{t-\\Delta t}^2$, together with the discrete update $x_{t-\\Delta t}=\\frac{\\mu_{t-\\Delta t}}{\\mu_t}x_t+(\\frac{\\mu_{t-\\Delta t}}{\\mu_t}\\sigma_t-\\gamma_t)\\sigma_t\\nabla_x\\log p_t(x_t)+\\sqrt{2\\sigma_t(\\sigma_{t-\\Delta t}-\\gamma_t)}\\,\\epsilon$. Choices of $\\gamma_t$ index the sampling method: the Markov forward process (DDPM and, as verified in the appendix, BFN) corresponds to $\\gamma_t=\\mu_t\\sigma_{t-\\Delta t}^2/(\\mu_{t-\\Delta t}\\sigma_t)$, the deterministic DDIM limit to $\\gamma_t=\\sigma_{t-\\Delta t}$, and StoMax to $\\gamma_t=0$, which maximizes the variance under this parameterization and makes each reverse step conditionally independent of its predecessor given the clean data. The paper verifies that, for the DDPM schedule, StoMax with $\\gamma_t=0$ also arises as the $\\beta(t)\\to\\infty$ limit of the general reverse SDE, connecting it to Langevin-like dynamics.","core_discovery":"The paper establishes that, within the one-parameter family of reverse sampling updates derived from the general diffusion SDE, the maximally stochastic choice $\\gamma_t = 0$ yields the best balance of quality and diversity for molecular generation. Concretely, replacing default samplers with StoMax improves molecule stability on QM9 from 82.0% to 87.9% for EDM and from 89.8% to 96.1% for UniGEM(EDM), and validity from 91.9% to 94.5% and 95.0% to 98.1% respectively; GeoBFN and UniGEM(BFN) also improve, with smaller but consistent gains. On GEOM-Drugs, StoMax lifts EDM validity from 92.6% to 99.7% and UniGEM(EDM) from 98.4% to 99.9%, and atom stability by roughly four to five points. The diversity metric V*U rises slightly for EDM-based models but falls for BFN-based ones, and interpolating between StoMax, DDPM, and DDIM shows stability and validity increasing monotonically with reverse-step stochasticity. The paper states plainly that a rigorous theory is still missing, and that in the continuous formulation StoMax corresponds to taking $\\beta(t)\\to\\infty$, outside the bounded-$\\beta$ condition under which the reverse SDE family provably shares the forward marginals.","pith_inferences":["A direct extension the paper does not test: if StoMax's benefit comes from its variance effect, the same update should improve other structured diffusion outputs, such as conformer ensembles or point clouds, where validity is hard to enforce.","The formal gap at $\\gamma_t=0$ suggests StoMax may sample from a mildly biased distribution whose bias happens to help valence and bond-counting metrics; a toy Gaussian experiment with an exact score would separate 'better sampling' from 'useful bias'.","Because StoMax's variance acts like temperature, one testable extension is an adaptive schedule that starts near StoMax and anneals toward DDPM late in the trajectory, which might recover the lost uniqueness without sacrificing stability.","If future theory identifies the effective stationary distribution of the StoMax update, the same calculation could prescribe an optimal noise level for each dataset, turning sampling design into a principled hyperparameter rather than a choice of extremes."],"forward_implications":["Switching any of the four tested generators to StoMax improves atom stability, molecule stability, and validity on QM9 without retraining, so sampling design is a model-agnostic performance lever.","On GEOM-Drugs StoMax brings validity close to 100% for EDM and UniGEM(EDM), suggesting gains grow on larger, more drug-like molecules.","Interpolation results imply that within this family, more reverse-step stochasticity monotonically buys quality at a small uniqueness cost, and StoMax is the best operating point on the V*U trade-off.","The paper's Langevin connection suggests that treating sampling as annealing dynamics rather than denoising discretization could be the right lens for molecular generation; this is presented as a hypothesis, not a proof."],"supporting_citations":[{"why":"Defines DDPM and its default first-order reverse sampler, the primary baseline sampling convention.","marker":"Ho et al., 2020"},{"why":"Defines Bayesian Flow Networks whose default continuous-data sampler is the other canonical strategy StoMax is compared against.","marker":"Graves et al., 2023"},{"why":"Supplies the EDM baseline, the QM9 evaluation protocol, and the stability and validity metrics used throughout.","marker":"Hoogeboom et al., 2022"},{"why":"Supplies the GeoBFN baseline whose default BFN sampler StoMax replaces.","marker":"Song et al., 2023"},{"why":"Supplies the UniGEM(EDM) and UniGEM(BFN) models on which StoMax delivers the largest gains.","marker":"Feng et al., 2025"},{"why":"Introduces the family of conditional reverse distributions parameterized by gamma and the DDIM deterministic limit and interpolation technique.","marker":"Song et al., 2021a"},{"why":"Provides the reverse-SDE family with bounded beta(t) that the paper discretizes to obtain the general sampling update.","marker":"Xue et al., 2024b"},{"why":"Provides the QM9 dataset used for the main unconditional generation experiments.","marker":"Ramakrishnan et al., 2014"},{"why":"Provides the GEOM-Drugs dataset used for the larger-scale unconditional generation experiments.","marker":"Axelrod & Gomez-Bombarelli, 2022"}],"fun_headline_variants":["StoMax sampling lifts molecular validity to near-perfect","Max-stochastic sampling outperforms defaults for molecule generation","StoMax: simple sampling strategy beats defaults in molecular generation","Sampling stochasticity strongly influences molecular generation quality","One-liner sampling change: StoMax improves molecular stability"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that running the StoMax update still generates molecules from the distribution the model actually learned, even though StoMax's extreme noise level falls outside the range where the standard theory guarantees the reverse process matches that distribution.","fun_headline_variants_meta":{"raw":{"variants":["StoMax sampling lifts molecular validity to near-perfect","Max-stochastic sampling outperforms defaults for molecule generation","StoMax: simple sampling strategy beats defaults in molecular generation","Sampling stochasticity strongly influences molecular generation quality","One-liner sampling change: StoMax improves molecular stability"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001352,"raw_usage":{"total_tokens":5480,"prompt_tokens":923,"completion_tokens":4557,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":539,"completion_tokens_details":{"reasoning_tokens":4492}},"tokens_in":539,"tokens_out":4557,"duration_ms":33891,"temperature":1.0,"reasoning_tokens":4492,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T23:44:46.042093+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct check is to run the StoMax update with $\\gamma_t=0$ on a Gaussian target whose score function is known exactly: if the update is unbiased, the sample mean and covariance will match the target's as the step size shrinks, and any systematic offset would show that StoMax samples a different distribution. A molecule-level version of the same test is to increase the number of sampling steps by an order of magnitude and see whether the diversity drop in Table 1 persists; if it persists, the loss is a bias of the StoMax discretization rather than a natural property of the learned density.","supporting_citations":[{"cited_title":"Unified generative modeling of 3d molecules with bayesian flow networks","cited_arxiv_id":null,"evidence_quote":"Supplies the GeoBFN baseline whose default BFN sampler StoMax replaces."},{"cited_title":"Unigem: A unified approach to generation and property prediction for molecules","cited_arxiv_id":null,"evidence_quote":"Supplies the UniGEM(EDM) and UniGEM(BFN) models on which StoMax delivers the largest gains."}],"review_version":1}