{"id":"98d7a750-d101-48f6-ac4a-bf99bf046904","arxiv_id":"2501.09064","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"RGDM generates data by reversing renormalization-group-style coarse-graining, using a colored-noise schedule and projection layers to sample coarse-to-fine, and outperforms a vanilla DDPM on protein and image benchmarks.","lead":"This paper introduces RGDM, a diffusion model that corrupts data with wavenumber-dependent colored noise inspired by the renormalization group, and generates samples by reversing that coarse-graining. It reports better sample quality or up to an order-of-magnitude faster sampling than a standard diffusion baseline on protein structure and image generation tasks.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Scale-separation property (Eq. 3) is assumed for real data but only tested to second order; if high-k modes are non-Gaussian and correlated with low-k modes, the projection layers can discard signal and the order-of-magnitude speedup is built on an invalid premise.","rationale":"The reader's weakest assumption identifies the same load-bearing point: Eq. (3) is exact for the RG effective model of the ansatz (2), but the forward process operates on real data. The projection layers are precisely what enable the order-of-magnitude speedup, so if Eq. (3) fails for natural data, the central claim of improved quality/speed is compromised. I checked whether any other concern is more fundamental. SM Sec. C shows the RGDM can represent any colored-noise schedule by choosing the regulator, so the RG derivation does not uniquely select the schedule; this weakens the 'no tuning' claim, but the empirical comparison still stands and is not the method's core mechanism. The fitted parameters v, m, Λ1, ΛT, kcutoff, and regulator are also overclaimed, but these are addressable with a revision. The scale-separation issue is the one whose failure would invalidate the design of the projection layers and the reverse sampling, not just the marketing. The proposed test—measuring mutual information/kurtosis of the discarded modes on real data—directly settles whether the concern is real. I therefore keep the reader's CONDITIONAL verdict and see no need to change it; the paper should add higher-order statistics of the discarded scales or an ablation without projection layers.","tokens_in":31181,"tokens_out":7367,"duration_ms":72518,"concrete_test":"On CIFAR-10, FFHQ, and protein Cα chains, compute the empirical mutual information I(φ_>; φ_<) between the high-pass (|k|>cΛt) and low-pass (|k|≤cΛt) DCT components at several t values along the actual RGDM schedule (Eqs. S96-S98), and compare the excess kurtosis of the high-pass coefficients to 0. If the mutual information is significantly above zero (or |kurtosis|>0.5) for the modes that Algorithm 4 would discard or Gaussian-resample, Eq. (3) is violated for natural data and the projection shortcut can lose non-Gaussian structure; the concern lands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Eq. (3) states that at every RG scale the high-wavenumber modes are exactly Gaussian and independent of the retained modes. In the exact RG this holds for the effective model SΛ built from the ansatz (2), but the RGDM's forward process (Eq. 8) is applied to arbitrary pdata, not to the Gibbs measure of Sdata. Consequently, the high-k components of ϕt are only Gaussian in the limit Kt(k)→0, not at finite Λt; and they are not exactly independent of ϕ<. The projection layers (Fig. 1e) and the reverse process (Algorithm 4 lines 6-7) discard or resample every mode with |k|>cΛt without training on them, relying entirely on Eq. (3). The only empirical evidence for Eq. (3) is Fig. 1b, which shows the second-order variance (power spectrum). Natural images and protein structures have well-documented non-Gaussian high-frequency statistics (edges, secondary-structure geometry); if those modes carry information not captured by low-k modes, the generated samples lose that structure, and the claimed acceleration at T=200 is achieved at a hidden quality cost. This is a correctness risk in the method's core, not merely an overclaim about tuning.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a renormalization-group-based diffusion model (RGDM) in which the forward noising process is derived from the Polchinski exact RG flow, yielding a wavenumber-dependent colored-noise schedule (Eqs. (8)-(9)) that destroys fine-scale modes before coarse-grained ones. The reverse process is trained with a denoising network that, via RG projection layers, removes high-wavenumber modes that are supposed to have been integrated out, and sampling proceeds coarse-to-fine. The method is applied to protein structure prediction and image generation, and the authors report that it consistently outperforms conventional DDPMs in sample quality and/or sampling speed, with substantially fewer steps, while reducing the need for data-dependent noise-schedule tuning.","tokens_in":31412,"tokens_out":16288,"duration_ms":177246,"significance":"If the central claims hold, the paper offers a principled multiscale alternative to standard diffusion models and gives a physics-based rationale for colored-noise schedules, which is a timely and useful contribution. The derivation of the forward process from the Polchinski equation is clean, the numerical protocol is careful (matched networks, matched step counts, code and trained models provided), and the empirical comparisons cover two distinct domains. The main weaknesses are overstatements about the exactness of scale separation and about the degree to which the schedule is free of data-dependent choices; these are correctable with more careful claims and a quantitative error estimate for the projection approximation.","major_comments":[{"comment":"Equation (3) is not an exact property of the distribution pΛ(φ) ∝ e^{−SΛ(φ)} with SΛ given by Eq. (4). For |k| ≫ Λ, KΛ(k) → 0, so the quadratic term in Eq. (4) behaves as k^2/KΛ(k), and the marginal variance of the high-k modes tends to zero, not to the variance 1/k^2 of pGS. The factorization in Eq. (3) becomes true only for the rescaled distribution p'_Λ(φ) ∝ e^{−SΛ(√KΛ φ)} introduced after Eq. (6), and even then only in the limit KΛ(k) → 0 on the eliminated modes. This distinction matters because Algorithm 4 and Fig. 2c justify discarding and resampling high-k modes on the basis of Eq. (3). The manuscript should state the approximate, asymptotic nature of this property and give the corresponding error bound.","section":"Renormalization group-based diffusion model, Eq. (3)"},{"comment":"The projection layers discard modes with |k| > cΛt, not modes with |k| > Λt as stated in the main text. With the values used in the experiments (c ≈ 54 for CIFAR-10 and c ≈ 63 for protein structures), the discarded modes satisfy √αbar_tk ≤ 1/c ≈ 0.02 for the regulator r(x) = x^{−1}, so the approximation is numerically well controlled. However, the paper does not provide this bound and instead asserts that these modes 'have no information about the data.' The only empirical evidence for scale separation is the second-order power spectrum in Fig. 1b; the stronger independence statement in Eq. (3) is not tested. I recommend replacing the exact-sounding justification with an explicit SNR/amplitude estimate and, if feasible, a sensitivity study in which the cutoff c is varied.","section":"Methods/Supplementary Eq. (S99), Algorithm 4"},{"comment":"The claim that 'all the noise schedules are unambiguously determined by the RG theory once a regulator is specified' is stronger than what the implementation actually does. The parameters v and m are fitted to Var[φk] (Table I), Λ1 and ΛT are chosen to match the DDPM's SNR endpoints (Table II), kcutoff is set by hand, the regulator is chosen separately for protein and image data, and the step-count parameters T0, τ, N0 are specified empirically. These are data-dependent hyperparameters in essentially the same sense as DDPM schedule choices, even if fewer in number and better motivated. The abstract and Discussion should either be softened or accompanied by a sensitivity analysis showing that the reported FID and protein metrics are insensitive to these choices.","section":"Supplementary Sec. B 3, Tables I, II, and Discussion"},{"comment":"The abstract's claim of 'accelerating sampling speed by an order of magnitude' is not directly supported by the reported comparison. The main text states that the RGDM produces quality comparable to a 'thousand-step DDPM' with 'hundreds of steps,' which, at the tested step counts (200, 300, 500, 1000), is roughly a factor of 3–5 rather than 10. Please report the actual FID values and the step count at which the RGDM reaches the DDPM T = 1000 quality, and adjust the abstract if the observed speedup factor is smaller than an order of magnitude in the tested range.","section":"Abstract and Fig. 4c"}],"minor_comments":[{"comment":"In Algorithm 4, newly introduced shell modes are sampled as φ_{t−1,k} ∼ N(0, βbar_tk), but consistency with the forward process would suggest N(0, βbar_{t−1,k}), or an explicit statement that K_t(k) is negligible in the shell so that the difference is immaterial. This point should be clarified.","section":"Fig. 2c and Algorithm 4"},{"comment":"The main text says the projection layers remove modes with |k| > Λt, while the implementation in Eq. (S99) removes modes with |k| > cΛt with c ≈ 54–63. The text should use the same cutoff notation as the algorithm.","section":"Fig. 1e and Algorithm 1"},{"comment":"The Methods section refers to 'the single-structure prediction accuracy in Fig. 3b,' but the CAMEO evaluation appears in Fig. 3c; please correct the cross-reference.","section":"Methods and Fig. 3 caption"},{"comment":"There are several typographical issues: 'frech ´et interception distance' should be 'Fréchet inception distance' (Fig. 4 caption), 'Adam optimzer' should be 'Adam optimizer' (Methods), and 'convex-diffusion' is presumably intended as 'convection-diffusion' or 'Fokker-Planck' in the relevant passages.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper is a solid and timely contribution, and the empirical protocol is careful. The main risk is not the numerical results but the theoretical framing: the exactness claims about scale separation and about the absence of data-dependent hyperparameters are stronger than what the derivations and protocol support. These issues are fixable within the manuscript's scope by reformulating Eq. (3), adding a quantitative bound for the projection cutoff, and tempering the abstract and Discussion. I therefore recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nQuick take on 2501.09064. The genuinely new thing is the derivation: they start from Polchinski's exact RG, rescale to a stable fixed point, and obtain a wavenumber-dependent noise schedule (Eqs. 8-9) together with a projection scheme that drops already-diffused high-k modes. That is a real conceptual step forward, and the protein-structure application with matched e3NN and matched step counts is a solid, reproducible piece of work—code is up.\n\nTwo things bother me. First, the claim of no data-dependent tuning does not hold up. The variance scale v and IR mass m are fitted to the data spectrum (Table I); the regulator r(x) is chosen differently for images vs proteins; and Lambda_1, Lambda_T are set by matching SNRs to the DDPM. So the RG theory fixes the schedule only after several free knobs have been turned. This is not fatal—the schedule still has fewer knobs than typical diffusion—but the advertised property is weaker than claimed.\n\nSecond, they cite Blurring Diffusion Models (Hoogeboom & Salimans) and Inverse Heat Dissipation (Rissanen et al.) but never benchmark against them. Those are essentially the same colored-noise, coarse-to-fine idea; the RG framing gives a nice unified view (SM Sec. C even shows any colored-noise schedule can be represented), but the empirical superiority claim over \"conventional diffusion models\" needs those baselines to be credible.\n\nOn the stress-test concern about Eq. (3): I think the worry is real but not demonstrated. The exact factorization p_Lambda = p_eff p_GS is an idealization for the effective action; for arbitrary p_data the high-k modes are not exactly Gaussian and independent of the low-k modes, and the projection layers discard them without training. The paper only shows the second-order spectrum (Fig. 1b) as evidence. Still, at the moment a mode is dropped its SNR is already low, and the image and protein results are decent, so this is a gap in justification rather than a shown failure. A referee should ask for higher-order statistics or an ablation that keeps those modes.\n\nVerdict: this deserves a serious referee. It is a well-posed idea with clean math, reproducible experiments, and honest citation of prior work. The revision should add the missing colored-noise baselines, temper the tuning-free claim, and address the scale-separation approximation explicitly. I would cite it for the Polchinski derivation and the projection scheme.","headline":"A genuinely RG-derived colored-noise diffusion schedule with clean math and honest experiments, but the tuning-free claim is overstated and the closest prior baselines (Blurring Diffusion, Inverse Heat Dissipation) are never benchmarked.","tokens_in":31980,"tokens_out":3176,"would_cite":true,"duration_ms":37874,"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":"A diffusion model whose noise schedule is fixed by exact renormalization-group flows generates higher-quality images and protein structures than standard DDPMs, often with an order of magnitude fewer sampling steps and no data-dependent…","keywords":["renormalization group","diffusion model","noise schedule","colored noise","coarse-to-fine generation","protein structure prediction","image generation","scale separation"],"falsifier":"Take the RGDM's forward process and compute the bispectrum of its intermediate distributions $p_t$; if modes with $|k|>\\Lambda_t$ are measurably non-Gaussian or correlated with the retained modes at any $t$, the scale-separation property (Eq. (3))—the premise that allows discarding those modes—fails. A complementary check is to high-pass-filter generated images and compare edge statistics between RGDM and DDPM at matched step counts; if the RGDM's advantage vanishes, its gains come from treating fine structure as Gaussian noise rather than from coarse-to-fine scheduling.","tokens_in":30909,"feed_emoji":"⚛️","tokens_out":11188,"duration_ms":108982,"temperature":0.7,"pith_summary":"Diffusion models generate data by progressively adding noise and then learning to reverse the process, but their noise schedules are usually tuned by hand and they need many steps. This paper claims that the schedule can instead be derived from the exact renormalization group: a forward process that eliminates information scale by scale, from fine details to coarse structure. Reversing these renormalization-group flows yields a model that generates images and protein structures coarse-to-fine. The authors report that this model consistently beats standard denoising diffusion probabilistic models on image quality and protein-structure metrics, often reaching comparable quality with an order of magnitude fewer sampling steps and without data-dependent schedule tuning. If correct, the result turns a physics framework for changing scales into a practical principle for making diffusion models faster and more sample-efficient.","feed_headline":"Scale-by-scale noise beats standard diffusion at a tenth of the steps","feed_subtitle":"The schedule is fixed by renormalization-group theory and gives sharper images and protein structures with fewer sampling steps.","key_machinery":"The engine is the exact renormalization-group flow equation (Eq. (6) in the paper), together with its rescaled convex-diffusion form (Eq. (7)). Given a one-dimensional regulator $r(x)$, the cutoff function is $K_t(k)=r(k^2/\\Lambda_t^2)/(1+r(k^2/\\Lambda_t^2))$, and this determines the colored-noise schedule $\\bar{\\alpha}_{tk}=K_t(k)$, $\\bar{\\beta}_{tk}=k^{-2}(1-\\bar{\\alpha}_{tk})$, in which modes are erased from high to low wavenumber as $\\Lambda_t$ decreases. The scale-separation property $p_\\Lambda(\\phi)=p_{\\mathrm{eff},\\Lambda}(\\phi_<)p_{\\mathrm{GS}}(\\phi_>)$ is what makes the projection layers legitimate: they delete Fourier modes with $|k|>\\Lambda_t$ before and after the denoising network, shrinking the effective input and output dimensions at each step and making both training and sampling cheaper.","core_discovery":"The paper's central claim is that the forward noising process of a diffusion model should be identified with an exact renormalization-group coarse-graining. For data modeled as a field $\\phi$, a cutoff function $K_t(k)$ with a chosen regulator $r(x)$ determines how much each Fourier mode is erased at time $t$: $\\bar{\\alpha}_{tk}=K_t(k)$ and $\\bar{\\beta}_{tk}=k^{-2}(1-K_t(k))$. Because the exact RG guarantees scale separation, the erased high-wavenumber modes become Gaussian and independent of the retained modes, so the model can discard them entirely during training and sampling. Reversing this flow generates data coarse-to-fine, and the empirical claim is that this consistently outperforms the standard DDPM, which denoises white noise on all modes at once, on protein structure prediction (RMSD, TM-score, GDT-TS, GDT-HA) and image generation (FID on CIFAR-10 and FFHQ), often improving quality and/or cutting the required steps by an order of magnitude. Since the schedule follows from the RG once a regulator is specified, the model removes the heuristic, data-dependent tuning of the noise schedule.","pith_inferences":["Inference: Because the schedule is determined by a single regulator, the model opens a simple testbed for choosing $r(x)$ from measured data spectra; selecting the regulator automatically from the dataset's variance profile could further close the gap to domain-optimized schedules, which the paper does not do.","Inference: The coarse-to-fine construction should carry over to other approximately scale-invariant data, such as 3D point clouds or audio spectrograms, with the projection layers pruned per scale; this is a natural transfer not tested in the paper.","Inference: If the Gaussian scale-separation assumption holds only at the power-spectrum level, sharp non-Gaussian features such as image edges or local protein backbone geometry are the most likely place for the model to degrade; measuring higher-order spectra of generated samples would reveal this.","Inference: The coverage result for colored schedules suggests that the practical advantage of the RGDM over tuned DDPMs may come less from the schedule itself than from the projection layers and the coarse-to-fine inductive bias, a distinction the experiments do not isolate."],"forward_implications":["On CIFAR-10 and FFHQ, the RGDM reaches or beats the DDPM's FID with an order of magnitude fewer generation steps, and the gap grows as $T$ shrinks.","On CAMEO protein targets, the RGDM's sampled structures score better than the DDPM on RMSD, TM-score, GDT-TS, and GDT-HA at the same step count, while still trailing state-of-the-art deterministic predictors.","Once a regulator $r(x)$ is chosen, the noise schedule is fixed by RG theory; no data-dependent schedule tuning is needed, and a simple $r(x)=x^{-1}$ works for image generation.","The projection layers, justified by scale separation, cut the effective dimensionality of the denoising network's input and output, which stabilizes training: DDPM validation FID saturates while RGDM keeps improving.","In the continuous-time picture, the RGDM's schedule covers every wavenumber-dependent (colored) noise schedule by an appropriate choice of $K_t(k)$, so the RGDM furnishes a unified parametrization of such schedules."],"supporting_citations":[{"why":"Supplies the exact renormalization-group flow equation that the forward noising process of the RGDM implements.","marker":"[21]"},{"why":"Provides the convex-diffusion and optimal-transport form of the RG flow used to derive the forward diffusion equation and its Gaussian fixed point.","marker":"[26]"},{"why":"Defines the denoising diffusion probabilistic model baseline whose noise schedule, UNet architecture, and performance the RGDM is compared against.","marker":"[4]"},{"why":"Establishes the nonequilibrium-thermodynamics diffusion-model formalism that the reverse-process sampling scheme builds on.","marker":"[1]"},{"why":"Supplies the functional-RG effective-action formalism used to write the effective model and derive the flow equations in the Supplementary Information.","marker":"[35]"},{"why":"Documents the approximately $1/k^2$ spectral scaling of natural images that motivates the data-model ansatz and the choice of Gaussian fixed point.","marker":"[23]"}],"fun_headline_variants":["RG-guided diffusion cuts steps tenfold","Physics-informed noise schedule accelerates diffusion","Coarse-to-fine generation beats standard diffusion","Renormalization group flow speeds up diffusion models"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that at every stage of coarse-graining, the fine details that have already been erased are exactly Gaussian noise with no information about the remaining coarse structure; the paper checks this only through the power spectrum, not through higher-order statistics.","fun_headline_variants_meta":{"raw":{"variants":["RG-guided diffusion cuts steps tenfold","Physics-informed noise schedule accelerates diffusion","Coarse-to-fine generation beats standard diffusion","Renormalization group flow speeds up diffusion models"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000385,"raw_usage":{"total_tokens":2069,"prompt_tokens":1015,"completion_tokens":1054,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":631,"completion_tokens_details":{"reasoning_tokens":1000}},"tokens_in":631,"tokens_out":1054,"duration_ms":8202,"temperature":1.0,"reasoning_tokens":1000,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T20:10:38.321431+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the RGDM's forward process and compute the bispectrum of its intermediate distributions $p_t$; if modes with $|k|>\\Lambda_t$ are measurably non-Gaussian or correlated with the retained modes at any $t$, the scale-separation property (Eq. (3))—the premise that allows discarding those modes—fails. A complementary check is to high-pass-filter generated images and compare edge statistics between RGDM and DDPM at matched step counts; if the RGDM's advantage vanishes, its gains come from treating fine structure as Gaussian noise rather than from coarse-to-fine scheduling.","supporting_citations":[{"cited_title":"Renormalization and effective lagrangians","cited_arxiv_id":null,"evidence_quote":"Supplies the exact renormalization-group flow equation that the forward noising process of the RGDM implements."},{"cited_title":"& Rezchikov, S","cited_arxiv_id":null,"evidence_quote":"Provides the convex-diffusion and optimal-transport form of the RG flow used to derive the forward diffusion equation and its Gaussian fixed point."},{"cited_title":"& Abbeel, P","cited_arxiv_id":null,"evidence_quote":"Defines the denoising diffusion probabilistic model baseline whose noise schedule, UNet architecture, and performance the RGDM is compared against."},{"cited_title":"& Ganguli, S","cited_arxiv_id":null,"evidence_quote":"Establishes the nonequilibrium-thermodynamics diffusion-model formalism that the reverse-process sampling scheme builds on."},{"cited_title":"& Sch\\\"utz, F","cited_arxiv_id":null,"evidence_quote":"Supplies the functional-RG effective-action formalism used to write the effective model and derive the flow equations in the Supplementary Information."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Documents the approximately $1/k^2$ spectral scaling of natural images that motivates the data-model ansatz and the choice of Gaussian fixed point."}],"review_version":1}