Generative diffusion model with inverse renormalization group flows

Kanta Masuki; Yuto Ashida

REVIEW 4 major objections 4 minor 4 cited by

Generative diffusion model with inverse renormalization group flows

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2501.09064 v1 pith:A4QA4I4E submitted 2025-01-15 cond-mat.stat-mech cond-mat.dis-nncs.LGphysics.app-phphysics.bio-ph

Kanta Masuki , Yuto Ashida This is my paper · ORCID

classification cond-mat.stat-mechcond-mat.dis-nncs.LGphysics.app-phphysics.bio-ph

keywords renormalizationgroupdiffusionmodelnoiseschedulecoloredcoarse-to-finegenerationproteinstructurepredictionimagescaleseparation

topics Protein Folding

open problems Protein Folding

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear colleague,

Quick 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.

Two 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.

Second, 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.

On 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.

Verdict: 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.

Referee Report

4 major / 4 minor

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.

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 (4)

[Renormalization group-based diffusion model, Eq. (3)] 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.
[Methods/Supplementary Eq. (S99), Algorithm 4] 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.
[Supplementary Sec. B 3, Tables I, II, and Discussion] 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.
[Abstract and Fig. 4c] 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.

minor comments (4)

[Fig. 2c and Algorithm 4] 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.
[Fig. 1e and Algorithm 1] 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.
[Methods and Fig. 3 caption] 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.
[Throughout] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the RGDM noise schedule is derived from the Polchinski RG flow equation, and the reported quality/speed gains are external empirical benchmarks rather than reductions to fitted inputs.

full rationale

The central forward process is not defined by fiat. Equation (7) is derived in SM Sec. A5 from the Polchinski RG equation (Eq. S68) by a rescaling of the effective distribution, and Eqs. (8)-(9) are the explicit solution of the resulting Fokker-Planck/Langevin equation (SM Eqs. S87-S90). The data distribution enters only as the t=0 initial condition, so the schedule is not constructed from the target output metric. The empirical claims are supported by held-out evaluations (CIFAR-10, FFHQ, CAMEO targets) using FID, RMSD, TM-score, GDT-TS, and GDT-HA; these are external, falsifiable benchmarks, not quantities fitted during training. The parameters v and m are explicitly fitted to the second-order spectrum Var[phi_k] (SM Table I and Fig. S5), and the paper does not disguise this as a prediction. SM Sec. C even concedes that any wavenumber-dependent noise schedule can be represented by some RG cutoff K_Lambda(k) = v_tk/(v_tk + k^2), so the RG derivation does not uniquely force the winning schedule; the optimality claim rests on the numerical comparisons, not on a definitional identity. Equation (3)'s scale-separation property is exact for the effective model S_Lambda built from the ansatz (2), while applying it to arbitrary natural data is an unverified modeling assumption (only second-order statistics are checked in Fig. 1b); this is a correctness risk, not a circular step. The only self-citations (refs. 62-63) appear as background for functional RG methods and are not load-bearing for the RGDM construction or its benchmark results.

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

The central derivation rests on the RG flow of a Gaussian reference action, which is standard mathematics, plus the domain assumption that natural data follow a scale-invariant k^-2 spectrum and that high-frequency modes become Gaussian. Several free parameters (v, m, Lambda endpoints, kcutoff, regulator shape, tau) are fitted or chosen heuristically, so the claim of an unambiguous schedule should be read as conditional on those choices. No new physical entities are introduced.

free parameters (6)

v (variance scale) = protein: (3.8)^2/2.3211, CIFAR-10: (3π/20)^2, FFHQ: (3π/20)^2
Chosen so that v/(k^2+m^2) matches Var[phi_k] of each dataset (SM Table I, Fig. S5). This is data-dependent fitting.
m (IR mass regulator) = protein: π/100, CIFAR-10: π/20, FFHQ: π/40
Fitted together with v to match the spectral variance and to regularize the k^-2 singularity at k=0 (SM Table I).
Lambda_1 and Lambda_T (SNR endpoints) = CIFAR-10: 47.1215 and 0.00290; FFHQ: 23.5608 and 0.00145
Chosen so that the signal-to-noise ratio at t=1 and t=T is the same order as the DDPM baseline (SM Table II), a data- and baseline-dependent choice.
kcutoff (RG projection cutoff) = CIFAR-10: π/20, FFHQ: π/40
Set to approximately the IR cutoff Lambda_IR to determine which Fourier modes are discarded (SM Table III).
tau, T0, N0 in the step number T(N) = protein: tau=8.17, T0=80, N0=32
Tuned so the RG scale at the final step scales as Lambda_T ∝ N^-1; in protein experiments chosen by matching mutual information to the DDPM (SM Table IV).
regulator function r(x) = image: r(x)=1/x; protein: r(x)=(exp(ln^2(x+1)-1))^{-1}
The only claimed ambiguity of the exact RG is the regulator; the authors choose different regulators per task and state the optimal choice is an open problem (Discussion).

assumptions (6)

domain assumption Natural data distributions are described by p(phi) ∝ exp(-S_data(phi)) with S_data = (1/2)∫(∇phi)^2 + V(phi), i.e., a scale-invariant k^-2 spectrum and an unspecified interaction V.
Introduced in Eq. (2) and justified by power-law spectral densities in Fig. 1b; this is an empirical ansatz, not derived.
domain assumption The scale-separation property p_Lambda(phi) = p_eff,Lambda(phi_<) p_GS(phi_>) holds for the RG flow, so high-wavenumber modes are Gaussian and independent.
Used to justify the projection layers and the Gaussian resampling of high-k modes (Eq. (3), Methods). It is proven for the exact RG of the Gaussian reference, but for real data it is an approximation.
standard math The exact RG (Polchinski equation) applies to the data action S_data.
The derivation of Eq. (6) from Polchinski (Ref. 21) is standard statistical field theory; the authors reproduce it in SM Sec. A4.
domain assumption The denoising DNN (UNet or e3NN) is sufficiently expressive to learn the colored noise.
Standard assumption in deep generative modeling; no guarantee for finite data.
domain assumption At t=T the distribution has converged to the Gaussian prior p_GS.
Relies on Lambda_T << Lambda_IR, which is enforced by choosing Lambda_T; an approximation.
standard math The conditional forward process is Gaussian with the specified alpha and beta.
Follows from the linear Langevin equation (S87)-(S90) in the SM; standard SDE theory.

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Pith. "Pith review of Generative diffusion model with inverse renormalization group flows." pith.science (2026). https://pith.science/paper/A4QA4I4E

@misc{pith2026250109064,
  author       = {Pith},
  title        = {Pith review of: Generative diffusion model with inverse renormalization group flows},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A4QA4I4E}},
  note         = {Machine review of arXiv:2501.09064}
}

read the original abstract

Diffusion models represent a class of generative models that produce data by denoising a sample corrupted by white noise. Despite the success of diffusion models in computer vision, audio synthesis, and point cloud generation, so far they overlook inherent multiscale structures in data and have a slow generation process due to many iteration steps. In physics, the renormalization group offers a fundamental framework for linking different scales and giving an accurate coarse-grained model. Here we introduce a renormalization group-based diffusion model that leverages multiscale nature of data distributions for realizing a high-quality data generation. In the spirit of renormalization group procedures, we define a flow equation that progressively erases data information from fine-scale details to coarse-grained structures. Through reversing the renormalization group flows, our model is able to generate high-quality samples in a coarse-to-fine manner. We validate the versatility of the model through applications to protein structure prediction and image generation. Our model consistently outperforms conventional diffusion models across standard evaluation metrics, enhancing sample quality and/or accelerating sampling speed by an order of magnitude. The proposed method alleviates the need for data-dependent tuning of hyperparameters in the generative diffusion models, showing promise for systematically increasing sample efficiency based on the concept of the renormalization group.

Figures

Figures reproduced from arXiv: 2501.09064 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

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**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

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Forward citations

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ENTRY address archive archivePrefix author booktitle chapter edition editor eprint howpublished institution journal key month note number organization pages publisher primaryClass school series title type url volume year label INTEGERS output.state before.all mid.sentence afte...
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" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 gl...

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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