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REVIEW 3 major objections 6 minor 96 references

Unleashing High-Quality Image Generation in Diffusion Sampling Using Second-Order Levenberg-Marquardt-Langevin

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper proposes Levenberg-Marquardt-Langevin, a training-free second-order sampler that approximates the diffusion Hessian as a rank-one outer product of the noise prediction and reports consistent image-quality gains at negligible…

desk verdict The proposed Hessian sampler degenerates to the DPM-Solver baseline under its own stated hyperparameters, so the reported FID gains cannot be caused by the mechanism. read the letter →

arxiv 2505.24222 v1 pith:VD5J4XIK submitted 2025-05-30 cs.CV cs.LG

Fangyikang Wang , Hubery Yin , Lei Qian , Yinan Li , Shaobin Zhuang , Huminhao Zhu , Yilin Zhang , Yanlong Tang
show 4 more authors
Chao Zhang Hanbin Zhao Hui Qian Chen Li
This is my paper
classification cs.CVcs.LG
keywords diffusionmodelssamplingLevenberg-MarquardtNewton-Langevinlow-rankHessianapproximationdampingmechanismtraining-freesamplerDPM-Solver
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

The paper aims to show that diffusion-sampling quality can be improved by adding second-order (Hessian) information at each denoising step, with no retraining and almost no extra computation. Its central move is to approximate the Hessian of the log-density at each noise level by a rank-one outer product of the network's noise prediction, stabilize it with an identity damping term, and invert it cheaply through the Sherman-Morrison formula. This 'Levenberg-Marquardt-Langevin' step rotates and rescales the noise prediction before a DPM-Solver advance, and the authors report consistent FID and perceptual improvements on CIFAR-10, CelebA-HQ, SD-1.5, SD2-base, SD-XL, and PixArt-α. A sympathetic reader would care because the method is a drop-in arithmetic change to existing samplers, so if the claim holds, it upgrades the quality of essentially every pretrained diffusion model for free.

What carries the argument

The load-bearing object is the 'LM-approximated Hessian geometry': the rank-one matrix $H_{\mathrm{LM}} = \frac{1}{\sigma(t)^2\|\varepsilon_\theta\|^2}\varepsilon_\theta\varepsilon_\theta^\top$ from Proposition 1, regarded as the Hessian of the log-density, plus its damped version $H_{\mathrm{LM}} + \lambda I$. Damping converts a singular outer product into an invertible matrix whose inverse is computed in $O(d)$ time via the Sherman-Morrison identity $I - \frac{u u^\top}{\lambda + \|u\|^2}$, replacing the quadratic-complexity Hessian inversion that makes Newton-Langevin infeasible at diffusion scale. In Algorithm 1 this inverse acts as a preconditioner on the (EMA-mixed) noise prediction, followed by a unit-spectrum normalization, then a DPM-Solver step advances the latent state. This inverse-preconditioned update is the method's entire second-order content, and the theoretical sections are devoted to showing this approximation is accurate, the damped dynamics targets the right measure, and convergence is exponentially fast.

What would settle it

Run Algorithm 1 with the averaging coefficient set to exactly zero (no mixing of the previous step's prediction) and compare FID with the reported κ=1e-8 results on CIFAR-10 at 10 NFEs. If the scores are identical, the Hessian-guided update is not the source of the improvement; a direct numerical check of Algorithm 1 steps 11-13 will also show whether the normalized output equals the input prediction at κ=1e-8.

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Extended reading notes

Core claim

The central claim, stated on the paper's own terms, is that the Hessian of the diffused log-density admits a tractable low-rank form: Proposition 1 asserts $\nabla^2_{x_t}\log p_t(x_t) \approx \frac{1}{\sigma(t)^2\|\varepsilon_\theta\|^2}\varepsilon_\theta\varepsilon_\theta^\top$, a scaled outer product of the noise prediction. Adding $\lambda I$ makes this rank-one, singular matrix invertible, and the Sherman-Morrison formula gives the inverse as $\frac{1}{\lambda'\sigma(t)^2\|\varepsilon_\theta\|^2}\bigl(I - \frac{\varepsilon_\theta\varepsilon_\theta^\top}{\lambda' + \|\varepsilon_\theta\|^2}\bigr)$, where the bracketed matrix is the geometric essence carried into Algorithm 1. The sampler applies this damped inverse to an EMA-mixed noise prediction, restores the original norm, and hands the result to DPM-Solver, with theory (Propositions 2-4) bounding the approximation error in Hilbert-Schmidt norm, identifying the stationary measure, and giving exponential $\chi^2$-ergodic convergence. The paper's experiments then claim that this training-free change improves FID and perceptual metrics across six pretrained models at nearly identical wall-clock cost.

Load-bearing premise

The claimed quality gains rest entirely on the step that applies the Hessian geometry actually changing the noise-prediction direction; at the tiny averaging coefficient (κ=1e-8) used in the main experiments, that step mathematically leaves the prediction unchanged, so the gains depend on the coefficient being effectively larger than the paper reports.

Editorial extensions

If this is right

  • Any pretrained diffusion model, including latent and transformer-based text-to-image models, can adopt LML without retraining, auxiliary networks, or model-specific assumptions.
  • Wall-clock measurements in the paper show LML adds only fractions of a second even at 100 function evaluations, so the quality gain, if real, is essentially free at inference time.
  • Because LML only rewrites the noise prediction before the solver step, it composes with DPM-Solver, DDIM, ControlNet conditioning, and other pipeline components rather than replacing them.
  • At low NFE budgets (5-10 steps), where first-order samplers lose the most FID, LML reports its largest relative gains, which is the operating regime for interactive or batch image generation.

Reading between the lines

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

  • The same damped low-rank preconditioner could plausibly be attached to other ODE solvers, such as UniPC or DDIM, or combined with optimized timestep schedules; the paper only tests it with DDIM and DPM-Solver, so a compounding effect is an open testable possibility.
  • The mirror-duality view flagged in Remark 1 suggests the construction is not specific to score-based diffusion: a Levenberg-Marquardt-Langevin dual-space sampler can likely be written down for flow matching and rectified flow models.
  • An adaptive schedule for the damping coefficient $\lambda$ or the mixing coefficient $\kappa$, tuned per timestep rather than fixed globally as in the paper, is a natural extension that the authors' own limitation section invites.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper proposes a training-free, second-order diffusion sampler called Levenberg-Marquardt-Langevin (LML). The method approximates the Hessian of the log-density of the diffused distribution by a scaled outer product of the noise prediction (Proposition 1), stabilizes it with a damping term, and applies its inverse to the noise prediction at each step before feeding the result to DPM-Solver (Algorithm 1). The authors report significant FID and perceptual-metric improvements over DDIM, PNDM, DPM-Solver, DPM-Solver++, and UniPC across CIFAR-10, CelebA-HQ, SD-1.5, SD2-base, SD-XL, and PixArt-α, with negligible computational overhead, and provide theoretical results on approximation error and convergence.

Significance. If the mechanism worked as described, the contribution would be significant: a training-free, nearly zero-overhead sampler that improves generation quality across a wide range of diffusion models by exploiting second-order geometry. The paper is clearly written, the algorithm is precisely specified, and the experimental protocol is extensive, including wall-clock time measurements and Pareto curves. These are strengths that make the central flaw easy to identify. Unfortunately, the claimed improvement is internally inconsistent with the stated hyperparameters: with the paper's chosen κ=1e-8, the LML update is numerically the identity map on the noise prediction, so the sampler is indistinguishable from DPM-Solver. The reported gains therefore cannot be attributed to the proposed Hessian guidance, undermining the central claim.

major comments (3)
  1. [Algorithm 1 (steps 11–13) and Supp. B.2] With the EMA coefficient fixed to κ=1×10^-8 in Supp. B.2, the mixture eε_i = κ ε_{i+1} + (1-κ) ε_i is numerically identical to ε_i in float32. Then step 11 gives fH^{-1}_i = I - eε_i eε_i^T / (λ + ||eε_i||²), step 12 yields fH^{-1}_i ε_i = (λ/(λ+||ε_i||²)) ε_i, and step 13 rescales this to ||ε_i||, returning exactly ε_i. Thus the algorithm reduces to DPM-Solver with the unmodified noise prediction. This is directly contradicted by Table 1, which reports LML improving CIFAR-10 FID at 10 NFEs from 11.13 (DPM-Solver) to 6.54, and by Table 3 showing large gains on SD models. The improvement cannot be produced by the described mechanism; either a much larger κ was used in practice, or the baseline differs in an undocumented way.
  2. [§3.2, Eq. (8) and §3.4, step 13] The normalization in step 13 removes any scalar factor introduced by the damped inverse, so the Hessian guidance can only affect the update through a change in the direction of ε_i. When eε_i = ε_i, the direction is unchanged exactly. With κ=1e-8, the directional perturbation is O(κ), i.e., about 1e-8, which is far below the precision needed to explain the large experimental differences reported in Tables 1–4. The paper provides no analysis or ablation showing that such a tiny perturbation can produce the observed gains; its own Supp. B.2 statement that this κ value 'contributes minimally to performance enhancement' is inconsistent with the magnitude of the claimed improvements.
  3. [Supp. A.2, Eqs. (13)–(16)] The proof of Proposition 1 assumes that p_t(xt) is approximately Gaussian with mean α_t y_θ(xt,t), where y_θ is the learned denoiser. This is not justified: y_θ is not the true posterior mean in general, and the derivation uses ε_θ to express both the score and the Jacobian, making the approximation circular. Even if this approximation were accepted, the resulting Hessian form does not rescue the identity-problem in Algorithm 1, because the algorithm normalizes away the scalar factor. The theoretical claim about second-order geometry is therefore not established, and it is not connected to the actual numerical behavior of the sampler under the reported settings.
minor comments (6)
  1. [§3.3, first paragraph] The phrase 'at each nose level' appears to be a typo for 'noise level'.
  2. [Supp. C.2, caption of Figure 11] The dataset name is misspelled as 'CIAFR-10'; it should be 'CIFAR-10'.
  3. [Table 4 caption] The benchmark name appears as 'T2I-BC' in the caption and 'T2I-CB' in the main text; please standardize the abbreviation.
  4. [Figure 10, Supp. B.2] The κ-axis labels appear corrupted (e.g., '1 × 10 8'); presumably they denote 1e-8, 2e-8, etc., but as printed they are ambiguous.
  5. [§3.2, after Eq. (8)] The notation λ' is introduced and then the paper says it will 'not distinguish between λ and λ''; this can confuse the reader when λ appears in Algorithm 1, where the damping parameter is not accompanied by the schedule-dependent factor.
  6. [Proposition 2 and Supp. A.6] The statement of Proposition 2 uses quantities δ1, δ2, δ3, and D_y without defining them in the main text; the proof in Supp. A.6 only partially clarifies these definitions, and the bound is not interpretable as written.

Circularity Check

1 steps flagged · score 8.0 of 10

With κ=1e-8 the LML update is the identity mapping on the noise prediction, so the reported FID gains cannot be produced by the claimed Hessian mechanism; the central 'prediction' reduces to the baseline by construction.

  1. self definitional [Algorithm 1, steps 7 and 11-13; Supp. B.2]
    "eεi =κ∗ε i+1 + (1−κ)∗ε i ... fH −1 i =I− eεi eε⊤ i λ+∥eεi∥2 ... ε LM i = fH −1 i εi ... ε LM i = ∥εi∥ ∥εLM i ∥ εLM i ... we have chosen to fixκ= 1×10 −8 for all tests in Tables 1 and 3. This is a very small value that contributes minimally to performance enhancement."

    Setting κ=0 makes eε_i = ε_i. Then fH^{-1}_i ε_i = [I − ε_i ε_i^T/(λ+∥ε_i∥^2)] ε_i = (λ/(λ+∥ε_i∥^2)) ε_i, a positive scalar multiple of ε_i. Step 13 rescales this vector back to the original norm ∥ε_i∥, returning exactly ε_i. With the stated κ=1e-8 in float32, eε_i differs from ε_i by 1e-8(ε_{i+1}−ε_i), at or below machine precision, so the 'Hessian-guided' direction is numerically identical to the unmodified noise prediction. Algorithm 1 therefore reduces to DPM-Solver with ε_i, and the large FID improvements in Tables 1 and 3 cannot originate from the LML Hessian mechanism; the claimed mechanism is vacuous by construction.

full rationale

The central claim is that an LML-approximated Hessian geometry improves diffusion sampling quality. But Algorithm 1's construction makes the Hessian-guided gradient equal to the original noise prediction whenever the EMA coefficient is zero, and the paper fixes κ=1e-8. Mathematically, fH^{-1}_i ε_i is a positive scalar multiple of ε_i, and the normalization in step 13 restores the original norm, so the final vector is ε_i itself. The perturbation from κ=1e-8 is far below float32 precision relative to typical noise-prediction magnitudes, so numerically the algorithm is indistinguishable from feeding ε_i directly into DPM-Solver. The reported improvements—e.g., CIFAR-10 FID from 30.87 to 17.28 at 5 NFEs and from 17.45 to 6.54 at 10 NFEs—therefore cannot be explained by the Hessian step described in the paper; they would have to come from an undocumented implementation difference. This is a case where the method's central 'prediction' reduces to its input by construction, not a mere overclaim or a minor self-citation issue. The paper also cites the authors' own [84] for the Fisher-analytical form used in Proposition 2, but that self-citation is secondary to the identity-by-construction problem. Score 8 reflects that the main empirical result is internally inconsistent with the stated hyperparameters and that the claimed Hessian guidance is, at the stated setting, a vacuous transformation.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The method's free parameters (λ and κ) are both fitted or set to near-zero values that trivialize the claimed mechanism. The axioms are standard diffusion assumptions plus an ad hoc convexity condition for the convergence proof. There are no invented physical entities. The central mathematical content is the low-rank Hessian approximation, whose validity rests on the Gauss-Newton approximation and the sign of the Hessian, both of which are problematic.

free parameters (2)
  • damping coefficient λ = 0.0008 to 0.00005 on CIFAR-10 (Table 6); 0.001 on SD-1.5 (Table 7); tuned per dataset and per NFE
    λ directly controls the strength of the damped Hessian inverse. The paper uses a binary search over FID to choose λ for each setting (Supp B.2), so the reported performance is fitted to the evaluation metric.
  • EMA coefficient κ = 1e-8 for Tables 1 and 3; other values in ablations (Supp Figure 10)
    κ is described as contributing minimally to performance. With κ=1e-8 the mixture is numerically indistinguishable from the current noise prediction, and the LML update becomes identity.
assumptions (3)
  • domain assumption The learned noise predictor ε_θ approximates the true score well enough that the Gaussian form p_t(x_t) ≈ (1/√(2πσ_t)) exp(-‖x_t - α_t y_θ‖²/(2σ_t²)) holds (Supp A.2, Eq. 13).
    This is a standard assumption in diffusion modeling, but the paper uses it to derive the Hessian approximation without verifying the error from a potentially imperfect denoiser.
  • domain assumption The Gauss-Newton approximation omitting the second-derivative term r(x_t) ∂²r/∂x_t² in Eq. 16 is valid for diffusion models.
    This is the core approximation that turns the Hessian into a rank-one outer product. The paper provides an error bound (Prop. 2) that depends on several unverified constants (δ1, δ2, δ3, D_y) and is not numerically quantified.
  • ad hoc to paper The mirror Poincaré condition with φ = log p(·) + λ‖·‖²/2 holds for the target distribution (Assumption 1 in Supp A.8).
    The convergence proof (Prop. 4) requires this condition. Since log p is typically concave and λ is small, φ may not be convex, so the assumption is not obviously satisfied and is introduced only to make the theorem work.

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Cite this review

Pith. "Pith review of Unleashing High-Quality Image Generation in Diffusion Sampling Using Second-Order Levenberg-Marquardt-Langevin." pith.science (2026). https://pith.science/paper/VD5J4XIK

@misc{pith2026250524222,
  author       = {Pith},
  title        = {Pith review of: Unleashing High-Quality Image Generation in Diffusion Sampling Using Second-Order Levenberg-Marquardt-Langevin},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VD5J4XIK}},
  note         = {Machine review of arXiv:2505.24222}
}
read the original abstract

The diffusion models (DMs) have demonstrated the remarkable capability of generating images via learning the noised score function of data distribution. Current DM sampling techniques typically rely on first-order Langevin dynamics at each noise level, with efforts concentrated on refining inter-level denoising strategies. While leveraging additional second-order Hessian geometry to enhance the sampling quality of Langevin is a common practice in Markov chain Monte Carlo (MCMC), the naive attempts to utilize Hessian geometry in high-dimensional DMs lead to quadratic-complexity computational costs, rendering them non-scalable. In this work, we introduce a novel Levenberg-Marquardt-Langevin (LML) method that approximates the diffusion Hessian geometry in a training-free manner, drawing inspiration from the celebrated Levenberg-Marquardt optimization algorithm. Our approach introduces two key innovations: (1) A low-rank approximation of the diffusion Hessian, leveraging the DMs' inherent structure and circumventing explicit quadratic-complexity computations; (2) A damping mechanism to stabilize the approximated Hessian. This LML approximated Hessian geometry enables the diffusion sampling to execute more accurate steps and improve the image generation quality. We further conduct a theoretical analysis to substantiate the approximation error bound of low-rank approximation and the convergence property of the damping mechanism. Extensive experiments across multiple pretrained DMs validate that the LML method significantly improves image generation quality, with negligible computational overhead.

Figures

Figures reproduced from arXiv: 2505.24222 by the authors.

Figure 1
Figure 1. Schematic comparison between our LML method with baselines. While previous works mainly focus on intriguing designs on [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. The relation between optimization algorithms and MCMC sampling algorithms. We initially wanted to develop a diffusion [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Visual comparison of images generated by our LML method and other methods, using the pre-trained LDM on CelebA￾HQ 256×256 [39] with the same seeds. It shows that our LML method contributes to more vivid and detailed generated images. the second-order partial derivatives. The low-rank approxi￾mate Hessian of diffusion models that we obtained is shown below. It is important to note that it includes noise sched￾ule rel… view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Qualitative comparison of our LML method and other methods in text-guided image generation. The evaluation was per￾formed on SD-15 [67], using 10 NFEs and the same seeds. with LML at each noise level. At each noise level, we ini￾tially compute the LM low-rank approxima…
Figure 5
Figure 5. Figure 5: Qualitative comparison of our LML method and other methods in text-guided image generation. The evaluation was per￾formed on SD2-base [67], using 10 NFEs and the same seeds. 4. Theoretical Analysis In this section, we present rigorous theoretical analyses to substantia…
Figure 6
Figure 6. Figure 6: This line chart compares the log-FID scores ( [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: LML integrates seamlessly with ControlNet. [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: Comparison of Pareto curves between LML and baseline on SD-v1.5 and SD-v2-base on 30k COCO images, across various [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]
Figure 10
Figure 10. Figure 10: The performance of the LML method on CIFAR-10 [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]
Figure 9
Figure 9. Figure 9: The performance of the LML method on CIFAR-10 gen [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]
Figure 11
Figure 11. Figure 11: Comparison of the CIFAR-10 generation task performance between our LML method and the baseline methods under 10 NFEs. [PITH_FULL_IMAGE:figures/full_fig_p018_11.png]

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.