Pith. sign in

REVIEW 3 major objections 3 minor 60 references

Instability in Diffusion ODEs: An Explanation for Inaccurate Image Reconstruction

T0 review · 3 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Diffusion reconstruction is intrinsically unstable: as image dimension grows, the probability of instability tends to one.

desk verdict Good mechanism, shaky theorem: the instability intuition is plausible and the experiments are suggestive, but the δ→0 step in Theorem 4.3 quietly substitutes π_gen for π_real and the asymptotic conclusion does not follow. read the letter →

arxiv 2506.18290 v1 pith:XF6BL65F submitted 2025-06-23 cs.LG

Han Zhang , Jinghong Mao , Shangwen Zhu , Zhantao Yang , Lianghua Huang , Yu Liu , Deli Zhao , Ruili Feng
show 1 more author
Fan Cheng
This is my paper · ORCID
classification cs.LG
keywords diffusionreconstructionprobabilityflowODEintrinsicinstabilityinversiondistributionsparsityJacobiansingularvaluesasymptoticerroramplification
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 reconstruction inverts an image to noise through the probability-flow ODE and then regenerates it by solving the same ODE backward. This paper argues that even in the ideal, continuous-time setting this round trip fails in a strong sense: the generation mapping can amplify infinitesimal perturbations in the inverted noise. It defines instability through the Jacobian of the generation mapping and shows that instability is tied to the sparsity of the generation distribution, where most probability mass sits in scattered small regions and the surrounding space has low density. Under stated assumptions, the paper proves that the probability of encountering such instability tends to one as the image dimension grows, implying that numerical errors, however small, can be amplified into large reconstruction errors in high dimensions.

What carries the argument

The central object is the intrinsic instability coefficient E_F(x,u) = ||J_F(x)u||/||u|| and its geometric average bar_E_F(x), the geometric mean of the singular values of the Jacobian J_F(x). The carrying identity is the density-ratio formula from Lemma D.1: (bar_E_{phi_1}(x))^n >= |prod_i lambda_i(J(1))| = p0(x0) / p1(phi_1(x0)), meaning that the expansion of infinitesimal volumes along the ODE flow equals the ratio of the initial Gaussian density to the generated density. Because the generation distribution pi_gen is sparse, real images often fall where p_gen is tiny, making this ratio large; probability preservation then forces large gradients in the vector field and hence large singular values. This is what converts distribution sparsity into instability and finally into reconstruction error amplification.

What would settle it

For a trained diffusion model on images of dimension n, estimate the geometric mean of the singular values of the generation map's Jacobian at the inverted noise $G^{{-1}}$(x) for real images x. If, for some fixed M > 1, the measured fraction of images with bar_E_G($G^{{-1}}$(x)) > M does not approach 1 as n increases, Theorem 4.3 is contradicted; a cheaper check is whether real image mass concentrates in high-density regions of pi_gen, which would keep the epsilon term bounded away from zero.

Watch

Extended reading notes

Core claim

The central claim is Theorem 4.3: for the ideal diffusion generation mapping G built from the probability-flow ODE, and for real images sampled from a distribution pi_real, the probability P_M = pi_real({x : bar_E_G($G^{{-1}}$(x)) > M}) satisfies P_M -> 1 as n -> infinity for every fixed M > 1, provided Assumptions 4.1 to 4.4 hold. Here bar_E_G is the geometric average of the singular values of the Jacobian of G, and $G^{{-1}}$ is the ideal diffusion inversion. In words: for high-dimensional image data, the generation mapping almost surely has an unbounded amplification factor at the inverted noise of a real image, so even infinitesimal perturbations are magnified. The proof splits the desired probability into two vanishing pieces: the chance that a real image avoids the low-density regions of the generation distribution, and the chance that its inverted noise falls far outside a Gaussian concentration ball.

Load-bearing premise

The asymptotic proof rests on Assumption 4.2, which requires that every point of the unit cube [0,1]^n is a possible real image with density at least a fixed positive amount, so that real images can land in the low-density regions of the generation distribution.

Editorial extensions

If this is right

  • Large reconstruction errors in diffusion models are not solely numerical artifacts; they are amplified by an intrinsic property of the probability-flow ODE generation mapping.
  • Under Assumptions 4.1 to 4.4, the probability that the geometric-average instability coefficient exceeds any fixed M > 1 converges to one as the dimension n tends to infinity.
  • The mechanism is the sparsity of the generation distribution: low-density regions of pi_gen must be images of large Gaussian regions, which requires large Jacobian singular values.
  • Numerical experiments on toy mixture-of-Gaussians settings and on Stable Diffusion 3.5 and FLUX show a positive correlation between measured instability coefficients and reconstruction error.
  • Improvements that only reduce ODE discretization error or make inversion numerically exact cannot remove reconstruction error if the underlying generation mapping remains unstable.

Reading between the lines

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

  • If the asymptotic result transfers to latent diffusion spaces, where the relevant dimension is the latent dimension rather than pixel count, then even exact inversion samplers will still show uncorrectable reconstruction failure whenever the latent distribution is sparse.
  • A testable implication of the density-ratio argument is that training procedures which smooth the generation distribution, such as adding noise regularization or enforcing Lipschitz bounds on the vector field, should reduce the instability probability.
  • The same push-forward density-ratio argument applies to any generative flow with a sparse target distribution, so similar instability may appear in flow matching or other ODE-based generative models whenever the data lie on a low-dimensional manifold surrounded by low-density regions.
  • One could directly test the paper's mechanism by measuring whether reconstruction failures concentrate on inputs whose inverted noise falls in low-density regions of pi_gen, rather than on inputs with large discretization error.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper argues that errors in diffusion-based image reconstruction are amplified by an intrinsic instability of the PF-ODE generation map, caused by the sparsity of the generation distribution. It defines an intrinsic instability coefficient and a geometric-average metric, provides toy and Stable Diffusion/FLUX experiments showing a positive correlation between instability and reconstruction error, and claims a theorem (Theorem 4.3) that, under assumptions on the real and generated distributions, the probability of instability tends to one as the data dimensionality grows. The main theoretical contribution is the asymptotic analysis in Section 4, with the proofs in Appendix D.

Significance. If Theorem 4.3 were correct, the result would be significant: it would provide a rigorous mechanism by which ideal diffusion reconstruction of high-dimensional images is almost surely unstable, shifting the interpretation of reconstruction failures from purely numerical to intrinsic. The paper also offers useful empirical evidence—correlation between the instability coefficient and reconstruction error in Stable Diffusion 3.5 and FLUX, including with a second-order Heun solver—and a universal approximation result for the mixture-of-Gaussian-neighbors class. However, the central asymptotic theorem contains a load-bearing proof gap and rests on an unrealistic distributional premise, and the supporting Theorem 3.1 has an inequality-direction problem. The empirical correlation is suggestive but does not by itself establish the amplification mechanism.

major comments (3)
  1. [Appendix D.5 / Eq. (11)] The proof that δ→0 in Theorem 4.3 is not established. The quantity δ is defined as π_real({x: ‖G^{-1}(x)‖² > 2n+3√(2n)}), but the tail computation in Appendix D.5 is performed under π_gen and then transferred to π_real via absolute continuity. A change of variables x=G(z) gives δ_n = ∫_{‖z‖²>2n+3√(2n)} γ(z)[p_real(G(z))/p_gen(G(z))] dz, and under Assumptions 4.3–4.4 the density p_gen can be exponentially small on most of the cube while p_real is only bounded above, so the density ratio need not be uniformly bounded in n. Absolute continuity alone provides no quantitative control, so δ→0 is equivalent to an unstated strong closeness condition between π_real and π_gen. Since P_M ≥ 1−ε−δ, the conclusion P_M→1 collapses without this step.
  2. [Assumption 4.2] Assumption 4.2, that the real image density p_real satisfies p_real ≥ C0 > 0 on the entire cube [0,1]^n, contradicts the low-dimensional manifold structure of natural images and is adopted precisely to force ε→0. The paper's claim that Theorem 4.3 is based on the characteristics of image data (Section 4.2.1) is therefore not supported; the theorem depends on a premise that real images fill the pixel cube. This is a modeling assumption rather than an internal inconsistency, but it means the asymptotic result does not apply to the image distributions for which the instability problem is posed.
  3. [Theorem 3.1 / Eq. (5), Appendix D.2] The derivation in Appendix D.2 produces an upper bound on the Euler global truncation error, E_n ≤ h M2/(2L)(e^L−1), and substituting L ≥ log C gives E_n ≤ h M2(C−1)/(2 log C). This is an upper bound, so it cannot justify the claimed lower bound U ≥ h M2(C−1)/(2 log C)·C in Eq. (5). The direction of the inequality in Theorem 3.1 is therefore unsupported; the theorem as stated does not demonstrate that the reconstruction error is bounded below by the displayed quantity.
minor comments (3)
  1. [Section 4.2, page 7] The sentence 'In the next two subsections, we will first make reasonable assumptions ... in Section 4.1' should refer to Section 4.2, since the assumptions are discussed there.
  2. [Appendix D.4 header] The header 'Proof of Theorem 4.1' is repeated for the proof of the universal approximation result; it should read 'Proof of Theorem 4.2'.
  3. [Figure 4(c) and Figure A1] The correlation analyses report fit lines but no confidence intervals or significance tests; adding these would strengthen the empirical claim of a positive relationship.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the central probability bound is a conditional mathematical derivation, not a self-referential prediction.

full rationale

The paper's central derivation is not circular in the sense defined here. Theorem 4.1 is a genuine measure-theoretic bound: Lemma D.1 obtains the geometric-average instability coefficient from the continuity equation, Lemma D.2 is an elementary measure inequality, and the decomposition P_M >= 1 - epsilon - delta follows by algebra from these lemmas; none of these steps assumes P_M -> 1 as an input. Theorem 4.3 is a conditional asymptotic statement whose epsilon -> 0 part is derived from Assumptions 4.1-4.4 by explicit volume estimates, and the sparsity assumptions are stated rather than fitted. The only overlapping self-citation is reference [50], used only as an example application in the introduction, so it is not load-bearing. The main weakness of the paper is a rigor gap, not circularity: in Appendix D.5 the proof of delta -> 0 transfers a tail bound from pi_gen to pi_real using the sentence "Since pi_real(n) is absolutely continuous about pi_gen till the infinite-dimensional case," a condition that is not among Assumptions 4.1-4.4 and is stronger than ordinary finite-dimensional absolute continuity. This is an omitted proof or hidden assumption and should be weighed as a correctness risk, but it is not a circular reduction of the conclusion to the assumptions.

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

No fitted parameters are introduced; the central theorem is a formal consequence of the stated domain assumptions. The main burden is carried by Assumptions 4.2 and 4.4, which already assert the sparsity and uniform-like real distribution that produce the conclusion. The change-of-variables identity is standard mathematics, but its application to real image distributions is the fragile step.

assumptions (6)
  • standard math Standard change-of-variables and push-forward formula for ODE flows (Lemma D.1).
    Used to relate the geometric mean of Jacobian singular values of the generation map to the density ratio p_noise divided by p_data.
  • domain assumption Assumption 4.1: supp(pi_real) = [0,1]^n.
    Treats the real image distribution as occupying the entire pixel cube.
  • domain assumption Assumption 4.2: p_real(x) >= C0 > 0 on [0,1]^n.
    Assumes every pixel configuration has non-negligible probability; unrealistic for natural images and load-bearing for epsilon -> 0.
  • domain assumption Assumption 4.3: p_gen is a mixture of Gaussian neighbors, p_gen = sum a_i f_i * g_{w_i}.
    Models the generation distribution as concentrated around local neighborhoods with Gaussian smoothing.
  • domain assumption Assumption 4.4: sparsity assumption group (constant m, disjoint separated neighborhoods, small side lengths, low outside probability).
    Encodes the conclusion that generation density is sparse; drives the asymptotic guarantee but is not derived from data.
  • standard math Gaussian concentration in high dimensions (chi-square tail bound).
    Used for delta -> 0, but only applies to standard Gaussian noise, not to the pushforward of pi_real under G^{-1} without an additional unjustified assumption.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Instability in Diffusion ODEs: An Explanation for Inaccurate Image Reconstruction." pith.science (2026). https://pith.science/paper/XF6BL65F

@misc{pith2026250618290,
  author       = {Pith},
  title        = {Pith review of: Instability in Diffusion ODEs: An Explanation for Inaccurate Image Reconstruction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XF6BL65F}},
  note         = {Machine review of arXiv:2506.18290}
}
read the original abstract

Diffusion reconstruction plays a critical role in various applications such as image editing, restoration, and style transfer. In theory, the reconstruction should be simple - it just inverts and regenerates images by numerically solving the Probability Flow-Ordinary Differential Equation (PF-ODE). Yet in practice, noticeable reconstruction errors have been observed, which cannot be well explained by numerical errors. In this work, we identify a deeper intrinsic property in the PF-ODE generation process, the instability, that can further amplify the reconstruction errors. The root of this instability lies in the sparsity inherent in the generation distribution, which means that the probability is concentrated on scattered and small regions while the vast majority remains almost empty. To demonstrate the existence of instability and its amplification on reconstruction error, we conduct experiments on both toy numerical examples and popular open-sourced diffusion models. Furthermore, based on the characteristics of image data, we theoretically prove that the instability's probability converges to one as the data dimensionality increases. Our findings highlight the inherent challenges in diffusion-based reconstruction and can offer insights for future improvements.

Figures

Figures reproduced from arXiv: 2506.18290 by the authors.

Figure 1
Figure 1. Instability amplifies diffusion reconstruction errors. Given an initial point in the data distribution, the reconstruction process first undergoes diffusion inversion to produce the actual inverted noise. Numerical errors introduced by the inversion will cause the actual inverted noise to deviate from the theoretically ideal inverted noise. When the generation process exhibits instability, these numerical errors are… view at source ↗
Figure 2
Figure 2. Intuitive illustration of instability occurrence during the generation process. For typical image data, the generation distribution is inherently sparse, meaning that most of the probability mass is concentrated in scattered, small regions, while the majority of the space has low probability. In contrast, the prior distribution for the generation process, i.e., the Gaussian, concentrates its probability within a bou… view at source ↗
Figure 3
Figure 3. Experimental evidence on the existence of instability for a two-dimensional generation distribution. (a) visualizes the density function of the generation distribution, which is a mixture of Gaussians generated from a standard gaussian. The green region, after applying generation mapping, becomes a significantly larger red region. (b) visualizes the intrinsic instability coefficient along the y-axis. It is observed … view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Positive correlation between the reconstruction error and instability coefficient in a numerical case and Stable Diffusion 3.5 [11]. In numerical experiments, the diffusion model’s generation distribution is a two-dimensional mixture of Gaussians. (a) shows the probabi…

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

60 extracted references · 36 canonical work pages

  1. [1]

    M. S. Albergo, N. M. Boffi, and E. Vanden-Eijnden. Stochastic interpolants: A unifying framework for flows and diffusions.arXiv preprint arXiv:2303.08797, 2023

    arXiv 2023

  2. [2]

    B. D. Anderson. Reverse-time diffusion equation models.Stochastic Processes and their Applications, 12(3):313–326, 1982. ISSN 0304-4149. doi: https://doi.org/10.1016/ 0304-4149(82)90051-5. URL https://www.sciencedirect.com/science/article/ pii/0304414982900515

    arXiv 1982

  3. [3]

    Balaji, S

    Y . Balaji, S. Nah, X. Huang, A. Vahdat, J. Song, Q. Zhang, K. Kreis, M. Aittala, T. Aila, S. Laine, B. Catanzaro, T. Karras, and M.-Y . Liu. eDiff-I: Text-to-image diffusion models with an ensemble of expert denoisers, 2023. URLhttps://arxiv.org/abs/2211.01324

    arXiv 2023

  4. [4]

    Billingsley.Convergence of probability measures

    P. Billingsley.Convergence of probability measures. Wiley Series in Probability and Statistics: Probability and Statistics. John Wiley & Sons Inc., New York, second edition, 1999. ISBN 0-471-19745-9. A Wiley-Interscience Publication

    work page 1999

  5. [5]

    Black Forest Labs. Flux.1. https://blackforestlabs.ai, 2024. URL https:// blackforestlabs.ai. Accessed: 2024-12-03

    work page 2024

  6. [6]

    Chung, B

    H. Chung, B. Sim, D. Ryu, and J. C. Ye. Improving diffusion models for inverse problems using manifold constraints.NeurIPS, 35:25683–25696, 2022

    work page 2022

  7. [7]

    Chung, B

    H. Chung, B. Sim, and J. C. Ye. Come-closer-diffuse-faster: Accelerating conditional diffusion models for inverse problems through stochastic contraction. InCVPR, pages 12413–12422, 2022

    work page 2022

  8. [8]

    J. Dai, Y . Zhang, S. Chen, J. Yang, and L. Luo. Erddci: Exact reversible diffusion via dual-chain inversion for high-quality image editing.arXiv preprint arXiv:2410.14247, 2024

    work page Pith review arXiv 2024

Show all 60 references
  1. [9]

    Dhariwal and A

    P. Dhariwal and A. Nichol. Diffusion models beat GANs on image synthesis.NeurIPS, 2021

    2021

  2. [10]

    W. Dong, S. Xue, X. Duan, and S. Han. Prompt tuning inversion for text-driven image editing using diffusion models. InCVPR, pages 7430–7440, 2023

    2023

  3. [11]

    Esser, S

    P. Esser, S. Kulal, A. Blattmann, R. Entezari, J. Müller, H. Saini, Y . Levi, D. Lorenz, A. Sauer, F. Boesel, et al. Scaling rectified flow transformers for high-resolution image synthesis. In ICML, 2024

    2024

  4. [12]

    Folland.Real Analysis: Modern Techniques and Their Applications

    G. Folland.Real Analysis: Modern Techniques and Their Applications. Pure and Applied Math- ematics: A Wiley Series of Texts, Monographs and Tracts. Wiley, 2013. ISBN 9781118626399. URLhttps://books.google.com.hk/books?id=wI4fAwAAQBAJ

    2013

  5. [13]

    G. B. Folland.Real analysis: modern techniques and their applications, volume 40. John Wiley & Sons, 1999

    1999

  6. [14]

    Kodak lossless true color image suite

    Franzen, R. Kodak lossless true color image suite. http://r0k.us/graphics/kodak, 1999. URLhttp://r0k.us/graphics/kodak. Accessed: 2024-12-03

    1999

  7. [15]

    R. Gal, Y . Alaluf, Y . Atzmon, O. Patashnik, A. H. Bermano, G. Chechik, and D. Cohen-Or. An image is worth one word: Personalizing text-to-image generation using textual inversion.arXiv preprint arXiv:2208.01618, 2022

    2022 arXiv

  8. [16]

    Hertz, R

    A. Hertz, R. Mokady, J. Tenenbaum, K. Aberman, Y . Pritch, and D. Cohen-Or. Prompt-to- prompt image editing with cross attention control.arXiv preprint arXiv:2208.01626, 2022

    2022 arXiv

  9. [17]

    Ho and T

    J. Ho and T. Salimans. Classifier-free diffusion guidance.arXiv preprint arXiv:2207.12598, 2022

    2022 arXiv

  10. [18]

    J. Ho, A. Jain, and P. Abbeel. Denoising diffusion probabilistic models. InNeurIPS, 2020

    2020

  11. [19]

    S. Hong, K. Lee, S. Y . Jeon, H. Bae, and S. Y . Chun. On exact inversion of dpm-solvers. In CVPR, 2024. 10

    2024

  12. [20]

    Ikeda and S

    N. Ikeda and S. Watanabe.Stochastic Differential Equations and Diffusion Processes. North- Holland Mathematical Library. North Holland, 2014. ISBN 9781483296159. URL https: //books.google.com.hk/books?id=QZbOBQAAQBAJ

    2014

  13. [21]

    X. Ju, A. Zeng, Y . Bian, S. Liu, and Q. Xu. Pnp inversion: Boosting diffusion-based editing with 3 lines of code. InICLR, 2024

    2024

  14. [22]

    Karras, M

    T. Karras, M. Aittala, T. Aila, and S. Laine. Elucidating the design space of diffusion-based generative models.arXiv preprint arXiv:2206.00364, 2022

    2022 arXiv

  15. [23]

    Kawar, S

    B. Kawar, S. Zada, O. Lang, O. Tov, H. Chang, T. Dekel, I. Mosseri, and M. Irani. Imagic: Text-based real image editing with diffusion models, 2023. URL https://arxiv.org/abs/ 2210.09276

    2023 arXiv

  16. [24]

    Y . Li, H. Wang, Q. Jin, J. Hu, P. Chemerys, Y . Fu, Y . Wang, S. Tulyakov, and J. Ren. Snapfusion: Text-to-image diffusion model on mobile devices within two seconds, 2023. URL https: //arxiv.org/abs/2306.00980

    2023 arXiv

  17. [25]

    H. Lin, M. Wang, J. Wang, W. An, Y . Chen, Y . Liu, F. Tian, G. Dai, J. Wang, and Q. Wang. Schedule your edit: A simple yet effective diffusion noise schedule for image editing.arXiv preprint arXiv:2410.18756, 2024

    2024 arXiv

  18. [26]

    T.-Y . Lin, M. Maire, S. Belongie, J. Hays, P. Perona, D. Ramanan, P. Dollár, and C. L. Zitnick. Microsoft coco: Common objects in context. InECCV, pages 740–755. Springer, 2014

    2014

  19. [27]

    Lipman, R

    Y . Lipman, R. T. Chen, H. Ben-Hamu, M. Nickel, and M. Le. Flow matching for generative modeling. InICLR, 2023

    2023

  20. [28]

    X. Liu, C. Gong, and Q. Liu. Flow straight and fast: Learning to generate and transfer data with rectified flow. InICLR, 2023

    2023

  21. [29]

    X. Liu, X. Zhang, J. Ma, J. Peng, and Q. Liu. Instaflow: One step is enough for high- quality diffusion-based text-to-image generation, 2024. URL https://arxiv.org/abs/ 2309.06380

    2024 arXiv

  22. [30]

    C. Lu, Y . Zhou, F. Bao, J. Chen, C. Li, and J. Zhu. Dpm-solver: A fast ode solver for diffusion probabilistic model sampling in around 10 steps.arXiv preprint arXiv:2206.00927, 2022

    2022 arXiv

  23. [31]

    C. Lu, Y . Zhou, F. Bao, J. Chen, C. Li, and J. Zhu. Dpm-solver++: Fast solver for guided sampling of diffusion probabilistic models.arXiv preprint arXiv:2211.01095, 2022

    2022 arXiv

  24. [32]

    Mokady, A

    R. Mokady, A. Hertz, K. Aberman, Y . Pritch, and D. Cohen-Or. Null-text inversion for editing real images using guided diffusion models. InCVPR, pages 6038–6047, 2023

    2023

  25. [33]

    Peebles and S

    W. Peebles and S. Xie. Scalable diffusion models with transformers. InICCV, pages 4195–4205, 2023

    2023

  26. [34]

    Pernias, D

    P. Pernias, D. Rampas, M. L. Richter, C. J. Pal, and M. Aubreville. Wuerstchen: An efficient architecture for large-scale text-to-image diffusion models, 2023. URL https://arxiv.org/ abs/2306.00637

    2023 arXiv

  27. [35]

    Podell, Z

    D. Podell, Z. English, K. Lacey, A. Blattmann, T. Dockhorn, J. Müller, J. Penna, and R. Rombach. Sdxl: improving latent diffusion models for high-resolution image synthesis.arXiv preprint arXiv:2307.01952, 2023

    2023 arXiv

  28. [36]

    Ramesh, P

    A. Ramesh, P. Dhariwal, A. Nichol, C. Chu, and M. Chen. Hierarchical text-conditional image generation with clip latents.arXiv preprint arXiv:2204.06125, 2022

    2022 arXiv

  29. [37]

    Rombach, A

    R. Rombach, A. Blattmann, D. Lorenz, P. Esser, and B. Ommer. High-resolution image synthesis with latent diffusion models. InCVPR, 2022

    2022

  30. [38]

    Saharia, W

    C. Saharia, W. Chan, S. Saxena, L. Li, J. Whang, E. Denton, S. K. S. Ghasemipour, B. K. Ayan, S. S. Mahdavi, R. G. Lopes, et al. Photorealistic text-to-image diffusion models with deep language understanding.arXiv preprint arXiv:2205.11487, 2022. 11

    2022 arXiv

  31. [39]

    Schneider.Convex Bodies: The Brunn-Minkowski Theory

    R. Schneider.Convex Bodies: The Brunn-Minkowski Theory. Encyclopedia of Mathematics and its Applications. Cambridge University Press, 1993. ISBN 9780521352208. URL https: //books.google.com.hk/books?id=2QhT8UCKx2kC

    1993

  32. [40]

    J. Song, C. Meng, and S. Ermon. Denoising diffusion implicit models.arXiv preprint arXiv:2010.02502, 2020

    2010 arXiv

  33. [41]

    Y . Song, J. Sohl-Dickstein, D. P. Kingma, A. Kumar, S. Ermon, and B. Poole. Score-based generative modeling through stochastic differential equations.ICLR, 2021

    2021

  34. [42]

    Stein and R

    E. Stein and R. Shakarchi.Fourier Analysis: An Introduction. Princeton University Press, 2003. ISBN 9780691113845. URL https://books.google.com.hk/books?id=I6CJngEACAAJ

    2003

  35. [43]

    X. Su, J. Song, C. Meng, and S. Ermon. Dual diffusion implicit bridges for image-to-image translation.arXiv preprint arXiv:2203.08382, 2022

    2022 arXiv

  36. [44]

    Tao.An Introduction to Measure Theory

    T. Tao.An Introduction to Measure Theory. Graduate Studies in Mathematics. American Mathematical Society, 2021. ISBN 9781470466404. URL https://books.google.com. hk/books?id=k0lDEAAAQBAJ

    2021

  37. [45]

    Tumanyan, M

    N. Tumanyan, M. Geyer, S. Bagon, and T. Dekel. Plug-and-play diffusion features for text-driven image-to-image translation. InCVPR, pages 1921–1930, 2023

    1921

  38. [46]

    von Platen, S

    P. von Platen, S. Patil, A. Lozhkov, P. Cuenca, N. Lambert, K. Rasul, M. Davaadorj, D. Nair, S. Paul, W. Berman, Y . Xu, S. Liu, and T. Wolf. Diffusers: State-of-the-art diffusion models. https://github.com/huggingface/diffusers, 2022

    2022

  39. [47]

    Wallace, A

    B. Wallace, A. Gokul, and N. Naik. Edict: Exact diffusion inversion via coupled transformations. InCVPR, pages 22532–22541, 2023

    2023

  40. [48]

    F. Wang, H. Yin, Y . Dong, H. Zhu, C. Zhang, H. Zhao, H. Qian, and C. Li. Belm: Bidirectional explicit linear multi-step sampler for exact inversion in diffusion models.arXiv preprint arXiv:2410.07273, 2024

    2024 arXiv

  41. [49]

    F. Wang, H. Yin, Y .-J. Dong, H. Zhu, H. Zhao, H. Qian, C. Li, et al. Belm: Bidirectional explicit linear multi-step sampler for exact inversion in diffusion models.NeurIPS, 2025

    2025

  42. [50]

    J. Xiao, R. Feng, H. Zhang, Z. Liu, Z. Yang, Y . Zhu, X. Fu, K. Zhu, Y . Liu, and Z.-J. Zha. Dreamclean: Restoring clean image using deep diffusion prior. InICLR, 2024

    2024

  43. [51]

    Zhang, J

    G. Zhang, J. P. Lewis, and W. B. Kleijn. Exact diffusion inversion via bidirectional integration approximation. InECCV, pages 19–36. Springer, 2024. 12 Contents of Appendix A Pseudocode for diffusion reconstruction 1 B Additional experiments 1 B.1 Reconstruction error amplific...

    2024

  44. [52]

    Whileπ′ gen(B∞) = limn→∞πgen G(z) :∥z∥ 2≥2n+ 3 √ 2n = 0, we derive that limn→∞δ= limn→∞πreal G(z) :∥z∥ 2≥2n+ 3 √ 2n = limn→∞πreal(n)(Bn) = 0

    And the set Bn ={∥z∥ 2 ≥2n+ 3 √ 2n} when placed in the function space means ∥fz∥L2 ≥m(S)(2 + 3 q 2 n), fz is the step function associated with z, as a result we obtain Bn ⊂B n+1, so B∞ = lim n→∞Bn = S∞ n=1Bn and limn→∞πreal(n)(Bn) =π real′(B∞). Whileπ′ gen(B∞) = limn→∞πgen G(z...

  45. [53]

    Denote each initial point as x[i,j] , wherei denotes the index alongx-axis, andj denotes the index alongy-axis

    Uniformly sample initial points as on a 201×201 uniform grid of the area [−1,1]×[−1,1] . Denote each initial point as x[i,j] , wherei denotes the index alongx-axis, andj denotes the index alongy-axis. Thus,x[i,j] = (−1 + i 100,−1 + j 100)fori,j= 0,1,...,200

  46. [54]

    Each solution at t= 1 ,i.e., the generated sample, can be denoted as ˆG(x[i,j]) for each initial pointx[i,j]

    Numerically solve the PF-ODE in Equation (1) using the RK45 solver fromt= 1 tot= 0 . Each solution at t= 1 ,i.e., the generated sample, can be denoted as ˆG(x[i,j]) for each initial pointx[i,j]

  47. [55]

    When computing the instability coefficient shown in Figure 3(c), we use two points with larger input difference

    Estimate the intrinsic instability coefficient as EG(x[i,j],n y)≈ ∥ ˆG(x[i,j+ 1])− ˆG(x[i,j])∥/∥x[i,j+ 1]−x[i,j]∥. When computing the instability coefficient shown in Figure 3(c), we use two points with larger input difference. The results demonstrate that the obtained instabi...

  48. [56]

    Uniformly sample initial data from[−1,1]×[−1,1]

  49. [57]

    Compute the reconstructed samples using the diffusion reconstruction process: first obtain the inverted noise ˆz= ˆG−1(x) for each initial data x, and then regenerate the data as ˆx= ˆG(ˆz)

  50. [58]

    Calculate the reconstruction errorR(x)for each sample

  51. [59]

    Estimate the intrinsic instability coefficient by 1) applying a small perturbation noise n to each inverted noise ˆz, and then 2) regenerating the data under perturbation as˜x= ˆG(ˆz+n) , 11 and 3) finally resulting in the estimation of intrinsic instability coefficient asEG(ˆ...

  52. [60]

    Statistically analyze the correlation betweenR(x)and the intrinsic instability coefficient. This procedure allows us to empirically assess the relationship between instability coefficients and reconstruction inaccuracies, thereby validating the theoretical insights discussed i...

    2014

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.