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REVIEW 4 major objections 4 minor 41 references

Adapting Noise to Data: Generative Flows from 1D Processes

T0 review · 4 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read Generative flows can learn their own noise from data using one-dimensional quantile functions.

desk verdict A genuinely new way to learn per-coordinate latent noise in flow matching, with an honest experimental story; the main issues are a real inconsistency in Proposition 5 and an abstract that promises weather experiments that are not in the paper. read the letter →

arxiv 2510.12636 v5 pith:Q5TV5AKJ submitted 2025-10-14 stat.ML cs.LGmath.AP

Jannis Chemseddine , Gregor Kornhardt , Richard Duong , Gabriele Steidl This is my paper · ORCID
classification stat.MLcs.LGmath.AP MSC 68T0749Q22
keywords flowmatchingquantilefunctionslearnedlatentnoiseWassersteindistanceheavy-taileddistributionsgenerativemodelsoptimaltransportconsistency
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 claims that the default Gaussian latent in flow-based generative models is a choice, not a requirement, and that learning the latent noise from data is both possible and useful. It parameterizes each coordinate of the latent by a learnable quantile function, a monotone map from (0,1) into R, and matches the resulting product distribution to the data by minimizing the Wasserstein distance. When combined with flow matching, this learned noise adapts to heavy tails and compact supports, shortens the transport paths the network must learn, and adds negligible computational cost. The same quantile machinery extends to consistency models, giving a unified way to turn noise selection into a data-driven design element.

What carries the argument

The engine is the quantile function: for a 1D measure, the monotone map Q from (0,1) to R defined by the inverse of the cumulative distribution function. Since W_2^2 between two 1D measures is the integrated squared difference of their quantile functions, learning the latent distribution reduces to learning monotone maps. The paper parameterizes each coordinate by a rational-quadratic spline with per-coordinate scale and bias, which guarantees strict monotonicity and cheap analytic derivatives. The loss is LAN(phi) = W_2^2(mu0, nu_phi) combined with the optimal-coupling flow matching loss LOT-CFM; a minibatch OT coupling connects latent and data samples, and a log-Jacobian regularizer preven

What would settle it

On a strongly correlated dataset (for example CIFAR-10 with the same training budget), if sweeping the regularization parameter beta and quantile loss weight lambda yields no FID improvement over a Gaussian latent while the learned W2 latent distance is decreasing, the claim that data-adapted noise improves sampling would be refuted. A simpler targeted calculation: for a 2D Gaussian with correlation rho, compute W_2^2 between the target and the best product measure; if this distance stays large for high rho, the latent cannot shorten paths by design.

Watch

Extended reading notes

Core claim

The central claim is that the terminal noise distribution in a flow model should be fitted to the data rather than fixed in advance. Restricting to independent one-dimensional components, the paper represents each coordinate's distribution by its quantile function Q^i_phi and learns these maps so that Law(Q_phi(U)) is as close as possible to the target in squared Wasserstein distance, W_2^2(mu0, Law(Q_phi(U))). Because W_2^2 between two 1D measures equals the L2 distance of their quantile functions, this objective is tractable and differentiable. The learned latent is then used inside an optimal-transport flow matching loss, so the velocity field and the noise are trained jointly. The paper

Load-bearing premise

The method assumes that a product distribution of independent one-dimensional marginals can get close enough to the data in Wasserstein distance; when the data has strong cross-coordinate correlations, this assumption fails and the learned noise brings only limited gains, as the paper itself reports on CIFAR-10.

Editorial extensions

If this is right

  • Flow matching models trained with learned quantile latents converge faster and can use smaller networks, because the transport paths from latent to data are shorter.
  • Heavy-tailed and compactly supported targets become reachable without hand-designing a heavy-tailed or uniform latent; the noise adapts to the tails automatically.
  • The framework plugs into standard Flow Matching and consistency objectives, so existing pipelines can adopt learned noise with minimal overhead (a few percent in the reported image experiments).
  • On highly correlated data such as CIFAR-10, the independence restriction caps the gain; improvements there require correlation-aware noise beyond product measures, as the paper states.
  • Future work, such as time-dependent quantile functions, could learn the entire transport path rather than only the endpoint distribution.

Reading between the lines

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

  • The W2-projection view suggests a practical diagnostic: compare each learned marginal quantile to the empirical marginal quantile of the data; where they diverge, the product-measure restriction is the bottleneck.
  • If the quantile-learning trick proves stable, it could replace hand-tuned noise schedules in tabular or scientific domains where heavy tails are the norm, even when standard image benchmarks show little gain.
  • A natural next step is to couple the per-coordinate quantiles with a copula or triangular map, letting the latent capture cross-coordinate correlations while keeping the 1D machinery; the paper's own image experiments point in this direction.
  • Because quantile interpolants are self-consistent, the same learned noise could be distilled into few-step samplers, making the speed advantage of shorter paths even more direct.
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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

4 major / 4 minor

Summary. The paper introduces a framework for generative flow models in which the latent noise distribution is learned rather than fixed to a Gaussian. The key construction decomposes a multi-dimensional noising process into independent one-dimensional components, parameterizes each component through a learnable quantile function, and optimizes the latent together with a conditional flow-matching objective. The loss combines an optimal-transport flow-matching term with the Wasserstein-2 distance between the data distribution and the factorized latent distribution (Eqs. (22)–(23)). Three one-dimensional noising processes are presented as examples: the Wiener process, the Kac process, and an MMD gradient flow toward a uniform distribution. Experiments on synthetic targets (GMM, funnel, checkerboard), MNIST, and CIFAR-10 are reported, with the main quantitative gains on MNIST and the funnel, and marginal gains on CIFAR-10. The paper also sketches how quantile interpolants can be used in consistency models such as IMM.

Significance. The central idea is attractive: replacing the fixed Gaussian latent by learnable per-coordinate quantile functions is simple, computationally light, and naturally covers heavy-tailed or compactly supported targets. If the derivations are corrected, the framework could be a useful toolkit for flow matching and consistency models, and the explicit connection to 1D processes (Wiener, Kac, MMD-uniform) is conceptually interesting. The availability of code and the clear exposition of the optimization are strengths. However, the significance is currently weakened by internal inconsistencies in two main propositions, an unsupported claim in the abstract about weather experiments, and the acknowledged limitation that the product-measure latent class cannot capture cross-coordinate correlations, which limits the practical impact on realistic image data.

major comments (4)
  1. [Section 3.3, Proposition 5 (Eqs. (16)–(18))] The stated velocity field is inconsistent with the Appendix. Eq. (17) gives v_t(x) = x/b (exp(t/b) - 1), while Appendix A, Eq. (25) with a=-b and x0=0 gives v_t(x) = x/[b (exp(t/b) - 1)]. These are reciprocals. The correct velocity for the process U_t = b(1 - exp(-t/b))U is x/[b(exp(t/b)-1)], and a direct computation gives ||v_t||^2_{L2(mu_t)} = (1/3) exp(-2t/b), not 2b/3 exp(-2t/b). The Appendix's action computation also contains an algebra error. This proposition is the basis of the MMD-flow example and must be corrected.
  2. [Section 3.2, Proposition 4] The Kac velocity field as written is not compatible with the symmetric density flow it accompanies. For the stated symmetric measure mu_t, the product mu_t v_t must be odd in x for the continuity equation to hold; hence v_t should be odd at points where the density is positive. The given formula v_t(x) = x/t + (r_t/c) I0(beta r_t)/I0'(beta r_t) contains an even additive term and evaluates at x=0 to t I0(beta c t)/I0'(beta c t) > 0. This is either a typo (the second term should likely involve x/r_t, for example) or a different non-minimal velocity is being used without explanation. Since the Kac process is one of the three showcase examples, this needs to be corrected and checked against reference [10].
  3. [Abstract and Section 5] The abstract states that numerical results on 'heavy-tailed weather and image datasets' confirm the method, but Section 5 contains no weather dataset or weather experiment. The experiments are on synthetic distributions (GMM, funnel, checkerboard) and MNIST/CIFAR-10. Either the missing weather experiments should be added, or the abstract should be revised to describe what is actually reported. This is a claim-support mismatch that currently overstates the empirical content.
  4. [Section 4.2 (Eq. (22)) and Section 5.2 (Fig. 14)] The 'shorter transport paths' statement is partly circular: LAN(phi) in Eq. (22) is exactly E_{(x,y)~pi_phi} ||y-x||^2 for the optimal coupling, so minimizing LAN is by definition minimizing the average squared straight-line path length. The empirical contribution is that the optimization is tractable and helps the downstream flow matching, not that shorter paths are an unexpected discovery. More substantively, the restriction to the product class S prevents the latent from encoding cross-coordinate correlations, and the CIFAR-10 results show only marginal FID gains (100 Euler steps: 6.26 best learned vs 6.44 baseline). The paper is transparent about this, but the broad claims in the title, abstract, and introduction should be qualified so that the reader understands the demonstrated benefits are on low-dimensional or margin-dominated targets.
minor comments (4)
  1. [Section 5.2, Figure 14] The table header appears garbled ('1 2 3 4 Baseline') and it is not clear which columns correspond to which beta values. The table should list the beta values explicitly.
  2. [Algorithm 1 and Appendix E.3] The regularization term L_reg from E.3 is used in the image experiments but is not included in Algorithm 1. Also, the main text refers to the regularization parameter as beta while E.3 uses lambda_reg; unify the notation.
  3. [Section F.1] The phrase 'after a 25k training epochs' should be 'steps'. In the funnel paragraph, the sentence 'Note however we still train our method' is incomplete.
  4. [References] Reference [16] has a typo: 'C. Meng, , C. D. Manning' should be 'C. Meng, C. D. Manning'.

Circularity Check

1 steps flagged · score 4.0 of 10

Path-shortening is Eq. (22) by construction; self-citations to authors' prior theorems are present but the central learned-quantile claim has independent FID support.

  1. fitted input called prediction [Section 4.2, Eq. (22); Section 5.1]
    "LAN(ϕ)=W2^2(μ0,νϕ). (22) ... so minimizing (22) shortens the average segment ∥y−x∥2 that the model must predict along the straight line (1−t)x+ty, thereby improving conditioning."

    The transport path length in the straight-line FM objective is exactly the quantity minimized: with πϕ ∈ Πo(μ0,νϕ), E_{(x,y)∼πϕ}∥y−x∥^2 = W2^2(μ0,νϕ) = LAN(ϕ). Therefore presenting the learned latent as 'closer to the data in the Wasserstein sense' and 'transport paths are substantially shorter' (Sec. 5.1, Fig. 1) is a restatement of the training objective, not an independent empirical discovery. The independent content lies in the FID/velocity-training improvements, not in the shortening itself.

full rationale

Score 4. The main circular step is the treatment of Wasserstein-closeness / path-shortening as an experimentally demonstrated benefit: Eq. (22) defines LAN(ϕ) as W2^2(μ0,νϕ), and the paper itself notes that under the optimal coupling this equals E∥y−x∥^2, exactly the straight-line segment length the model must predict. Hence the observed shortening is by construction. This does not make the method vacuous: the quantile-learning objective, Algorithm 1, and the FM losses are self-contained, and the MNIST FID ablations (Fig. 13), the CIFAR-10 table (Fig. 14), and the funnel heavy-tail comparison (Fig. 7) are external evaluation metrics — not the training loss itself — giving independent empirical grounding to the central claim that learned quantile noise improves generative modeling. The Kac (Prop. 4) and Uniform/MMD (Prop. 5) example processes are delegated to the authors' own prior results ([10], [11]); these are parameter-free stated theorems rather than fitted values, so under the review rules they are real evidence and are not counted as circular, though the paper would be stronger if the proofs were reproduced. The CIFAR-10 product-measure limitation is a scope issue, not circularity.

Assumptions & free parameters 6 free parameters · 7 assumptions · 0 invented entities

The central claim rests on standard quantile/Wasserstein theory plus two process-level results taken from the authors' own prior work (Kac and MMD gradient flows). The main novel free parameter is the entire learned quantile function, which is fitted to data; the product-measure restriction is the key modeling assumption that both enables tractability and limits the method on correlated data.

free parameters (6)
  • Per-coordinate RQS quantile parameters (bin widths, heights, knot slopes, affine scale/bias) = Learned from data on each dataset
    The central method fits Q_phi to minimize the Wasserstein distance between data and latent; all expressiveness of the learned latent comes from these parameters.
  • Joint loss weight lambda = 5 (CIFAR-10); not reported for other datasets
    Eq. (23) weights LAN relative to the flow-matching loss and controls how strongly the latent is pushed toward the data.
  • Regularization weight beta for negative log-Jacobian = 3 or 4 (CIFAR-10)
    Penalizes degenerate high-dimensional quantile solutions; the paper varies beta in Section 5.2 and Appendix E.3.
  • RQS hyperparameters = bins=32, bound=50 or 500, min width/height=1e-3, min slope=1e-5
    Hand-chosen; control the flexibility and tail behavior of the learned quantile functions.
  • Schedule functions f and g = f(t)=1-t, g(t)=t
    Linear interpolation chosen throughout the learned-latent experiments; other schedules are described but not explored.
  • Uniform process scale b = b=1 in Appendix B.2
    Illustrative noising process used to connect the MMD gradient flow to standard FM; not used in the learned-latent experiments.
assumptions (7)
  • standard math The quantile map is an isometric embedding of (P2(R), W2) into L2(0,1): W2^2(mu,nu)=∫|Q_mu-Q_nu|^2.
    Used to define the LAN objective and the quantile-process/interpolant construction in Section 4.1.
  • standard math Conditional flow matching loss equals the unconditional loss up to a constant, and the conditional velocity formula (6) holds for mean-reverting processes.
    Basis of the training objective; cited from [22,39].
  • domain assumption The Kac process distribution and velocity field from Proposition 4 are correct as proven in the authors' prior paper [10].
    Used as an example noising process; the proof is referenced rather than repeated, and the cited authors overlap with the present paper.
  • domain assumption The MMD gradient flow existence, uniqueness, and subdifferential formulas from [11, Theorem 4.5 / Lemma 4.3] are correct.
    Used to derive the Uniform process in Proposition 5; the cited work shares authors with this paper.
  • ad hoc to paper The latent distribution is restricted to the product class S of independent one-dimensional marginals.
    This is the core modeling choice that makes the quantile method tractable but limits representation of cross-coordinate correlations; acknowledged in Sections 4.2 and 5.2.
  • standard math Quantile interpolants satisfy the self-consistency and marginal-preservation properties of Proposition 6/12.
    Used to argue compatibility with consistency models such as IMM; proof given in Appendix C.
  • domain assumption Minibatch optimal-transport couplings are a sufficiently accurate surrogate for the true W2 coupling during training.
    Algorithm 1 uses a batch assignment as pi_phi; in high dimensions with batch size 128 this coupling is noisy and motivates the regularization and freezing schedule.

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Cite this review

Pith. "Pith review of Adapting Noise to Data: Generative Flows from 1D Processes." pith.science (2026). https://pith.science/paper/Q5TV5AKJ

@misc{pith2026251012636,
  author       = {Pith},
  title        = {Pith review of: Adapting Noise to Data: Generative Flows from 1D Processes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q5TV5AKJ}},
  note         = {Machine review of arXiv:2510.12636}
}
read the original abstract

The default Gaussian latent in flow-based generative models poses challenges when learning certain distributions such as heavy-tailed ones. We introduce a general framework for learning data-adaptive parametric prior distributions (latent noise) using one-dimensional quantile functions, optimized via the Wasserstein distance between noise and data. The quantile-based prior parameterization naturally adapts to both heavy-tailed and compactly supported distributions and shortens transport paths. Numerical results on heavy-tailed weather and image datasets confirm the method's flexibility and effectiveness achieved with negligible computational overhead.

Figures

Figures reproduced from arXiv: 2510.12636 by the authors.

Figure 1
Figure 1. FM via optimal coupling with Gaussian noise (left) and our learned noise (right). Samples [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Three realisations of a standard Wiener process (left), the Kac process (middle), and the [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. A generated trajectory from a Flow Matching model trained using the conditional density [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: Samples (1M) from the distribution of the two-dimensional Kac process K1 for (a, c) = (9, 3). We can clearly ob￾serve the atomic compo￾nent of its distribution derived in (13). The Kac process was recently used in generative modeling by some of the authors in [10]. It …
Figure 5
Figure 5. Figure 5: A generated trajectory from a Flow Matching model trained using the conditional density [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 6
Figure 6. Figure 6: A generated trajectory from a Flow Matching model trained using the conditional density [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: Sampling of Neal’s funnel with different latent distributions [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: The CDF Rµ and quantile function Qµ of a standard normal dis￾tribution µ. In [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]
Figure 9
Figure 9. Figure 9: A generated trajectory from the learned quantile latent (left) to the unevenly weighted [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]
Figure 10
Figure 10. Figure 10: Samples (1M) from our learned latent of the funnel distribu￾tion. Color shows endpoint norm. Funnel Distribution. The funnel distribution, shown in [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]
Figure 11
Figure 11. Figure 11: MNIST Dataset: From Left to Right: Generated samples, samples from the learned [PITH_FULL_IMAGE:figures/full_fig_p017_11.png]
Figure 12
Figure 12. Figure 12: Comparison of the empirical and learned probability density functions and their quantile [PITH_FULL_IMAGE:figures/full_fig_p018_12.png]
Figure 13
Figure 13. Figure 13: Ablation study over capacity of the U-Net for sampling from the MNIST dataset. In [PITH_FULL_IMAGE:figures/full_fig_p018_13.png]
Figure 14
Figure 14. Figure 14: CIFAR results for different choices of regularization parameter and for the baseline. The [PITH_FULL_IMAGE:figures/full_fig_p019_14.png]
Figure 15
Figure 15. Figure 15: A generated sample path from the learned quantile latent to the checkerboard. The [PITH_FULL_IMAGE:figures/full_fig_p031_15.png]
Figure 16
Figure 16. Figure 16: Flow Matching with optimal coupling using Gaussian noise (left) and our learned noise [PITH_FULL_IMAGE:figures/full_fig_p034_16.png]

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Reference graph

Works this paper leans on

41 extracted references · 7 linked inside Pith

  1. [10]

    Duong, J

    R. Duong, J. Chemseddine, P. Friz, and G. Steidl. Telegrapher’s generative model via Kac flows.arXiv preprint arXiv::2506.20641, 2025

    arXiv 2025

  2. [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

  3. [2]

    Ambrosio, N

    L. Ambrosio, N. Gigli, and G. Savaré.Gradient Flows. Lectures in Mathematics ETH Zürich. Birkhäuser, Basel, 2nd edition, 2008

    2008

  4. [3]

    Bartosh, D

    G. Bartosh, D. Vetrov, and C. A. Naesseth. Neural flow diffusion models: Learnable forward process for improved diffusion modelling, 2025. 19

    2025

  5. [4]

    Blessing, J

    D. Blessing, J. Berner, L. Richter, and G. Neumann. Underdamped diffusion bridges with applications to sampling. InInternational Conference on Learning Representations (ICLR), 2025

    2025

  6. [5]

    Blessing, X

    D. Blessing, X. Jia, and G. Neumann. End-to-end learning of gaussian mixture priors for diffusion sampler, 2025

    2025

  7. [6]

    Cattaneo

    C. Cattaneo. Sur une forme de l’équation de la chaleur éliminant le paradoxe d’une propagation instantanée.Comptes Rendus., 247, 1958

    1958

  8. [7]

    R. T. Q. Chen. torchdiffeq, 2018

    2018

Show all 41 references
  1. [8]

    M. Chester. Second sound in solids.Physical Review, 131, 1963

    1963

  2. [9]

    DiffusionmodelsbeatGANsonimagesynthesis

    P.DhariwalandA.Q.Nichol. DiffusionmodelsbeatGANsonimagesynthesis. InA.Beygelzimer, Y. Dauphin, P. Liang, and J. W. Vaughan, editors,Advances in Neural Information Processing Systems, 2021

    2021

  3. [11]

    Duong, V

    R. Duong, V. Stein, R. Beinert, J. Hertrich, and G. Steidl. Wasserstein gradient flows of MMD functionals with distance kernel and Cauchy problems on quantile functions.ArXiv:2408.07498, 2024

    2024 arXiv

  4. [12]

    Durkan, A

    C. Durkan, A. Bekasov, I. Murray, and G. Papamakarios. Neural spline flows.Advances in Neural Information Processing Systems, 32, 2019

    2019

  5. [13]

    Flamary, N

    R. Flamary, N. Courty, A. Gramfort, M. Z. Alaya, A. Boisbunon, S. Chambon, L. Chapel, A. Corenflos, K. Fatras, N. Fournier, et al. Pot: Python Optimal Transport.Journal of Machine Learning Research (JMLR), 22(1):3571–3578, 2021

    2021

  6. [14]

    Gregory and R

    J. Gregory and R. Delbourgo. Piecewise rational quadratic interpolation to monotonic data. IMA Journal of Numerical Analysis, 2(2):123–130, 1982

    1982

  7. [15]

    P. L. Hagemann and S. Neumayer. Stabilizing invertible neural networks using mixture models. Inverse Problems, 37(7):085002, 2021

    2021

  8. [16]

    W. Han, C. Meng, , C. D. Manning, and S. Ermon. DistillKac: Few-step image generation via damped wave equations.arXiv preprint arXiv:2509.215113, 2025

    2025

  9. [17]

    Hertrich, M

    J. Hertrich, M. Gräf, R. Beinert, and G. Steidl. Wasserstein steepest descent flows of discrep- ancies with Riesz kernels.Journal of Mathematical Analysis and Applications, 531(1):127829, 2024

    2024

  10. [18]

    Heusel, H

    M. Heusel, H. Ramsauer, T. Unterthiner, B. Nessler, and S. Hochreiter. GANs trained by a two time-scale update rule converge to a local Nash equilibrium. In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors,Advances in Neural I...

    2017

  11. [19]

    M. Kac. A stochastic model related to the telegrapher’s equation.Rocky Mountain Journal of Mathematics, 4, 1974. 20

    1974

  12. [20]

    Kapusniak, P

    K. Kapusniak, P. Potaptchik, T. Reu, L. Zhang, A. Tong, M. Bronstein, A. J. Bose, and F. Di Giovanni. Metric flow matching for smooth interpolations on the data manifold.arXiv preprint arXiv:2405.14780, 2024

    2024 arXiv

  13. [21]

    D. Kim, S. Shin, K. Song, W. Kang, and I.-C. Moon. Soft truncation: A universal training technique of score-based diffusion model for high precision score estimation.ICML, 2022

    2022

  14. [22]

    Lipman, R

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

    2023

  15. [23]

    Q. Liu. Rectified flow: A marginal preserving approach to optimal transport.arXiv preprint arXiv:2209.14577, 2022

    2022 arXiv

  16. [24]

    R. M. Neal. Slice sampling.The Annals of Statistics, 31(3):705–767, 2003

    2003

  17. [25]

    Pandey, J

    K. Pandey, J. Pathak, Y. Xu, S. Mandt, M. Pritchard, A. Vahdat, and M. Mardani. Heavy-tailed diffusion models, 2024

    2024

  18. [26]

    Pidstrigach

    J. Pidstrigach. Score-based generative models detect manifolds.NeurIPS, 2022

    2022

  19. [27]

    Pooladian, H

    A.-A. Pooladian, H. Ben-Hamu, C. Domingo-Enrich, B. Amos, Y. Lipman, and R. T. Q. Chen. Multisample flow matching: Straightening flows with batch couplings, 2023

    2023

  20. [28]

    S. S. Sahoo, A. Gokaslan, C. De Sa, and V. Kuleshov. Diffusion models with learned adaptive noise. InAdvances in Neural Information Processing Systems (NeurIPS), 2024

    2024

  21. [29]

    Salmona, V

    A. Salmona, V. D. Bortoli, J. Delon, and A. Desolneux. Can push-forward generative models fit multimodal distributions?Advances in Neural Information Processing Systems, 35:0766–10779, 2022

    2022

  22. [30]

    Shariatian, U

    D. Shariatian, U. Simsekli, and A. Durmus. Heavy-tailed diffusion with denoising Lévy probabilistic models, 2025

    2025

  23. [31]

    Shariatian, U

    D. Shariatian, U. Simsekli, and A. O. Durmus. Heavy-tailed diffusion with denoising levy probabilistic models.ICLR 2025, 2025

    2025

  24. [32]

    Sohl-Dickstein, E

    J. Sohl-Dickstein, E. Weiss, N. Maheswaranathan, and S. Ganguli. Deep unsupervised learning using nonequilibrium thermodynamics. In F. Bach and D. Blei, editors,Proceedings of the 32nd International Conference on Machine Learning, volume 37 ofProceedings of Machine Learning Re...

    2015

  25. [33]

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

    2010 arXiv

  26. [34]

    Song and S

    Y. Song and S. Ermon. Generative modeling by estimating gradients of the data distribution. ArXiv 1907.05600, 2019

    1907 arXiv

  27. [35]

    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

  28. [36]

    R. C. Tautz and I. Lerche. Application of the three-dimensional telegraph equation to cosmic-ray transport.Research in Astronomy and Astrophysics, 2016. 21

    2016

  29. [37]

    A. Tong, K. Fatras, N. Malkin, G. Huguet, Y. Zhang, J. Rector-Brooks, G. Wolf, and Y. Bengio. Improving and generalizing flow-based generative models with minibatch optimal transport, 2024

    2024

  30. [38]

    Vernotte

    P. Vernotte. Les paradoxes de la theorie continue de l’équation de la chaleur.Comptes Rendus., 246, 1958

    1958

  31. [39]

    Wald and G

    C. Wald and G. Steidl. Flow Matching: Markov kernels, stochastic processes and transport plans. InVariational and Information Flows in Machine Learning and Optimal Transport, Oberwolfach Seminars. Vol. 56, pages 185–254. Birkhäuser, 2025

    2025

  32. [40]

    Zhang, H

    T. Zhang, H. Zheng, J. Yao, X. Wang, M. Zhou, Y. Zhang, and Y. Wang. Long-tailed diffusion models with oriented calibration. InThe Twelfth International Conference on Learning Representations, 2024

    2024

  33. [41]

    inductive bootstrapping

    L. Zhou, S. Ermon, and J. Song. Inductive moment matching.ICML 2025, 2025. A Uniform Process and MMD Gradient Flow We prove the proposition more general forν = U [a, b]and a flow starting inx0 ∈ [a, b]meaning that we show the following. Proposition 9.The Wasserstein gradient f...

    2025

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.