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REVIEW 2 major objections 4 minor 46 references

A single flow-matching objective can generate heavy-tailed data by treating the source as a mixture of Gaussian flows indexed by a random clock path.

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2026-08-02 03:32 UTC pith:JA5XQIMZ

load-bearing objection HTFM is a worthwhile heavy-tailed flow matching framework with a novel random-clock mechanism and unified Gaussian/α-stable/Student-t sources; the math is mostly clean and results promising, but the signature truncation lacks a theoretical bound, the main tables have no error bars, and Thm 3.2(ii) has a normalization typo. the 2 major comments →

arxiv 2607.13841 v1 pith:JA5XQIMZ submitted 2026-07-15 cs.LG stat.ML

Heavy-Tailed Flow Matching via Random Clocks

Zhouhao Yang , Yezhen Wang , Kenji Kawaguchi , Vladimir Braverman , Haoyang Cao This is my paper
classification cs.LG stat.ML
keywords heavy-tailed flow matchingrandom clockGaussian scale mixturealpha-stableStudent-tpath logsignaturetail calibrationlow-NFE sampling
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper tries to establish that heavy-tailed data generation — where rare events matter — can be done with flow matching without abandoning Gaussian training targets. The trick: treat the heavy-tailed source as a Gaussian scale mixture produced by a path-valued random clock; condition on the realized clock path, do standard conditional flow matching in that Gaussian space, then average over clocks. The paper claims this recovers Gaussian, alpha-stable, and Student-t sources, works with cheap sampling, and improves mode coverage, quality, and tail statistics on imbalanced and weather benchmarks. Why care: one architecture can calibrate tail heaviness by changing the clock law.

Core claim

HTFM portrays heavy-tailed sources as mixtures of clock-conditioned Gaussian sources. For a fixed clock path T, the source and the affine flow are Gaussian, and the endpoint-conditioned velocity target has the same algebraic form as in Gaussian flow matching; marginalizing over the clock recovers Gaussian scale mixtures including Gaussian, alpha-stable, and Student-t laws. The velocity field conditions on a truncated logsignature of the clock path, adding negligible overhead, so the model adapts to the realized conditional Gaussian space while keeping low-NFE sampling. The paper reports that heavy-tailed clocks beat the Gaussian clock, that the clock family matters beyond tail-decay exponent

What carries the argument

The random clock is a nondecreasing path-valued latent variable that controls the conditional source covariance. Given a realized clock path, the source is Gaussian and the flow is affine, so the conditional velocity target is the familiar flow-matching expression; averaging over clocks creates heavy-tailed marginals. The path is summarized by a truncated logsignature feature, which gives the neural velocity field finite-dimensional access to the clock with negligible cost.

Load-bearing premise

The network never sees the full clock path — it sees only a low-truncation logsignature summary — and the paper provides no guarantee that this summary is sufficient for a novel clock path at inference; the claimed clock-adaptation gains rest on that empirical approximation.

What would settle it

Train two HTFM variants with clock laws that share the same terminal value and first-order logsignature summary but differ in higher-order path structure, then compare FID and tail metrics. If the truncated feature is insufficient, the model would fail to distinguish those clocks and would underperform a full-path-conditioned version.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Flow matching can be made heavy-tailed while keeping the exact Gaussian conditional regression target, so no family-specific score formulas or stable-density evaluations are needed.
  • The same architecture and training objective cover Gaussian, alpha-stable, and Student-t sources; only the clock law and tail parameter change, giving a practical tail-control interface.
  • Heavy-tailed clocks improve mode coverage, sample quality, and tail-statistic recovery over the Gaussian-clock variant while retaining the low-NFE sampling advantage of flow matching.
  • Clock family is meaningful beyond tail heaviness: at matched polynomial tail-decay rates, different clock families behave differently, so the clock law is a real design knob.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the conditional-Gaussian trick is more general than the paper's symmetric cases; allowing the clock to also affect a conditional mean would extend the same mechanism to asymmetric heavy-tailed laws, a direction the paper mentions but does not test.
  • Editorial inference: because the network sees only a low-order logsignature of the clock path, performance may drop for clock realizations whose higher-order path structure matters; a testable prediction is that higher signature orders help on highly non-monotone clock paths.
  • Editorial inference: if the clock law were learned from data rather than prescribed, the framework could become an adaptive latent description of local tail behavior, but that would require new identifiability and estimation assumptions.
  • Editorial inference: the pathwise geodesic theorem suggests an optimal transport reading that remains valid even when the marginalized source has infinite second moment, which may connect to other heavy-tailed generative formulations.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper introduces Heavy-Tailed Flow Matching via Random Clocks (HTFM), a framework in which a random nondecreasing path-valued clock conditions the source covariance of a flow-matching model. Conditioned on the clock path, the source and interpolation are Gaussian, so the endpoint-conditioned flow-matching target has the standard affine form. Marginalizing over the clock yields Gaussian scale mixtures covering Gaussian, alpha-stable, and Student-t sources. To make the path conditioning practical, the paper replaces the full clock path by a truncated logsignature feature and injects it into the velocity network. The authors prove a pathwise conditional-marginal equivalence, derive the recovered marginals, and present experiments on 2D imbalanced alpha-stable mixtures, CIFAR10-LT, and HRRR weather fields, reporting gains over Gaussian flow matching and several heavy-tailed baselines, especially at low NFE. The paper also provides extensive appendices with proofs, ablations, and experimental details.

Significance. If the claims hold, HTFM provides a unified and computationally cheap way to train flow-matching models with heavy-tailed source distributions while retaining Gaussian conditional targets, and it offers a practical tail-calibration interface by changing only the clock law. The theoretical development is mostly standard but clearly presented, with proofs in the appendix. The empirical study covers three rather different benchmarks and includes ablations on signature order, NFE, solver, and flow type. The paper is also explicit about the conditions under which the recovered marginals hold. The main risk is the gap between the full-path objective used in the theory and the truncated-signature objective used in practice; this is acknowledged in the reader's report and is not addressed by a bound or a precise approximation argument.

major comments (2)
  1. [Theorem 3.2(ii)] The statement uses the Laplace normalization E[e^{-sT_t}] = exp(-(ρ+1)t s^{ρ/2}) with ρ=α/2. With this normalization, A(T)=∫_0^1 T_s ds has Laplace exponent proportional to λ^{ρ/2}=λ^{α/4}, so the characteristic function in Eq. (C.4) would not correspond to an α-stable law with scale β_t. The proof in Appendix C.5 actually uses the standard normalization E[e^{-sT_t}] = exp(-(ρ+1)t s^ρ). The theorem statement should be corrected to s^ρ; otherwise the claimed recovered marginal is not the stated α-stable law.
  2. [Section 3.2, Eq. (3.7)] The practical objective (3.7) replaces the full clock path in (3.4) by the truncated logsignature ℓ_m(T), but no theorem bounds the error of this replacement. Proposition D.1 gives uniform approximation only for continuous functionals on compact path sets and does not cover the α-stable subordinator clocks used in the experiments, nor does it control the regression gap or the ODE integration error. Concretely, for m=1 the level-1 logsignature of the time-augmented clock is (1,T_1); two clock paths with identical T_1 but different area A(T) yield identical conditioning, while under (3.9) and Σ_A(T)=2A(T)I_d the pathwise optimal field u_t^T depends on A(T). Thus test-time clocks with the same ℓ_1(T) but different area will be integrated with the same field despite different optimal fields. Table 3 is one benchmark and does not establish sufficiency. Please add a truncation error analysis o
minor comments (4)
  1. [Throughout] The paper frequently types 'CIF AR10-LT' with a stray space; please use 'CIFAR10-LT' consistently. Similarly, 'c` adl` ag' appears in several places and should be typeset as 'càdlàg' or 'CADLAG'.
  2. [Section 3.1] The heading 'T raining objective' contains an unwanted space; it should read 'Training objective'.
  3. [Appendix D, Proposition D.1] The phrase 'on which the signature separates paths up to tree-like equivalence' is vague. Since the clocks in the experiments are not compact, the practical relevance of the proposition should be stated more precisely, ideally with the compact truncation used.
  4. [Appendix E.4] The description of the α-stable clock feature says the two channels are 'standardize[d] along time' before computing the logsignature. This means ℓ_1(T) is not literally (1,T_1), and the exact relation between the standardized feature and the area functional should be stated explicitly to support the claims in Section 3.2.

Circularity Check

0 steps flagged

No significant circularity: the random-clock construction is a self-contained conditional-Gaussian derivation, and the acknowledged signature-truncation limitation is an approximation concern, not an equation-level circularity.

full rationale

The derivation chain is constructive rather than circular. The paper defines a clock-conditioned affine path (Eq. 3.8), derives the conditional Gaussian law (Eq. 3.9) and the explicit endpoint-conditioned velocity target (Eq. 3.10), and proves the pathwise conditional-marginal equivalence (Proposition 3.1) with a self-contained Pythagorean-identity proof in Appendix C.3. Theorem 3.2 then verifies, by explicit characteristic-function and density calculations, that specific clock laws recover Gaussian, α-stable, and Student-t marginals; this is a construction, not a fitted parameter renamed as a prediction. The paper does not rely on any load-bearing self-citation: its references to standard flow matching, Gaussian scale mixtures, and signature theory are external and are not invoked as unverified uniqueness constraints. The truncated logsignature conditioning (Eq. 3.7) is an empirical approximation, and the paper itself flags the compactness limitation in Appendix D: Proposition D.1 is stated for compact path sets while heavy-tailed clocks need not be compact. That is a correctness/robustness gap, not circularity, because the training target and the pathwise theory are defined with respect to the full clock path before the feature truncation is introduced. The per-benchmark sweep of the tail parameter is a model-selection practice rather than an equation-level reduction; it may raise overfitting concerns but does not make any prediction equivalent to its input by construction. Overall, the paper's central derivation is self-contained and no step reduces to its own assumptions.

Axiom & Free-Parameter Ledger

4 free parameters · 6 axioms · 1 invented entities

The framework rests mainly on standard Gaussian scale mixture theory and standard flow-matching projection arguments. The genuinely new part is the path-valued clock and its logsignature conditioning, but the paper does not supply a formal guarantee that low-order features suffice, and the empirical tail parameters are chosen per benchmark on the evaluation split.

free parameters (4)
  • Tail parameter p_alpha for alpha-stable clock = 1.6, 1.7, 1.8, 1.9 (selected per benchmark)
    Chosen by sweeping the stability index of the stable subordinator; the best value per benchmark is bolded. This is a hyperparameter selected on the evaluation split, not derived.
  • Tail parameter p_nu for Student-t clock = 1.7, 2.0, 3.0 (selected per benchmark)
    Chosen by sweeping the inverse-gamma clock parameter; again a per-benchmark hyperparameter selection.
  • Logsignature truncation order m = 1 by default; 2 in ablation
    The velocity field receives a low-order logsignature feature; the paper uses m=1 because higher order did not give consistent gains. This is a hand-chosen approximation of the infinite-dimensional clock path.
  • VE signature-linear mixing weights lambda_alpha, lambda_beta = 1/4
    Used in the clock-aware trajectory ablation (Appendix D.2); chosen by hand, not learned.
axioms (6)
  • standard math Gaussian scale-mixture representations of alpha-stable and Student-t laws
    Invoked in Section 2.2, Appendix A.3, and Theorem 3.2 to express heavy-tailed sources as mixtures of Gaussians.
  • standard math Pathwise L2 projection identity (conditional-marginal equivalence)
    Proposition 3.1 and Appendix C.3 restate the standard flow-matching conditional-marginal equivalence pathwise in the clock variable, requiring finite conditional second moments.
  • domain assumption Integrability and differentiability conditions for exchanging expectations
    Appendix C.3 assumes the pathwise risks and the additive constant C_t(T) are integrable; without this, the equivalence of objectives can fail for very heavy outer clock laws.
  • domain assumption Random clock independent of data and base Gaussian
    Section 3 assumes T and X_1 are independent and that X_0 is conditionally Gaussian given T, which is needed for Eq. (3.9) and for the target formula (3.10).
  • domain assumption Signature universality on compact path families
    Proposition D.1 justifies finite-dimensional signature features, but only on compact sets; the paper does not prove the same approximation for the non-compact heavy-tailed clock paths used in experiments.
  • ad hoc to paper Low-order logsignature is a sufficient conditioning summary
    The paper assumes m=1 captures the dominant conditional-scale variation; this is supported only by the empirical ablation in Table 3, not by a theorem.
invented entities (1)
  • Random clock T[0,1] no independent evidence
    purpose: A path-valued latent variable that controls the conditional covariance of the Gaussian source and, optionally, the interpolation schedule.
    The random clock is a modeling construct with no directly observable counterpart. Its value is measured only through the generated data distribution; there is no falsifiable prediction attached to the clock itself.

pith-pipeline@v1.3.0-alltime-deepseek · 33857 in / 13396 out tokens · 124791 ms · 2026-08-02T03:32:34.058401+00:00 · methodology

0 comments
read the original abstract

Heavy-tailed data arise in many domains where rare events carry disproportionate importance, such as imbalanced image datasets, financial returns, and weather extremes. Standard diffusion and flow-matching models typically begin from Gaussian noise or Gaussian source distributions, which yield tractable training targets but provide a poor inductive match for heavy-tailed data. We propose Heavy-Tailed Flow Matching via Random Clocks (HTFM), a framework that portrays heavy-tailed sources as mixtures of clock-conditioned Gaussian sources. Conditioning on a given clock path, the source distribution and flow are Gaussian; marginalizing over the clock gives a Gaussian scale mixture covering Gaussian, $\alpha$-stable, and Student-t families. To make the clock-conditioned vector field practical, we encode the path-valued clock using truncated logsignature features, allowing the velocity field to adapt to the realized conditional space with negligible overhead. Empirically, on 2D imbalanced $\alpha$-stable mixtures, CIFAR10-LT, and HRRR weather fields, HTFM improves mode coverage, sample quality, and tail-statistic recovery over Gaussian flow matching and competitive heavy-tailed baselines, while retaining the low-NFE sampling advantage of flow matching. Moreover, the random-clock formulation further provides a practical tail-control interface: by varying only the clock law or tail parameter, the same architecture can calibrate the ``heaviness'' of generated tails across different distribution families.

Figures

Figures reproduced from arXiv: 2607.13841 by Haoyang Cao, Kenji Kawaguchi, Vladimir Braverman, Yezhen Wang, Zhouhao Yang.

Figure 1
Figure 1. Figure 1: Overview of random-clock conditioning for heavy-tailed flow matching. The right col [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: NFE-vs-FID curves on CIFAR10-LT. One panel per decay degree [PITH_FULL_IMAGE:figures/full_fig_p037_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: compares HTFM-α samples generated under clocks of different decay degrees (rows: p = ∞, pα = 1.8, pα = 1.5) against ground-truth samples (top row) across five 2D toy targets (columns: checkerboard, two moons, swiss roll, rings, olympic rings). The same back￾bone, training schedule, and NFE budget are used for every cell; only the random-clock law is swept [PITH_FULL_IMAGE:figures/full_fig_p038_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Generated samples on the 2D imbalanced α-stable mixture at NFE=20. Rows: αdata = 1.8 (top), αdata = 1.5 (bottom). Columns: real data, HTFM, DLPM, DLIM. W1 distances to the real data are reported below each generated panel. 39 [PITH_FULL_IMAGE:figures/full_fig_p039_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: CIFAR10-LT generated samples from HTFM-α. Each column corresponds to one CIFAR-10 class; from left to right: airplane, automobile, bird, cat, deer, dog, frog, horse, ship, truck. This is also the CIFAR10-LT head-to-tail order, i.e. per-class training-sample counts decrease from the leftmost to the rightmost column. Sample fidelity broadly tracks the per￾class training-set size: the head classes airplane an… view at source ↗

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Reference graph

Works this paper leans on

46 extracted references · 11 linked inside Pith

  1. [1]

    Building normalizing flows with stochastic interpolants.arXiv preprint arXiv:2209.15571, 2022

    Michael S Albergo and Eric Vanden-Eijnden. Building normalizing flows with stochastic interpolants.arXiv preprint arXiv:2209.15571, 2022

    Pith/arXiv arXiv 2022

  2. [2]

    D. F. Andrews and C. L. Mallows. Scale mixtures of normal distributions.Journal of the Royal Statistical Society: Series B (Methodological), 36(1):99–102, 1974

    1974

  3. [3]

    Barndorff-Nielsen, J

    O. Barndorff-Nielsen, J. Kent, and M. Sørensen. Normal variance-mean mixtures and z-distributions.International Statistical Review, 50(2):145–159, 1982

    1982

  4. [4]

    Generative modelling with jump-diffusions.arXiv preprint arXiv:2503.06558, 2025

    Adrian Baule. Generative modelling with jump-diffusions.arXiv preprint arXiv:2503.06558, 2025

    arXiv 2025

  5. [5]

    A north american hourly assimilation and model forecast cycle: The rapid refresh.Monthly Weather Review, 144(4):1669–1694, 2016

    Stanley G Benjamin, Stephen S Weygandt, John M Brown, and ... A north american hourly assimilation and model forecast cycle: The rapid refresh.Monthly Weather Review, 144(4):1669–1694, 2016

    2016

  6. [6]

    Learning imbal- anced datasets with label-distribution-aware margin loss

    Kaidi Cao, Colin Wei, Adrien Gaidon, Nikos Arechiga, and Tengyu Ma. Learning imbal- anced datasets with label-distribution-aware margin loss. InAdvances in Neural Informa- tion Processing Systems, 2019

    2019

  7. [7]

    A primer on the signature method in machine learning

    Ilya Chevyrev and Andrey Kormilitzin. A primer on the signature method in machine learning. InSignature Methods in Finance: An Introduction with Computational Applica- tions, pages 3–64. Springer, 2025

    2025

  8. [8]

    Springer, 2001

    Stuart Coles, Joanna Bawa, Lesley Trenner, and Pat Dorazio.An introduction to statistical modeling of extreme values, volume 208. Springer, 2001

    2001

  9. [9]

    Empirical properties of asset returns: stylized facts and statistical issues

    Rama Cont. Empirical properties of asset returns: stylized facts and statistical issues. Quantitative Finance, 1(2):223–236, 2001

    2001

  10. [10]

    Tail-gan: Learning to simulate tail risk scenarios.Management Science, 72(4):2917–2936, 2026

    Rama Cont, Mihai Cucuringu, Renyuan Xu, and Chao Zhang. Tail-gan: Learning to simulate tail risk scenarios.Management Science, 72(4):2917–2936, 2026

    2026

  11. [11]

    Universal approximation theorems for continuous functions of c` adl` ag paths and l´ evy-type signature models.Finance and Stochastics, 29(2):289–342, 2025

    Christa Cuchiero, Francesca Primavera, and Sara Svaluto-Ferro. Universal approximation theorems for continuous functions of c` adl` ag paths and l´ evy-type signature models.Finance and Stochastics, 29(2):289–342, 2025

    2025

  12. [12]

    The high-resolution rapid refresh (hrrr): An hourly updating convection-allowing forecast model

    David C Dowell, Curtis R Alexander, Eric P James, Stephen S Weygandt, Stanley G Ben- jamin, Geoffrey S Manikin, Benjamin T Blake, John M Brown, Joseph B Olson, Ming Hu, et al. The high-resolution rapid refresh (hrrr): An hourly updating convection-allowing forecast model. part i: Motivation and system description.Weather and Forecasting, 37(8):1371–1395, 2022

    2022

  13. [13]

    Springer Science & Business Media, 2013

    Paul Embrechts, Claudia Kl¨ uppelberg, and Thomas Mikosch.Modelling extremal events: for insurance and finance, volume 33. Springer Science & Business Media, 2013

    2013

  14. [14]

    Generative adversarial nets

    Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. InAdvances in Neural Information Processing Systems, volume 27, 2014

    2014

  15. [15]

    Rarest rainfall events will see the greatest relative increase in magnitude under future climate change.Communications Earth & Environment, 3(1):235, 2022

    Gaby J Gr¨ undemann, Nick van de Giesen, Lukas Brunner, and Ruud van der Ent. Rarest rainfall events will see the greatest relative increase in magnitude under future climate change.Communications Earth & Environment, 3(1):235, 2022. 12

    2022

  16. [16]

    Gans trained by a two time-scale update rule converge to a local nash equilib- rium

    Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter. Gans trained by a two time-scale update rule converge to a local nash equilib- rium. InAdvances in Neural Information Processing Systems, 2017

    2017

  17. [17]

    Flexible tails for normalizing flows

    Tennessee Hickling and Dennis Prangle. Flexible tails for normalizing flows. InProceedings of the 42nd International Conference on Machine Learning, volume 267 ofProceedings of Machine Learning Research, pages 23155–23178. PMLR, 2025

    2025

  18. [18]

    Denoising diffusion probabilistic models

    Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models. Advances in neural information processing systems, 33:6840–6851, 2020

    2020

  19. [19]

    Tails of Lipschitz tri- angular flows

    Priyank Jaini, Ivan Kobyzev, Yaoliang Yu, and Marcus Brubaker. Tails of Lipschitz tri- angular flows. InProceedings of the 37th International Conference on Machine Learning, volume 119 ofProceedings of Machine Learning Research, pages 4673–4681. PMLR, 2020

    2020

  20. [20]

    Deep signature transforms.Advances in neural information processing systems, 32, 2019

    Patrick Kidger, Patric Bonnier, Imanol Perez Arribas, Cristopher Salvi, and Terry Lyons. Deep signature transforms.Advances in neural information processing systems, 32, 2019

    2019

  21. [21]

    Patrick Kidger and Terry J. Lyons. Signatory: differentiable computations of the signature and logsignature transforms, on both CPU and GPU. InInternational Conference on Learning Representations, 2021

    2021

  22. [22]

    Kingma and Max Welling

    Diederik P. Kingma and Max Welling. Auto-encoding variational bayes. InInternational Conference on Learning Representations, 2014

    2014

  23. [23]

    Kiraly and Harald Oberhauser

    Franz J. Kiraly and Harald Oberhauser. Kernels for sequentially ordered data.Journal of Machine Learning Research, 20(31):1–45, 2019

    2019

  24. [24]

    Optimal flow matching: Learning straight trajectories in just one step.Advances in Neural Information Processing Systems, 37:104180–104204, 2024

    Nikita Kornilov, Petr Mokrov, Alexander Gasnikov, and Alexander Korotin. Optimal flow matching: Learning straight trajectories in just one step.Advances in Neural Information Processing Systems, 37:104180–104204, 2024

    2024

  25. [25]

    Neural operator: Learning maps between func- tion spaces with applications to pdes.Journal of Machine Learning Research, 24(89):1–97, 2023

    Nikola Kovachki, Zongyi Li, Burigede Liu, Kamyar Azizzadenesheli, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Neural operator: Learning maps between func- tion spaces with applications to pdes.Journal of Machine Learning Research, 24(89):1–97, 2023

    2023

  26. [26]

    Learning multiple layers of features from tiny images

    Alex Krizhevsky and Geoffrey Hinton. Learning multiple layers of features from tiny images. Technical report, University of Toronto, 2009

    2009

  27. [27]

    Learning from the past, predicting the statistics for the future, learning an evolving system.arXiv preprint arXiv:1309.0260, 2013

    Daniel Levin, Terry Lyons, and Hao Ni. Learning from the past, predicting the statistics for the future, learning an evolving system.arXiv preprint arXiv:1309.0260, 2013

    Pith/arXiv arXiv 2013

  28. [28]

    Cauchy diffusion: A heavy-tailed denoising diffu- sion probabilistic model for speech synthesis

    Qi Lian, Yu Qi, and Yueming Wang. Cauchy diffusion: A heavy-tailed denoising diffu- sion probabilistic model for speech synthesis. InProceedings of the AAAI Conference on Artificial Intelligence, volume 39, pages 24549–24557, 2025

    2025

  29. [29]

    Flow matching for generative modeling.arXiv preprint arXiv:2210.02747, 2022

    Yaron Lipman, Ricky TQ Chen, Heli Ben-Hamu, Maximilian Nickel, and Matt Le. Flow matching for generative modeling.arXiv preprint arXiv:2210.02747, 2022

    Pith/arXiv arXiv 2022

  30. [30]

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

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

    Pith/arXiv arXiv 2022

  31. [31]

    Flow straight and fast: Learning to generate and transfer data with rectified flow.arXiv preprint arXiv:2209.03003, 2022

    Xingchao Liu, Chengyue Gong, and Qiang Liu. Flow straight and fast: Learning to generate and transfer data with rectified flow.arXiv preprint arXiv:2209.03003, 2022. 13

    Pith/arXiv arXiv 2022

  32. [32]

    Large-scale long-tailed recognition in an open world

    Ziwei Liu, Zhongqi Miao, Xiaohang Zhan, Jiayun Wang, Boqing Gong, and Stella X Yu. Large-scale long-tailed recognition in an open world. InProceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 2537–2546, 2019

    2019

  33. [33]

    Rough paths, signatures and the modelling of functions on streams.arXiv preprint arXiv:1405.4537, 2014

    Terry Lyons. Rough paths, signatures and the modelling of functions on streams.arXiv preprint arXiv:1405.4537, 2014

    Pith/arXiv arXiv 2014

  34. [34]

    The Kolmogorov–Smirnov test for goodness of fit.Journal of the American Statistical Association, 46(253):68–78, 1951

    Frank J Massey Jr. The Kolmogorov–Smirnov test for goodness of fit.Journal of the American Statistical Association, 46(253):68–78, 1951

    1951

  35. [35]

    Neural rough differ- ential equations for long time series

    James Morrill, Cristopher Salvi, Patrick Kidger, and James Foster. Neural rough differ- ential equations for long time series. InProceedings of the 38th International Conference on Machine Learning, volume 139 ofProceedings of Machine Learning Research, pages 7829–7838. PMLR, 2021

    2021

  36. [36]

    Heavy-tailed diffusion models.arXiv preprint arXiv:2410.14171, 2024

    Kushagra Pandey, Jaideep Pathak, Yilun Xu, Stephan Mandt, Michael Pritchard, Arash Vahdat, and Morteza Mardani. Heavy-tailed diffusion models.arXiv preprint arXiv:2410.14171, 2024

    Pith/arXiv arXiv 2024

  37. [37]

    Four- CastNet: A global data-driven high-resolution weather model using adaptive Fourier neural operators.arXiv preprint arXiv:2202.11214, 2022

    Jaideep Pathak, Shashank Subramanian, Peter Harrington, Sanjeev Raja, Ashesh Chat- topadhyay, Morteza Mardani, Thorsten Kurth, David Hall, Zongyi Li, Kamyar Azizzade- nesheli, Pedram Hassanzadeh, Karthik Kashinath, and Animashree Anandkumar. Four- CastNet: A global data-driven high-resolution weather model using adaptive Fourier neural operators.arXiv pre...

    Pith/arXiv arXiv 2022

  38. [38]

    Multisample flow matching: Straightening flows with minibatch couplings.arXiv preprint arXiv:2304.14772, 2023

    Aram-Alexandre Pooladian, Heli Ben-Hamu, Carles Domingo-Enrich, Brandon Amos, Yaron Lipman, and Ricky TQ Chen. Multisample flow matching: Straightening flows with minibatch couplings.arXiv preprint arXiv:2304.14772, 2023

    Pith/arXiv arXiv 2023

  39. [39]

    Improved sampling algorithms for l´ evy-itˆ o diffusion models

    Vadim Popov, Assel Yermekova, Tasnima Sadekova, Artem Khrapov, and Mikhail Sergee- vich Kudinov. Improved sampling algorithms for l´ evy-itˆ o diffusion models. InThe Thir- teenth International Conference on Learning Representations

  40. [40]

    Springer, 2007

    Sidney I Resnick.Heavy-tail phenomena: probabilistic and statistical modeling. Springer, 2007

    2007

  41. [41]

    Sonia I Seneviratne, Xuebin Zhang, Muhammad Adnan, Wafae Badi, Claudine Dereczyn- ski, A Di Luca, Subimal Ghosh, Iskhaq Iskandar, James Kossin, Sophie Lewis, et al. Weather and climate extreme events in a changing climate.Climate change 2021: The physical science basis: Working group I contribution to the sixth assessment report of the intergovernmental p...

    2021

  42. [42]

    Heavy-tailed diffusion with denoising levy probabilistic models

    Dario Shariatian, Umut Simsekli, and Alain Oliviero Durmus. Heavy-tailed diffusion with denoising levy probabilistic models. InThe Thirteenth International Conference on Learn- ing Representations

  43. [43]

    Score-based generative modeling through stochastic differential equations

    Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. arXiv preprint arXiv:2011.13456, 2020

    Pith/arXiv arXiv 2011

  44. [44]

    Improving and generalizing flow-based generative models with minibatch optimal transport.arXiv preprint arXiv:2302.00482, 2023

    Alexander Tong, Kilian Fatras, Nikolay Malkin, Guillaume Huguet, Yanlei Zhang, Jarrid Rector-Brooks, Guy Wolf, and Yoshua Bengio. Improving and generalizing flow-based generative models with minibatch optimal transport.arXiv preprint arXiv:2302.00482, 2023. 14

    Pith/arXiv arXiv 2023

  45. [45]

    Score-based generative models with l´ evy processes.Advances in Neural Information Processing Systems, 36:40694– 40707, 2023

    Eun Bi Yoon, Keehun Park, Sungwoong Kim, and Sungbin Lim. Score-based generative models with l´ evy processes.Advances in Neural Information Processing Systems, 36:40694– 40707, 2023

    2023

  46. [46]

    Long-tailed diffusion models with oriented calibration

    Tianjiao Zhang, Huangjie Zheng, Jiangchao Yao, Xiangfeng Wang, Mingyuan Zhou, Ya Zhang, and Yanfeng Wang. Long-tailed diffusion models with oriented calibration. InThe twelfth international conference on learning representations, 2024. 15 Appendix Contents A Additional Discussions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...

    arXiv 2024