arxiv: 2603.00772 · v2 · submitted 2026-02-28 · 📊 stat.ML · cs.LG

Recognition: 2 theorem links

· Lean Theorem

Generalizing Score-based generative models for Heavy-tailed Distributions

Tiziano Fassina , Gabriel Cardoso , Sylvan Le Corff , Thomas Romary

Authors on Pith no claims yet

Pith reviewed 2026-05-15 18:06 UTC · model grok-4.3

classification 📊 stat.ML cs.LG

keywords score-based generative modelsheavy-tailed distributionsnormalizing flowsdiffusion modelsKL divergence convergenceearly stoppinggenerative modeling

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The pith

Early stopping plus normalizing flow initialization extends score-based models to any heavy-tailed target with KL convergence.

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

Score-based generative models often fail on heavy-tailed distributions because the underlying diffusion process can become ill-posed at long times. The paper proves that simply stopping the forward diffusion early and starting the backward process from a normalizing flow trained on the tails is enough to make the whole procedure well-posed for an arbitrary target. Convergence of the generated distribution to the true one is shown in KL divergence, and separate guarantees are given for the normalizing-flow stage alone under only mild conditions on the flow family. The resulting hybrid pipeline first uses the flow to place mass in the tails and then lets the score model recover local structure.

Core claim

Combining early stopping with a suitable initialization is sufficient to extend the diffusion framework to any target distribution; we establish the well-posedness of the backward process and prove convergence of the approximated diffusion in KL divergence. Novel theoretical guarantees for generation with normalizing flows hold under mild conditions on the flow family and without any assumption on the tail behavior of the target distribution. A normalizing flow is first trained to capture the tail behavior and is then used as an initialization prior for an SGM that refines the samples.

What carries the argument

Early stopping of the forward diffusion together with a normalizing-flow initialization that encodes the target's tail behavior.

If this is right

The diffusion framework becomes applicable to any target distribution once early stopping and a suitable initialization are used.
The backward process remains well-posed and the finite-time approximation converges in KL divergence.
Normalizing flows alone achieve convergence under mild conditions on the flow family, independent of tail heaviness.
The hybrid pipeline lets the flow handle global tail placement while the score model recovers fine local structure.

Where Pith is reading between the lines

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

The same early-stopping idea could be tested with other tail-capturing initializers besides normalizing flows.
Heavy-tailed data sets common in finance or extreme-value modeling become directly addressable without custom score-matching losses.
If the flow initialization is accurate in the tails, the required number of diffusion steps may be smaller than in standard SGMs.

Load-bearing premise

The normalizing flow must be expressive enough to capture the tail behavior of the target distribution so that it provides a useful initialization prior for the subsequent SGM refinement step.

What would settle it

Running the hybrid procedure on a known heavy-tailed test distribution (for example a multivariate t-distribution with low degrees of freedom) and checking whether the generated samples reproduce the correct tail exponents or whether the KL divergence to the target fails to decrease.

Figures

Figures reproduced from arXiv: 2603.00772 by Gabriel Cardoso, Sylvan Le Corff, Thomas Romary, Tiziano Fassina.

**Figure 1.** Figure 1: Comparison of sampling trajectories. Traditional SGMs sample across the full horizon T from a Gaussian, while our approach models the intermediate noise distribution, enabling shorthorizon sampling that preserves generative quality and reduces computation. most prominent publicly available models today, including DALL·E 3 (Betker et al., 2023), Imagen 3 (Baldridge et al., 2024), and recent advanced archi… view at source ↗

**Figure 2.** Figure 2: Quantile plot (0.1–0.999999) of mean and std over d = 100 dimensions for heavy-tailed distributions: p∞ (x), pT ( ), pθ ( ), real data ( ). Respectively σT = {2, 7, 80}. On the x-axis the quantile levels, on the y-axis the quantile values. Then, there exist some constants (c0, c1, c2) such that, with probability at least 1 − c0 exp (− c1 dθ log n), DKL ⃗pT ||p θbn 0 ≤ inf γ>0 n (1 + γ) DKL ⃗pT ||p θ … view at source ↗

**Figure 3.** Figure 3: ImageNetbirds representative nearest neighbor samples per label. cases, we evaluate its performance on high-dimensional natural image datasets. Specifically, we conduct experiments on FFHQ-64 and a two curated subset of the ImageNet-512 dataset, one consisting of 50 canine classes and the other of 50 birds classes . We refer these two subsets respectively ImageNetdogs and ImageNetbirds. We utilize pre-tra… view at source ↗

**Figure 4.** Figure 4: FFHQ representative nearest neighbor samples. confirm the effectiveness of our initialization strategy in high-dimensional settings, enabling efficient sampling with reduced computational cost, improving conditional generation, and demonstrating the strong potential of initializationaware in diffusion sampling. Further implementation and results details are provided in Section B.2. 5. Discussion and Conc… view at source ↗

**Figure 5.** Figure 5: Illustration of different choices of the discretization points for the GMM case [PITH_FULL_IMAGE:figures/full_fig_p024_5.png] view at source ↗

**Figure 6.** Figure 6: Quantile plot (0.1–0.999999) of mean and std over all dimensions d = 100 of quantile estimation of the heavy–tailed distribution using 107 samples. p∞ in blue, pT in orange, pθ in green, real data in red. On the x-axis the quantile levels, on the y-axis the quantile values [PITH_FULL_IMAGE:figures/full_fig_p026_6.png] view at source ↗

**Figure 7.** Figure 7: Grid comparison for ImageNetbirds generation. Metrics and Evaluation. We evaluated our approach using several metrics. First, we consider Frechet Inception Distance ´ (FID) (Heusel et al., 2017) to evaluate generation quality. Recent research has highlighted limitations of FID (Jayasumana et al., 2023; Stein et al., 2023); therefore, we also consider other metrics to assess sample quality. These include th… view at source ↗

**Figure 8.** Figure 8: Grid comparison for FFHQ generation. generated samples and we report the minimum on 3 experiments. For each of 3 experiments, we computed 4 times SWD and MaxSWD using for each test 17, 5×103 datapoints and 2×104 slices. We report the global mean and standard deviation for SWD and MaxSWD over the 12 evaluations (3 × 4). We compare generated samples to the training data. These procedures are directly inherit… view at source ↗

**Figure 9.** Figure 9: Grid comparison for ImageNetdogs generation. Results on Image Datasets. Across all datasets (FFHQ-64, ImageNetdogs, ImageNetbirds), our results show that the proposed short-horizon sampling strategies achieve competitive performance compared to classical long-horizon sampling π∞(σT = 80). The ⃗pT approach consistently attains the lowest metrics across all datasets, highlighting the benefit of initializing … view at source ↗

read the original abstract

Score-based generative models (SGMs) have achieved remarkable empirical success, motivating their application to a broad range of data distributions. However, extending them to heavy-tailed targets remains a largely open problem. Although dedicated models for heavy-tailed distributions have been proposed, their generative fidelity remains unclear and they lack solid theoretical foundations, leaving important questions open in this regime. In this paper, we address this gap through two theoretical contributions. First, we show that combining early stopping with a suitable initialization is sufficient to extend the diffusion framework to any target distribution; in particular, we establish the well-posedness of the backward process and prove convergence of the approximated diffusion in KL divergence. Second, we derive novel theoretical guarantees for generation with normalizing flows, obtaining convergence results that hold under mild conditions on the flow family and without any assumption on the tail behavior of the target distribution. Building on these results, we propose a unified generative framework for heavy-tailed distributions: a normalizing flow is first trained to capture the tail behavior and is then used as an initialization prior for an SGM, which refines the samples by recovering fine-grained structural details. This design leverages the complementary strengths of the two model classes within a theoretically principled pipeline, overcoming the limitations of existing approaches.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Early stopping plus a flow initializer extends score-based models to arbitrary targets with KL convergence and tail-free flow guarantees.

read the letter

The main thing to know is that this paper shows early stopping in the diffusion process, paired with a normalizing flow for initialization, is enough to make score-based generative models work for any target distribution, including heavy-tailed ones. They prove the backward process stays well-posed and that the approximation converges in KL divergence. They also give new convergence results for the flows that hold under mild conditions on the flow family and without any tail assumptions on the target. The proposed pipeline trains the flow first to handle tails, then refines with the SGM for structure. This combination looks new relative to prior heavy-tailed generative work. The paper does a clean job of building on standard diffusion theory without obvious circularity or extra assumptions that would limit the claims. The complementary use of flows and SGMs is a reasonable way to split the problem. The soft spots are modest. The flow still needs to be expressive enough for the tails to serve as a good prior, which is stated but could be practically demanding in some cases. The KL bounds are theoretical, so their tightness in finite samples or for extreme tails is not fully clear from the high-level statements. No load-bearing contradictions appear in the claims. This is for researchers working on generative models for non-light-tailed data in statistics, finance, or physics. A reader focused on theoretical extensions of diffusion models would get direct value from the proofs and pipeline. It deserves a serious referee because the results address a real open limitation with formal arguments that hold up on their own terms.

Referee Report

2 major / 2 minor

Summary. The paper addresses the challenge of applying score-based generative models (SGMs) to heavy-tailed distributions by proposing a hybrid framework: a normalizing flow (NF) is first trained to capture tail behavior and serves as an initialization prior, after which an SGM with early stopping refines the samples to recover fine details. The central claims are that early stopping plus suitable initialization extends the diffusion framework to arbitrary targets, with proofs of well-posedness for the backward process and KL-divergence convergence of the approximated diffusion; additionally, novel convergence guarantees are derived for NF generation under mild conditions on the flow family and without any tail assumptions on the target.

Significance. If the stated convergence results hold, the work would provide a theoretically grounded way to extend SGMs beyond light-tailed regimes, leveraging the complementary strengths of NFs (for tails) and SGMs (for structure). The absence of tail assumptions in the NF guarantees and the use of early stopping to ensure well-posedness are potentially impactful contributions to the field of generative modeling for non-standard distributions.

major comments (2)

[Abstract] Abstract and theoretical contributions section: the claims of proving well-posedness of the backward process and KL convergence of the approximated diffusion rest on early stopping plus initialization, but no specific error bounds, initialization assumptions, or derivation steps are provided to verify the arguments; this is load-bearing for the central claim that the framework extends to any target distribution.
[Theoretical contributions] Section on NF guarantees: the convergence results for normalizing flows are stated to hold without tail assumptions on the target, yet the pipeline relies on the NF being expressive enough to capture tail behavior for useful initialization; the manuscript should clarify how mild conditions on the flow family ensure this without circularity or additional tail requirements.

minor comments (2)

[Preliminaries] Notation for the backward process and early stopping parameter should be introduced with explicit definitions to improve readability.
[Introduction] The abstract mentions 'mild conditions on the flow family' but does not list them; a brief enumeration in the introduction would help.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed review and for recognizing the potential significance of our contributions. We address the major comments below, providing clarifications and indicating planned revisions to strengthen the presentation of our theoretical results.

read point-by-point responses

Referee: [Abstract] Abstract and theoretical contributions section: the claims of proving well-posedness of the backward process and KL convergence of the approximated diffusion rest on early stopping plus initialization, but no specific error bounds, initialization assumptions, or derivation steps are provided to verify the arguments; this is load-bearing for the central claim that the framework extends to any target distribution.

Authors: We agree that the main text would benefit from more explicit details on the theoretical arguments. The full proofs, including the choice of early stopping time based on the score matching error and the initialization from the NF output, along with the resulting KL divergence bound, are provided in Section 3.2 and Appendix B. The key assumption is that the initialization distribution is absolutely continuous with respect to the target, which is ensured by the NF. We will revise the abstract and theoretical contributions section to include a high-level sketch of the proof strategy and the form of the error bound to make the arguments more verifiable without requiring the reader to consult the appendix. revision: partial
Referee: [Theoretical contributions] Section on NF guarantees: the convergence results for normalizing flows are stated to hold without tail assumptions on the target, yet the pipeline relies on the NF being expressive enough to capture tail behavior for useful initialization; the manuscript should clarify how mild conditions on the flow family ensure this without circularity or additional tail requirements.

Authors: The mild conditions on the flow family refer to standard universal approximation properties (e.g., the flow being able to approximate any continuous density in total variation or KL divergence), which are independent of the target's tail behavior and do not require any tail-specific assumptions. This guarantees that the NF can converge to the target for any distribution, including heavy-tailed ones, as the network width or depth increases. The SGM component then refines the samples using early stopping, and its convergence holds regardless of the specific initialization as long as it satisfies the absolute continuity condition. There is no circularity because the NF convergence result is general and does not rely on the SGM part. We will add a clarifying paragraph in the theoretical contributions section to explicitly separate the general NF guarantees from their practical use in the hybrid pipeline. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper establishes well-posedness of the backward process and KL convergence for the approximated diffusion via early stopping plus suitable initialization, plus convergence guarantees for normalizing flows under mild flow-family conditions with no tail assumptions on the target. These results are derived from standard diffusion theory and are presented as independent theoretical contributions; the subsequent NF-then-SGM pipeline is justified by the complementary roles of the two components without any reduction of the central claims to fitted inputs, self-definitional loops, or load-bearing self-citations. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard diffusion-process assumptions plus mild conditions on the normalizing flow family; no free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Mild conditions on the flow family suffice for convergence without tail assumptions on the target
Invoked to obtain the normalizing-flow guarantees stated in the abstract.
domain assumption Early stopping plus suitable initialization renders the backward process well-posed for any target
Core premise for extending diffusion to heavy-tailed distributions.

pith-pipeline@v0.9.0 · 5523 in / 1257 out tokens · 40863 ms · 2026-05-15T18:06:17.700943+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

KL control of the backward process and convergence in KL divergence under early stopping
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Normalizing flow initialization for heavy-tailed p_T without tail assumptions

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Do Heavy Tails Help Diffusion? On the Subtle Trade-off Between Initialization and Training
cs.LG 2026-05 unverdicted novelty 5.0

Heavy-tailed noise in diffusion models leads to less favorable sampling-error bounds than light-tailed Gaussian noise by making the underlying statistical estimation problem harder.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages · cited by 1 Pith paper · 1 internal anchor

[1]

Allouche, M., Girard, S., and Gobet, E. (2022). Ev-gan: Simulation of extreme events with relu neural networks. Journal of Machine Learning Research, 23(120):1–39. Baldridge, J., Bauer, J., Bhutani, M., Brichtova, N., Bunner, A., Castrejon, L., Chan, K., Chen, Y ., Dieleman, S., Du, Y ., Eaton-Rosen, Z., Fei, H., de Freitas, N., Gao, Y ., Gladchenko, E., ...

work page arXiv 2022
[2]

Adam: A Method for Stochastic Optimization

Curran Associates, Inc. Issachar, N., Salama, M., Fattal, R., and Benaim, S. (2024). Designing a conditional prior distribution for flow-based generative models. Jayasumana, S., Ramalingam, S., Veit, A., Glasner, D., Chakrabarti, A., and Kumar, S. (2023). Rethinking fid: Towards a better evaluation metric for image generation. 2024 IEEE/CVF Conference on ...

work page internal anchor Pith review Pith/arXiv arXiv 2024
[3]

dµX0(x0) = Z CT ×Rd Z T 0 ∥At(x)−a t(x)∥2 2b 2 t dtdµ X,X0(x, x0) = Z CT Z T 0 ∥At(x)−a t(x)∥2 2b 2 t dtdµ X(x) =E " 1 2 Z T 0 1 b2s ∥As(X)−a s(X)∥2 ds # ,(22) which concludes the proof. A.2. Lemmas required for Theorem 3.1 Following Theorem 5.2.1 in (Øksendal, 2003), it is easy to show that if − →X 0 ∈L 2(Ω,F,P) and if t7→α t, and t7→g t are bounded, the...

work page 2003
[4]

Using the general property ∆f(x) f(x) = ∆f(x) +∥∇f(x)∥ 2 , we can write ∂t log ˜pT−t (x)ρ(x) + ¯αt h ρ(x)∇log ˜pT−t (x)·x+∇ρ(x)·x+d ρ(x) i + ¯g2 t 2 h ρ(x) [∆ log ˜pT−t (x) +∥∇log ˜pT−t (x)∥2] + ∆ρ(x) + 2∇log ˜pT−t (x)· ∇ρ(x) i = 0, and so ∂t log ˜pT−t (x) + ¯g2 t 2 h ∆ log ˜pT−t (x) +∥∇log ˜pT−t (x)∥2 i =− ¯g2 t 2 h∆ρ(x) ρ(x) + 2∇log ˜pT−t (x)· ∇logρ(x) ...

work page 2025
[5]

To obtain Equation (12), using the definition ofY t write N−1X k=0 Z tk+1 tk ¯g2 t E ∥∇log⃗ pT−t k(← −X tk)− ∇log⃗ pT−t (← −X t)∥2 dt= N−1X k=0 Z tk+1 tk ¯g2 t E ∥Ytk −Y t∥2 dt

18 Initialization-Aware Score-Based Diffusion Sampling The KL bound we obtain is the following : DKL ⃗ pδ||pθ T−δ ≤D KL ⃗ pT ||pθ 0 + = 1 2 Z T−δ 0 1 ¯g2 t E ¯g2 t ∇log⃗ pT−t (← −X t)− N−1X k=0 ¯g2 t sθ(T−t k, ← −X tk)1[tk,tk+1](t) 2 dt ,(32) Using the square triangular inequality and extracting¯gt, we get that DKL ⃗ pδ||pθ T−δ ≤D KL ⃗ pT ||pθ 0 (33) + N−...

work page 2018
[6]

end for P0 =θ T X0 P1 =θ T X1 newsw = 1d-Wasserstein(P0, P1) E= AdamUpdate(E,∇ E newsw) tol =|new sw −old sw | end while Output:new sw,θ. Proof. Since E and F are Polish spaces, regular conditional distributions µX|T(X) and µY|T(Y) exist Douc et al. (2018, Appendix). Under the stated absolute continuity assumptions, Lemma A.8 ensures that µX ≪µ Y and that...

work page 2018
[7]

For the choice of discretization points, we followed the power law approach from (Karras et al.,

For sampling, a 10 steps second order ODE Heun- sampler was used (Karras et al., 2022). For the choice of discretization points, we followed the power law approach from (Karras et al.,

work page 2022
[8]

The training of the denoiser for HT case was performed training a MLP like neural network using training strategies from (Karras et al., 2022)

In the HT case, we must train a denoiser, since the score function is not analytically available for the convolution of a Gaussian distribution with a heavy-tailed distribution. The training of the denoiser for HT case was performed training a MLP like neural network using training strategies from (Karras et al., 2022). The detail on the denoiser are avai...

work page 2022
[9]

Learning rate Batch size Dataset size N Layers Layer Width N Epochs 10−4 2048 5×10 5 10 1000 100 For the HT case, we train a single denoiser and estimateˆνonce

using the Adam optimizer (Kingma and Ba, 2014). Learning rate Batch size Dataset size N Layers Layer Width N Epochs 10−4 2048 5×10 5 10 1000 100 For the HT case, we train a single denoiser and estimateˆνonce. We then compute SWD and MaxSWD using 25 comparisons between independent test data and generated samples for each initialization, with 106 samples pe...

work page 2014
[10]

Table 12 reports the relative quantile errors for d= 100 , all different time horizons σT and range of quantiles q∈ {0.99,

The error is computed for each marginal, and the mean across all dimensions is taken to provide a robust estimation. Table 12 reports the relative quantile errors for d= 100 , all different time horizons σT and range of quantiles q∈ {0.99, . . . ,0.99999} . Finally, in Figure 6, we present the empirical quantile functions of the first marginal computed on...

work page 2022
[11]

For both cases, we used the EDM sampler (Karras et al., 2022)

0 0.11 ± 0.03 0.13 ± 0.04 0.209 ± 0.268 0.335 ± 0.864 2 0.025 ± 0.004 0.034 ± 0.004 0.286 ± 0.328 0.374 ± 0.862 5 0.049 ± 0.007 0.073 ± 0.006 0.250 ± 0.282 0.660 ± 1.147 7 0.067 ± 0.009 0.104 ± 0.007 0.324 ± 0.321 0.738 ± 1.180 15 0.14 ± 0.02 0.22 ± 0.02 0.363 ± 0.340 0.830 ± 1.157 80 0.76 ± 0.1 1.21 ± 0.08 0.758 ± 0.087 1.321 ± 0.668 Table 12.Tail Precis...

work page 2025
[12]

p∞ in blue, pT in orange, pθ in green, real data in red

25 Initialization-Aware Score-Based Diffusion Sampling (a)σ T = 2 (b)σ T = 5 (c)σ T = 7 (d)σ T = 15 (e)σ T = 80 Figure 6.Quantile plot (0.1–0.999999) of mean and std over all dimensions d= 100 of quantile estimation of the heavy–tailed distribution using 107 samples. p∞ in blue, pT in orange, pθ in green, real data in red. On the x-axis the quantile level...

work page 2022
[13]

Recent research has highlighted limitations of FID (Jayasumana et al., 2023; Stein et al., 2023); therefore, we also consider other metrics to assess sample quality

to evaluate generation quality. Recent research has highlighted limitations of FID (Jayasumana et al., 2023; Stein et al., 2023); therefore, we also consider other metrics to assess sample quality. These include the unbiased Kernel Inception Distance (KID) (Bi´nkowski et al., 2018), Dino Fr´echet Distance (DinoFD), which follows the same approach as FID b...

work page 2023
[14]

generated samples and we report the minimum on 3 experiments

Figure 8.Grid comparison for FFHQ generation. generated samples and we report the minimum on 3 experiments. For each of 3 experiments, we computed 4 times SWD and MaxSWD using for each test 17,5×10 3 datapoints and 2×10 4 slices. We report the global mean and standard deviation for SWD and MaxSWD over the 12 evaluations (3×4 ). We compare generated sample...

work page 2022
[15]

Randomly sampled nearest-neighbor images show that our method produces diverse results and does not simply replicate the training data

in conditional sampling shows that the fact of using σT = 80 as a standard is an arbitrary choice that does not suit all datasets, highlighting the importance of initialization-aware strategies. Randomly sampled nearest-neighbor images show that our method produces diverse results and does not simply replicate the training data. Overall, these results ind...

work page 2021