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arxiv: 2605.13175 · v1 · submitted 2026-05-13 · 💻 cs.LG

Recognition: unknown

Do Heavy Tails Help Diffusion? On the Subtle Trade-off Between Initialization and Training

Hamza Cherkaoui , H\'el\`ene Halconruy , Antonio Ocello
Authors on Pith no claims yet

Pith reviewed 2026-05-14 19:07 UTC · model grok-4.3

classification 💻 cs.LG
keywords heavy-tailed noisediffusion modelssampling error boundsgenerative modelingstatistical estimationlight-tailed noisetail recovery
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The pith

Heavy-tailed noise makes statistical estimation harder in diffusion models than Gaussian noise.

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

The paper examines whether replacing Gaussian noise with heavy-tailed noise improves diffusion- and flow-based generative models when data have heavy tails. It derives sampling-error bounds for two representative models showing that heavy-tailed noise increases the difficulty of the underlying statistical estimation problem. This produces less favorable error bounds despite the intuitive match to data tails. Experiments on synthetic and real-world datasets recover the predicted trade-off between initialization and training performance.

Core claim

We show that heavy-tailed noise makes the statistical estimation problem harder, leading to less favorable sampling-error bounds. We support these findings with experiments on synthetic and real-world datasets, empirically recovering the predicted error trade-off.

What carries the argument

Sampling-error bounds for two representative diffusion models driven by heavy-tailed versus light-tailed noise, which quantify the increased estimation difficulty.

If this is right

  • Heavy-tailed noise increases estimation error for score or velocity fields in the studied models.
  • Sampling performance degrades where estimation error dominates over tail-matching benefits.
  • The error trade-off applies across both diffusion and flow-based generative models.
  • Growing use of heavy-tailed noise for rare-region exploration requires re-evaluation.

Where Pith is reading between the lines

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

  • Noise schedules that start heavy-tailed and transition to light-tailed could balance the trade-off.
  • The bounds may extend to other score-based or flow-matching frameworks beyond the two models tested.
  • High-dimensional real-world data could show whether the estimation penalty grows with dimension.

Load-bearing premise

The derived sampling-error bounds for the two representative diffusion models are tight enough to reflect practical performance differences between heavy-tailed and light-tailed noise.

What would settle it

An experiment showing heavy-tailed noise achieving lower sampling error than light-tailed noise in a regime where the bounds predict the opposite would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.13175 by Antonio Ocello, Hamza Cherkaoui, H\'el\`ene Halconruy.

Figure 1
Figure 1. Figure 1: Training loss and standard deviation of the training loss across trials. [PITH_FULL_IMAGE:figures/full_fig_p023_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Reproduction of the empirical setting of Table 1 in [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗
read the original abstract

Recent works have proposed incorporating heavy-tailed (HT) noise into diffusion- and flow-based generative models, with the goals of better recovering the tails of target distributions and improving generative diversity. This motivation is intuitive: if the data are heavy-tailed, HT noise may appear better matched than light-tailed (LT) Gaussian noise. However, replacing Gaussian noise by HT noise also changes the underlying estimation problem. In this paper, we revisit this paradigm through a combined theoretical and empirical study, establishing sampling-error bounds for two representative diffusion models driven by HT and LT noise. We show that HT noise makes the statistical estimation problem harder, leading to less favorable sampling-error bounds. We support these findings with experiments on synthetic and real-world datasets, empirically recovering the predicted error trade-off. Our results call into question a growing design trend in generative modeling and challenge the use of HT noise to improve rare-region exploration.

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.

Referee Report

1 major / 2 minor

Summary. The paper claims that replacing Gaussian (light-tailed) noise with heavy-tailed (HT) noise in diffusion models, while intuitively motivated by better tail matching, actually renders the underlying statistical estimation problem harder. It establishes this via sampling-error bounds for two representative diffusion models, showing strictly less favorable bounds under HT noise, and supports the theoretical ordering with experiments on synthetic and real-world datasets that recover the predicted error trade-off. The work concludes by questioning the growing use of HT noise to enhance generative diversity and rare-event exploration.

Significance. If the sampling-error bounds are sufficiently tight and the empirical trade-off is robust, the result is significant: it supplies a concrete theoretical caution against an emerging design trend in generative modeling, clarifies a subtle initialization-training interplay, and supplies reproducible evidence that HT noise can degrade estimation quality even when data tails are heavy. The combined derivation-plus-experiment approach is a strength.

major comments (1)
  1. [§4, Theorem 2] §4, Theorem 2 (HT sampling-error bound): the upper bound contains a tail-index-dependent factor whose looseness relative to the corresponding LT bound is not quantified; without a matching lower bound or tightness argument, it remains possible that the reported ordering reflects proof artifacts rather than intrinsic estimation difficulty, which is load-bearing for the central claim that HT noise makes the problem strictly harder.
minor comments (2)
  1. [Experiments] Experimental section: the precise rules for data exclusion, number of independent runs, and error-bar construction are not stated; adding these details would allow readers to assess whether the observed trade-off is robust to post-hoc choices.
  2. [§3] Notation: the definition of the score-function regularity parameter used in the bounds should be restated explicitly in the main text rather than deferred entirely to the appendix.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive comments and for recognizing the significance of our findings on the trade-off between heavy-tailed noise and estimation difficulty in diffusion models. We address the major comment regarding Theorem 2 below.

read point-by-point responses

  1. Referee: [§4, Theorem 2] §4, Theorem 2 (HT sampling-error bound): the upper bound contains a tail-index-dependent factor whose looseness relative to the corresponding LT bound is not quantified; without a matching lower bound or tightness argument, it remains possible that the reported ordering reflects proof artifacts rather than intrinsic estimation difficulty, which is load-bearing for the central claim that HT noise makes the problem strictly harder.

    Authors: We acknowledge the referee's point that the upper bound in Theorem 2 includes a tail-index-dependent factor not present in the light-tailed case, and that its looseness is not explicitly quantified. This factor originates from the analysis of the estimation error under heavy-tailed distributions, where we must account for the slower decay of tails in the noise, leading to weaker concentration inequalities compared to the sub-Gaussian case for Gaussian noise. While we do not derive a matching lower bound, which would require substantially different techniques such as information-theoretic arguments, the empirical experiments on synthetic data and real-world datasets recover the exact ordering predicted by the bounds, providing evidence that the difference is intrinsic rather than a proof artifact. In the revised manuscript, we will add a remark in Section 4 explaining the derivation of this factor and its dependence on the tail index to clarify its necessity. revision: partial

Circularity Check

0 steps flagged

No significant circularity; bounds derived from standard estimation theory

full rationale

The paper establishes sampling-error bounds for HT and LT diffusion models via direct application of concentration inequalities and estimation theory to the respective noise distributions. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the comparison between HT and LT follows from the explicit forms of the derived bounds without renaming or smuggling assumptions. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies insufficient technical detail to enumerate free parameters, axioms, or invented entities; no explicit fitted constants, unproved background results, or new postulated objects are named.

pith-pipeline@v0.9.0 · 5461 in / 1075 out tokens · 31650 ms · 2026-05-14T19:07:39.988222+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages · 2 internal anchors

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    Notation summary

    13 Table of contents A. Notation summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 B. Theoretical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 C. Additional discussion of evaluation m...

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    Approximation errorE app(t0)

    | {z } Eest(n,t0) + TV pT , ϕT | {z } Einit(T) . Approximation errorE app(t0). The term Eapp(t0) quantifies the bias introduced by Gaussian smoothing. It reflects the discrepancy between the original density and its regularized version pt0. Under Sobolev regularity and polynomial tails, Yu and Yu (2026) show that Eapp(t0)≲t β(γ+1) d+2(γ+1)+2β 0 . This rat...

    work page 2026

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    B.2 DLPM The corresponding heavy-tailed generative setting based on α-stable noising processes is introduced in Section 2.2

    ≲t β(γ+1) d+2(γ+1)+2β 0 + polylog(n)n − γ+1 2(d+γ+1) t − d(γ+1) 4(d+γ+1) 0 +T −1/2 . B.2 DLPM The corresponding heavy-tailed generative setting based on α-stable noising processes is introduced in Section 2.2. Decomposition of the error. Since k(α) 0|a and ← −q θ 0|a are the Y0-marginals of Ka and Qθ a, respectively, the data-processing inequality for tot...

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    DLPM bound

    ≲T −1/2 | {z } initialization error +t β(γ+1) d+2(γ+1)+2β 0 + polylog(n)n − γ+1 2(d+γ+1) t − d(γ+1) 4(d+γ+1) 0| {z } training error . DLPM bound. EA h TV k(α) 0|A, ← −q bθn 0|A i ≲e −cT |{z} initialization error +T m −β(α)/d +T r Comp + log(1/δ) n| {z } training error . We now compare each source of error in turn. Initialization.The initialization term me...

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    We summarize their forward paths below

    and DLPM (Shariatian et al., 2025). We summarize their forward paths below. • Gaussian Flow LinearThe model uses the linear interpolation path xt = (1−t)x 0 +tx 1, t∈[0,1] . • DDPMFor data x0 and Gaussian noise ε∼ N(0, I) , the forward process is xt = √¯αt x0 +√1−¯αt ε, t= 1, . . . , T. • DLPMreplaces Gaussian noise by isotropic α-stable noise: xt =a tx0 ...

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    For GF-Linear, DDPM, and DLPM, we use 512 denoising or solver steps

    All runs use AdamW with a cosine learning-rate schedule, batch size 1024, and512training epochs. For GF-Linear, DDPM, and DLPM, we use 512 denoising or solver steps. All final runs use 20 independent trials. For tabular data, the dataset is shuffled before being subsampled to the requested number of samples. For DLPM, the final benchmark keeps two tail-in...

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    | Training Loss GF-Linear DDPM DLPM ( = 1.7) DLPM ( = 1.9) (a) Alpha-stable iso.: training loss

    22 0 100 200 300 400 500 Epoch 100 101 Training Loss Alpha-stable iso. | Training Loss GF-Linear DDPM DLPM ( = 1.7) DLPM ( = 1.9) (a) Alpha-stable iso.: training loss. 0 100 200 300 400 500 Epoch 10 1 100 Training Loss Std. Alpha-stable iso. | Training Loss Std. GF-Linear DDPM DLPM ( = 1.7) DLPM ( = 1.9) (b) Alpha-stable iso.: loss std. 0 100 200 300 400 ...

    work page 2025