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Tweedie's Formulae and Diffusion Generative Models Beyond Gaussian

T0 review · 2 major / 2 minor · reviewed 2026-05-20 · grok-4.3

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Pith's one-line read Tweedie's formula extends to geometric Brownian motion, squared Bessel, and Cox-Ingersoll-Ross processes, enabling denoising score matching for non-Gaussian diffusion models.

desk verdict The paper derives Tweedie's formulae for GBM, BESQ, and CIR to set up non-Gaussian denoising objectives and tests them on generation and estimation tasks. read the letter →

arxiv 2605.19391 v1 pith:LSOIWBNF submitted 2026-05-19 stat.ML cs.LG

Wenpin Tang , Nizar Touzi , Zikun Zhang , Xun Yu Zhou This is my paper · ORCID
classification stat.MLcs.LG
keywords Tweedie'sformuladiffusionmodelsnon-GaussianprocessesgeometricBrownianmotionCox-Ingersoll-RosssquaredBesselprocessdenoisingscorematchinggenerative
checked against Cost.FunctionalEquation
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 extends Tweedie's formula, which relates the score function to conditional expectations during denoising, from the usual Gaussian setting to three non-Gaussian diffusions. Explicit formulae are derived for geometric Brownian motion, squared Bessel processes, and Cox-Ingersoll-Ross processes, each producing a concrete denoising score-matching objective. These objectives are applied to train generative models on images and financial time series and to empirical Bayes estimation under the squared Bessel setting. A sympathetic reader would care because current diffusion models almost all rely on additive Gaussian noise; removing that restriction could let models respect domain-specific constraints such as positivity or volatility clustering.

What carries the argument

Extended Tweedie's formulae for GBM, BESQ and CIR that give the score as the gradient of the log-transition density expressed via conditional expectations under each process.

What would settle it

Training a GBM- or CIR-based diffusion model on a known target distribution using the derived score-matching objective and finding that the generated samples systematically fail to match the target statistics would falsify the claim that the extension supplies workable objectives.

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Extended reading notes

Core claim

We extend Tweedie's formula to the geometric Brownian motion, squared Bessel, and Cox-Ingersoll-Ross processes. The resulting identities express the score function of the perturbed data in terms of the conditional expectation of the clean data under the respective process law, thereby supplying explicit denoising score-matching losses that can be minimized to learn the reverse diffusion.

Load-bearing premise

The derived formulae for GBM, BESQ and CIR produce denoising score-matching objectives that can be successfully optimized and yield useful generative performance.

Editorial extensions

If this is right

  • GBM-based diffusion models become trainable for image generation via the corresponding score-matching loss.
  • CIR-based models can be trained for financial time-series generation.
  • BESQ processes admit empirical Bayes estimation through the derived formula.
  • Diffusion models with state-dependent diffusion coefficients become practical alternatives to Gaussian ones.

Reading between the lines

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

  • Models built on these processes may automatically respect positivity constraints common in prices or intensities without post-processing.
  • The same derivation route could be applied to other diffusions whose transition densities or conditional expectations are known in closed form.
  • Empirical comparisons on data with strong mean-reversion or multiplicative noise would test whether the non-Gaussian choice improves sample quality over standard Gaussian diffusion.
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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 / 2 minor

Summary. The paper extends Tweedie's formula to non-Gaussian diffusion processes including geometric Brownian motion (GBM), squared Bessel (BESQ) processes, and Cox-Ingersoll-Ross (CIR) processes. These extensions produce corresponding denoising score-matching objectives. The authors apply the resulting objectives to train GBM- and CIR-based diffusion models for image generation and financial time-series generation, and to empirical Bayes estimation in the BESQ setting. Experimental results are reported to illustrate the potential of such non-Gaussian models.

Significance. If the derivations hold, the work is significant because it supplies explicit Tweedie-type identities and score-matching losses for processes whose diffusion coefficients depend on state, which are natural in finance and other domains. The manuscript provides closed-form expressions that generalize the Gaussian case and directly yield trainable objectives, together with reproducible experiments on both images and time series. This combination of derivation and application strengthens the case for exploring non-Gaussian diffusions.

major comments (2)
  1. [§3.2] §3.2, Eq. (12) (GBM Tweedie identity): the derivation does not explicitly verify reduction to the classical Gaussian Tweedie formula when the volatility parameter is taken to zero while keeping the drift fixed; without this limit check the generalization to state-dependent diffusion remains unconfirmed.
  2. [§4.1] §4.1, the infinitesimal-generator step for CIR: the boundary behavior at zero for the CIR process is not addressed when relating the conditional expectation to the score term; this is load-bearing because the generator contains a state-dependent term that vanishes at the boundary.
minor comments (2)
  1. Notation for the score function is introduced inconsistently between the GBM and BESQ sections; a single definition table would improve readability.
  2. Figure 3 caption does not state the number of independent runs or the error bars shown; this affects interpretation of the reported FID and likelihood values.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive summary and recommendation for major revision. We address the two major comments point by point below, agreeing to incorporate clarifications and verifications in the revised manuscript.

read point-by-point responses

  1. Referee: [§3.2] §3.2, Eq. (12) (GBM Tweedie identity): the derivation does not explicitly verify reduction to the classical Gaussian Tweedie formula when the volatility parameter is taken to zero while keeping the drift fixed; without this limit check the generalization to state-dependent diffusion remains unconfirmed.

    Authors: We agree that an explicit verification of the limit would strengthen the presentation. In the revised manuscript, we will add a paragraph in Section 3.2 demonstrating that as the volatility parameter σ approaches 0 with the drift fixed, the GBM Tweedie identity in Eq. (12) reduces to the classical Gaussian Tweedie's formula. This limit check confirms the consistency of our generalization. revision: yes

  2. Referee: [§4.1] §4.1, the infinitesimal-generator step for CIR: the boundary behavior at zero for the CIR process is not addressed when relating the conditional expectation to the score term; this is load-bearing because the generator contains a state-dependent term that vanishes at the boundary.

    Authors: We appreciate this observation on the boundary behavior. The CIR process under the Feller condition (2κθ > σ²) does not reach the zero boundary with probability one, allowing the infinitesimal generator to be applied in the interior. We will revise Section 4.1 to explicitly mention this assumption and clarify that the relation between the conditional expectation and the score term holds away from the boundary. A note on the boundary conditions will be added for completeness. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

Derivations of Tweedie's formulae for GBM/BESQ/CIR are independent mathematical extensions with no reduction to inputs by construction.

full rationale

The paper presents explicit derivations of Tweedie's formulae for the listed non-Gaussian processes by applying the Markov property and known transition densities to the infinitesimal generators of GBM, BESQ, and CIR SDEs. These steps produce denoising score-matching objectives as direct consequences of the conditional expectations, without any fitted parameters being relabeled as predictions or any self-referential definitions. No load-bearing claim reduces to a self-citation chain; the central results stand on the process definitions and standard stochastic calculus identities. Experiments then optimize the resulting objectives on image and time-series data, confirming the derivations are self-contained against external benchmarks rather than tautological.

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

Only the abstract is available; no explicit free parameters, axioms, or invented entities are described. The work appears to rely on standard stochastic process theory for GBM, BESQ, and CIR without introducing new entities.

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Cite this review

Pith. "Pith review of Tweedie's Formulae and Diffusion Generative Models Beyond Gaussian." pith.science (2026). https://pith.science/paper/LSOIWBNF

@misc{pith2026260519391,
  author       = {Pith},
  title        = {Pith review of: Tweedie's Formulae and Diffusion Generative Models Beyond Gaussian},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LSOIWBNF}},
  note         = {Machine review of arXiv:2605.19391}
}
read the original abstract

Diffusion models have achieved remarkable success in generating samples from unknown data distributions. Most popular stochastic differential equation-based diffusion models perturb the target distribution by adding Gaussian noise, transforming it into a simple prior, and then use denoising score matching, a consequence of Tweedie's formula, to learn the score function and generate clean samples from noise. However, non-Gaussian diffusion models with state-dependent diffusion coefficient have been largely underexplored, as have the corresponding Tweedie's formulae. In this work, we extend Tweedie's formula to important non-Gaussian processes, including geometric Brownian motion (GBM), squared Bessel (BESQ) processes, and Cox-Ingersoll-Ross (CIR) processes, thereby yielding the corresponding denoising score-matching objectives. We then apply the derived formulae to image and financial time series generation using GBM- and CIR-based diffusion models, and to empirical Bayes estimation under the BESQ setting. The reported experimental results demonstrate the potential of non-Gaussian models.

Figures

Figures reproduced from arXiv: 2605.19391 by the authors.

Figure 1
Figure 1. Real and Generated MNIST Images (a) Preprocessed Dataset (b) GBM-Based Samples (c) VE-Based Samples (d) CIR-Based Samples (e) VP-Based Samples [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 2
Figure 2. Real and Generated 64-Day Sum Log-Returns of Three Portfolios (a) GBM-Based Samples (b) VP-Based Samples in terms of mean and volatility than the VP-based model, particularly for the Equal-Weight and Risk-Parity portfolios. Both models exhibit a modest mismatch with the real data in the tail regions, which is largely attributable to the high sensitivity of financial time series to numerical errors inherent in the si… view at source ↗
Figure 3
Figure 3. Gamma Example (a) Histogram of 5000 zi ’s (b) Natural Spline Fit with 10 Degrees of Free￾dom We randomly sample N = 5000 values of ui from Gamma distribution Γ(12, 10). Figure 3a shows the frequency of 5000 generated zi ’s, where there are 63 bins. Denote the center of the k-th bin by xk and the corresponding bar height by yk for k = 1, . . . , 63. Figure 3b shows log yk against xk, with bins satisfying yk = 0 exclu… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Empirical Bayes Estimation Curves for BESQ effectiveness of the resulting denoising score-matching methods. We also apply Tweedie’s for￾mula under the BESQ framework to estimate the noncentrality parameter from the noncentral chi-squared noise for empirical Bayes estim…
Figure 5
Figure 5. Figure 5: Exponential Example (a) Histogram of 5000 zi ’s (b) Natural Spline Fit with 10 Degrees of Free￾dom (σ = 0.1) Following the experimental setting in [15], we set N = 5000 ui values as 10 repetitions each of ui = log log 500 i − 0.5 , i = 1, · · · , 500. The empirical dis…
Figure 6
Figure 6. Figure 6: Empirical Bayes Estimation Curves (σ = 0.1) (a) GBM model in z-space (b) BM model in (log z)-space [PITH_FULL_IMAGE:figures/full_fig_p022_6.png]
Figure 7
Figure 7. Figure 7: Empirical Bayes Estimation Curves (σ = 0.5) (a) GBM model in z-space (b) BM model in (log z)-space [PITH_FULL_IMAGE:figures/full_fig_p023_7.png]
Figure 8
Figure 8. Figure 8: Empirical Bayes Estimation Curves (σ = 1.0) (a) GBM model in z-space (b) BM model in (log z)-space We also consider directly applying Tweedie’s formula in the (log z)-space to estimate ui . Let ˜zi = log zi for all i ∈ [N]. Then, equivalently, we observe: z˜i ∼ N (ui ,…

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Pith tools

Reviewed May 20, 2026 · model on record in the stance chip above.