# A Malliavin-Gamma calculus approach to Score Based Diffusion Generative models for random fields

arXiv: https://arxiv.org/abs/2505.13189 · HTML: https://mlsanity.com/paper/2505.13189/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 21 of its 44 references that are in the map.
- First posted on arXiv 19 May 2025.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 21 of 21)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Convergence of score-based generative modeling for general data distributions](https://mlsanity.com/paper/2209.12381/) (2022)
- [Convergence for score-based generative modeling with polynomial complexity](https://mlsanity.com/paper/2206.06227/) (2022)
- [Score-based Diffusion Models in Function Space](https://mlsanity.com/paper/2302.07400/) (2023)
- [KL Convergence Guarantees for Score diffusion models under minimal data assumptions](https://mlsanity.com/paper/2308.12240/) (2023)
- [Infinite-Dimensional Diffusion Models](https://mlsanity.com/paper/2302.10130/) (2023)
- [Multilevel Diffusion: Infinite Dimensional Score-Based Diffusion Models for Image Generation](https://mlsanity.com/paper/2303.04772/) (2023)
- [Continuous Ensemble Weather Forecasting with Diffusion models](https://mlsanity.com/paper/2410.05431/) (2024)
- [Beyond Log-Concavity and Score Regularity: Improved Convergence Bounds for Score-Based Generative Models in W2-distance](https://mlsanity.com/paper/2501.02298/) (2025)
- [An analysis of the noise schedule for score-based generative models](https://mlsanity.com/paper/2402.04650/) (2024)
- [Posterior samples of source galaxies in strong gravitational lenses with score-based priors](https://mlsanity.com/paper/2211.03812/) (2022)
- [Conditioning Diffusions Using Malliavin Calculus](https://mlsanity.com/paper/2504.03461/) (2025)
- [Wasserstein Convergence of Score-based Generative Models under Semiconvexity and Discontinuous Gradients](https://mlsanity.com/paper/2505.03432/) (2025)
- [A Unified Approach to Analysis and Design of Denoising Markov Models](https://mlsanity.com/paper/2504.01938/) (2025)
- [Manifold Diffusion Fields](https://mlsanity.com/paper/2305.15586/) (2023)
- [Malliavin Calculus for Score-based Diffusion Models](https://mlsanity.com/paper/2503.16917/) (2025)
- [Stochastic Super-resolution of Cosmological Simulations with Denoising Diffusion Models](https://mlsanity.com/paper/2310.06929/) (2023)
- [Removing Dust from CMB Observations with Diffusion Models](https://mlsanity.com/paper/2310.16285/) (2023)

## Built on by (most cited first, 2 of 2)

- [A Malliavin calculus approach to score functions in diffusion generative models](https://mlsanity.com/paper/2507.05550/) (2025)
- [SPDE Methods for Nonparametric Bayesian Posterior Contraction and Laplace Approximation](https://mlsanity.com/paper/2603.22468/) (2026)

## Abstract (arXiv)

We adopt a Gamma and Malliavin Calculi point of view in order to generalize Score-based diffusion Generative Models (SGMs) to an infinite-dimensional abstract Hilbertian setting. Particularly, we define the forward noising process using Dirichlet forms associated to the Cameron-Martin space of Gaussian measures and Wiener chaoses; whereas by relying on an abstract time-reversal formula, we show that the score function is a Malliavin derivative and it corresponds to a conditional expectation. This allows us to generalize SGMs to the infinite-dimensional setting. Moreover, we extend existing finite-dimensional entropic convergence bounds to this Hilbertian setting by highlighting the role played by the Cameron-Martin norm in the Fisher information of the data distribution. Lastly, we specify our discussion for spherical random fields, considering as source of noise a Whittle-Mat'ern random spherical field.