A Malliavin-Gamma calculus approach to Score Based Diffusion Generative models for random fields
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918153469034496 |
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| author | Greco, Giacomo |
| author_facet | Greco, Giacomo |
| contents | 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-Matérn random spherical field. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_13189 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Malliavin-Gamma calculus approach to Score Based Diffusion Generative models for random fields Greco, Giacomo Probability Machine Learning 62D05, 60G60, 60J46, 60H07, 60H30 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-Matérn random spherical field. |
| title | A Malliavin-Gamma calculus approach to Score Based Diffusion Generative models for random fields |
| topic | Probability Machine Learning 62D05, 60G60, 60J46, 60H07, 60H30 |
| url | https://arxiv.org/abs/2505.13189 |