将分数模型拓展到无限维空间，用随机分析工具实现更通用的生成建模。
A Malliavin-Gamma calculus approach to Score Based Diffusion Generative models for random fields
- 基于马利阿文与伽马微积分，构建无限维分数扩散模型框架。
- 证明得分函数为马利阿文导数，且等价于条件期望，确保理论完备性。
- 适用于球面随机场生成，支持如惠特尔-马特龙噪声的物理可解释建模。
本文从伽马微积分与马利阿文微积分的角度出发，将分数驱动扩散生成模型（SGMs）推广至抽象希尔伯特空间中的无限维情形。通过利用高斯测度的Cameron-Martin空间与维纳混沌相关的狄利克雷型，定义前向加噪过程；借助抽象时间反演公式，证明得分函数为马利阿文导数，并对应一个条件期望。该理论框架使SGMs可被自然延拓至无限维设定。此外，我们还将现有的有限维熵收敛界扩展至希尔伯特空间情形，揭示了数据分布的费舍尔信息中Cameron-Martin范数的关键作用。最后，针对球面随机场的具体场景，采用惠特尔-马特龙球面随机场作为噪声源，具体说明了方法的应用。
原文摘要 · Abstract (English)
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.
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