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

2025-05-19

AI Summary

This work addresses the theoretical challenge of extending score-based generative models (SGMs) to infinite-dimensional Hilbert spaces—particularly spherical random fields. Methodologically, it introduces the first infinite-dimensional notion of “score” by combining the Malliavin derivative with the Gamma operator, thereby reformulating both forward noising and reverse denoising via Malliavin–Gamma calculus; employs the Cameron–Martin norm to characterize Fisher information and derives a novel upper bound on entropy convergence in Hilbert space; and designs a Whittle–Matérn-type noise process tailored to spherical domains. Key contributions include: (i) a rigorous theoretical framework for infinite-dimensional SGMs; (ii) a proof that the generalized score coincides with the composition of the Malliavin derivative and conditional expectation; (iii) extension of finite-dimensional entropy convergence analysis to abstract Hilbert spaces; and (iv) instantiation and empirical validation on spherical random fields, establishing a new paradigm for generative modeling on high-dimensional non-Euclidean stochastic domains.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Stochastic Optimization
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

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.

Problem

• Generalize Score-based diffusion models to infinite-dimensional Hilbert spaces
• Define forward noising process using Dirichlet forms and Wiener chaoses
• Extend entropic convergence bounds to Hilbertian setting with Cameron-Martin norm

Innovation

• Generalize SGMs using Gamma and Malliavin Calculi
• Define forward noising with Dirichlet forms
• Extend entropic bounds to Hilbertian setting