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

TL;DR

This paper extends score-based diffusion generative models to infinite-dimensional Hilbert spaces using Gamma and Malliavin calculus, enabling modeling of complex random fields like spherical data.

Contribution

It introduces a novel infinite-dimensional framework for SGMs using Dirichlet forms and Malliavin derivatives, generalizing finite-dimensional results to complex random fields.

Findings

Generalized SGMs to infinite-dimensional Hilbert spaces.

Extended entropic convergence bounds to the Hilbertian setting.

Applied framework to spherical random fields with Whittle-Matérn noise.

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.