# Score-Based Diffusion Models in Infinite Dimensions: A Malliavin Calculus Perspective

## TL;DR

This paper extends score-based diffusion models to infinite-dimensional Hilbert spaces using Malliavin calculus, deriving a formula for the score function applicable to SPDEs with spatially correlated noise.

## Contribution

It introduces an infinite-dimensional score formula for SPDE-driven diffusion models, accommodating complex noise structures and preserving geometric properties.

## Key findings

- Derived a closed-form score function for infinite-dimensional SPDEs.
- Validated the score formula numerically for linear SPDEs in 1D and 2D.
- Extended the analysis beyond finite-dimensional models using operator-theoretic methods.

## Abstract

We study score-based diffusion modelling in infinite-dimensional separable Hilbert spaces through Malliavin calculus, extending the analysis of generative models beyond the finite-dimensional setting. The forward diffusion process is formulated as a linear stochastic partial differential equation (SPDE) driven by space--time coloured noise with a trace-class covariance operator, ensuring well-posedness in arbitrary spatial dimensions. Building on Malliavin calculus and an infinite-dimensional extension of the Bismut--Elworthy--Li formula, we derive a closed-form expression for the logarithmic derivative of the transition measure along Cameron--Martin directions, which serves as the natural infinite-dimensional analogue of the score function. Our operator-theoretic approach preserves the intrinsic geometry of Hilbert spaces and accommodates general trace-class operators, thereby incorporating spatially correlated noise without assuming semigroup invertibility. We validate the derived score formula numerically for several classes of linear SPDEs in both one and two spatial dimensions using spectral methods.