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

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.