Efficiency of Diffusion Models for Infinite-dimensional Data Generation: Dimension Independent Approximation and Estimation Errors

Despite the empirical success of diffusion models in generation of extremely high dimensional data such as images and videos, existing theoretical guarantees are largely confined to finite-dimensional spaces and suffer from the curse of dimensionality. This work provides the first rigorous statistical analysis of diffusion models in infinite-dimensional Hilbert spaces. To handle the absence of a Lebesgue measure in ${\mathbb{R}}^{\infty }$, we reformulate the score function using Radon-Nikodym derivatives relative to a Gaussian base measure, thereby eliminating the need for restrictive density lower-bound assumptions. By adopting the *$\gamma$-smooth function space* with different degrees of smoothness depending on directions and a novel Gaussian CDF transform, we derive polynomial-order, "dimension-independent” estimation error bounds for neural network score estimators. Our results establish a formal theoretical foundation for the scalability of diffusion models to infinite-dimensional data.