Hierarchical Functional Diffusion Models with Global and Local Statistics

Function generation is a promising and challenging problem in generative modeling. Many scientific and spatial data are more naturally viewed as random functions or fields, where the goal is to generate an entire function that can be evaluated at any locations and resolutions. Functional diffusion models provide a natural framework for this setting, but a central difficulty is that the Bayes optimal denoiser is conditioned on an infinite dimensional noisy function. Existing approaches typically rely either on finite dimensional latent representations, which may discard important predictive information, or on a large number of sampled function values, which can be computationally expensive. We propose a hierarchical functional diffusion model that addresses this tradeoff by separating global and local predictive structure. A moderate dimensional diffusion model first generates a global state that captures long-range dependence, after which a conditional functional diffusion model generates the residual using compressed local information around each query. We develop an information based error decomposition that separates discretization, localization, compression, and learning errors, and derive Wasserstein bounds for the resulting functional distribution. Our algorithm is particularly well suited to hierarchical Gaussian process mixture models, which allow globally non-Gaussian and multimodal distributions while requiring Gaussian structure only conditionally at the residual scale. The proposed framework provides a scalable, coarse-to-fine, and resolution free approach to generative modeling on function spaces.