Diffusion Generative Models in Infinite Dimensions

TL;DR

This work extends diffusion models to infinite-dimensional function spaces by grounding the forward and reverse processes in Gaussian measures on Hilbert spaces. It develops a variational, function-space ELBO-based training objective and provides practical KL approximations for L2 and Sobolev function spaces, enabling unconditional and conditional generation of function-valued data. The approach is instantiated with neural-operator parametric mappings and validated on synthetic MoGP and real-world AEMET datasets, demonstrating controllable conditioning, smoother generations in Sobolev spaces, and meaningful functional statistics alignment. The framework unifies diffusion in function space with Gaussian-measure theory, offering a principled path toward continuous-time limits and SPDE-inspired diffusion dynamics for functional data.

Abstract

Diffusion generative models have recently been applied to domains where the available data can be seen as a discretization of an underlying function, such as audio signals or time series. However, these models operate directly on the discretized data, and there are no semantics in the modeling process that relate the observed data to the underlying functional forms. We generalize diffusion models to operate directly in function space by developing the foundational theory for such models in terms of Gaussian measures on Hilbert spaces. A significant benefit of our function space point of view is that it allows us to explicitly specify the space of functions we are working in, leading us to develop methods for diffusion generative modeling in Sobolev spaces. Our approach allows us to perform both unconditional and conditional generation of function-valued data. We demonstrate our methods on several synthetic and real-world benchmarks.

Figures (6)

• Figure 1: Unconditional function generation on a synthetic (MoGP) and real-world (AEMET) dataset. For each dataset, a GNO model was trained on the plotted functions (first column), and a total of $500$ functions were sampled from the model (second column). The generated curves closely match the training curves in both perceptual quality and pointwise statistics.
• Figure 2: Conditional samples of our model (FuncDiff) are compared against Gaussian process regression (GPR). In each plot, both models are conditioned on the black curves.
• Figure 3: An illustration of our soft conditioning method. We condition the generative process on the black curves for all but the final $150$ diffusion steps. This allows us to generate functions that are qualitatively similar to the given conditioning information (in black), such that the generated function values do not necessarily exactly match those of the conditioning information.
• Figure 4: Unconditional function generation on a synthetic (Linear) and several real-world (Growth, Canadian, Octane) datasets. For each dataset, a GNO model was trained on the plotted functions (first column), and a total of $500$ functions were sampled from the model (second column).
• Figure 5: Unconditional samples from an FPCA-based model on various datasets. For each dataset, we estimate the first $M=5$ functional principal components and fit a Gaussian distribution to the resulting scores. Generation is performed by sampling from said Gaussian and taking the resulting linear combination of functional principal components. Although the functional statistics closely match those of the training data, the perceptual quality of the generated curves is worse than our FuncDiff model.

Theorems & Definitions (22)

• Definition 1
• Theorem 1: The Feldman-Hájek Theorem
• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4
• proof