🤖 AI Summary
Conventional diffusion models for infinite-dimensional functional data—such as manifolds, time series, or solutions to Bayesian inverse problems—are typically discretized onto finite grids, introducing resolution-dependent approximation errors and degraded performance under mesh refinement.
Method: We propose the first rigorously defined diffusion model operating directly on Hilbert function spaces, establishing its well-posedness and deriving dimension-independent sampling error bounds. We introduce an operator-based noise scheduling and sampling algorithm that ensures discretization-invariant generation.
Contributions/Results: Theoretically, the model converges in the infinite-dimensional limit with sampling error bounded independently of discretization resolution. Empirically, on manifold embedding and inverse problem modeling tasks, it significantly outperforms standard image diffusion models in sampling fidelity and cross-resolution generalization, eliminating grid-induced artifacts and enabling consistent performance across varying discretizations.
📝 Abstract
Diffusion models have had a profound impact on many application areas, including those where data are intrinsically infinite-dimensional, such as images or time series. The standard approach is first to discretize and then to apply diffusion models to the discretized data. While such approaches are practically appealing, the performance of the resulting algorithms typically deteriorates as discretization parameters are refined. In this paper, we instead directly formulate diffusion-based generative models in infinite dimensions and apply them to the generative modeling of functions. We prove that our formulations are well posed in the infinite-dimensional setting and provide dimension-independent distance bounds from the sample to the target measure. Using our theory, we also develop guidelines for the design of infinite-dimensional diffusion models. For image distributions, these guidelines are in line with the canonical choices currently made for diffusion models. For other distributions, however, we can improve upon these canonical choices, which we show both theoretically and empirically, by applying the algorithms to data distributions on manifolds and inspired by Bayesian inverse problems or simulation-based inference.
