  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.
