  This paper introduces $\infty$-Diff, a generative diffusion model defined in
an infinite-dimensional Hilbert space, which can model infinite resolution
data. By training on randomly sampled subsets of coordinates and denoising
content only at those locations, we learn a continuous function for arbitrary
resolution sampling. Unlike prior neural field-based infinite-dimensional
models, which use point-wise functions requiring latent compression, our method
employs non-local integral operators to map between Hilbert spaces, allowing
spatial context aggregation. This is achieved with an efficient multi-scale
function-space architecture that operates directly on raw sparse coordinates,
coupled with a mollified diffusion process that smooths out irregularities.
Through experiments on high-resolution datasets, we found that even at an
$8\times$ subsampling rate, our model retains high-quality diffusion. This
leads to significant run-time and memory savings, delivers samples with lower
FID scores, and scales beyond the training resolution while retaining detail.
