  We propose a new class of generative diffusion models, called functional
diffusion. In contrast to previous work, functional diffusion works on samples
that are represented by functions with a continuous domain. Functional
diffusion can be seen as an extension of classical diffusion models to an
infinite-dimensional domain. Functional diffusion is very versatile as images,
videos, audio, 3D shapes, deformations, \etc, can be handled by the same
framework with minimal changes. In addition, functional diffusion is especially
suited for irregular data or data defined in non-standard domains. In our work,
we derive the necessary foundations for functional diffusion and propose a
first implementation based on the transformer architecture. We show generative
results on complicated signed distance functions and deformation functions
defined on 3D surfaces.
