  Domain-to-domain translation involves generating a target domain sample given
a condition in the source domain. Most existing methods focus on fixed input
and output domains, i.e. they only work for specific configurations (i.e. for
two domains, either $D_1\rightarrow{}D_2$ or $D_2\rightarrow{}D_1$). This paper
proposes Multi-Domain Diffusion (MDD), a conditional diffusion framework for
multi-domain translation in a semi-supervised context. Unlike previous methods,
MDD does not require defining input and output domains, allowing translation
between any partition of domains within a set (such as $(D_1,
D_2)\rightarrow{}D_3$, $D_2\rightarrow{}(D_1, D_3)$, $D_3\rightarrow{}D_1$,
etc. for 3 domains), without the need to train separate models for each domain
configuration. The key idea behind MDD is to leverage the noise formulation of
diffusion models by incorporating one noise level per domain, which allows
missing domains to be modeled with noise in a natural way. This transforms the
training task from a simple reconstruction task to a domain translation task,
where the model relies on less noisy domains to reconstruct more noisy domains.
We present results on a multi-domain (with more than two domains) synthetic
image translation dataset with challenging semantic domain inversion.
