  Unsupervised domain translation (UDT) aims to find functions that convert
samples from one domain (e.g., sketches) to another domain (e.g., photos)
without changing the high-level semantic meaning (also referred to as
``content''). The translation functions are often sought by probability
distribution matching of the transformed source domain and target domain.
CycleGAN stands as arguably the most representative approach among this line of
work. However, it was noticed in the literature that CycleGAN and variants
could fail to identify the desired translation functions and produce
content-misaligned translations. This limitation arises due to the presence of
multiple translation functions -- referred to as ``measure-preserving
automorphism" (MPA) -- in the solution space of the learning criteria. Despite
awareness of such identifiability issues, solutions have remained elusive. This
study delves into the core identifiability inquiry and introduces an MPA
elimination theory. Our analysis shows that MPA is unlikely to exist, if
multiple pairs of diverse cross-domain conditional distributions are matched by
the learning function. Our theory leads to a UDT learner using distribution
matching over auxiliary variable-induced subsets of the domains -- other than
over the entire data domains as in the classical approaches. The proposed
framework is the first to rigorously establish translation identifiability
under reasonable UDT settings, to our best knowledge. Experiments corroborate
with our theoretical claims.
