  We consider the problem of unsupervised domain adaptation (UDA) between a
source and a target domain under conditional and label shift a.k.a Generalized
Target Shift (GeTarS). Unlike simpler UDA settings, few works have addressed
this challenging problem. Recent approaches learn domain-invariant
representations, yet they have practical limitations and rely on strong
assumptions that may not hold in practice. In this paper, we explore a novel
and general approach to align pretrained representations, which circumvents
existing drawbacks. Instead of constraining representation invariance, it
learns an optimal transport map, implemented as a NN, which maps source
representations onto target ones. Our approach is flexible and scalable, it
preserves the problem's structure and it has strong theoretical guarantees
under mild assumptions. In particular, our solution is unique, matches
conditional distributions across domains, recovers target proportions and
explicitly controls the target generalization risk. Through an exhaustive
comparison on several datasets, we challenge the state-of-the-art in GeTarS.
