  Given datasets from multiple domains, a key challenge is to efficiently
exploit these data sources for modeling a target domain. Variants of this
problem have been studied in many contexts, such as cross-domain translation
and domain adaptation. We propose AlignFlow, a generative modeling framework
that models each domain via a normalizing flow. The use of normalizing flows
allows for a) flexibility in specifying learning objectives via adversarial
training, maximum likelihood estimation, or a hybrid of the two methods; and b)
learning and exact inference of a shared representation in the latent space of
the generative model. We derive a uniform set of conditions under which
AlignFlow is marginally-consistent for the different learning objectives.
Furthermore, we show that AlignFlow guarantees exact cycle consistency in
mapping datapoints from a source domain to target and back to the source
domain. Empirically, AlignFlow outperforms relevant baselines on image-to-image
translation and unsupervised domain adaptation and can be used to
simultaneously interpolate across the various domains using the learned
representation.
