  Given two probability densities on related data spaces, we seek a map pushing
one density to the other while satisfying application-dependent constraints.
For maps to have utility in a broad application space (including domain
translation, domain adaptation, and generative modeling), the map must be
available to apply on out-of-sample data points and should correspond to a
probabilistic model over the two spaces. Unfortunately, existing approaches,
which are primarily based on optimal transport, do not address these needs. In
this paper, we introduce a novel pushforward map learning algorithm that
utilizes normalizing flows to parameterize the map. We first re-formulate the
classical optimal transport problem to be map-focused and propose a learning
algorithm to select from all possible maps under the constraint that the map
minimizes a probability distance and application-specific regularizers; thus,
our method can be seen as solving a modified optimal transport problem. Once
the map is learned, it can be used to map samples from a source domain to a
target domain. In addition, because the map is parameterized as a composition
of normalizing flows, it models the empirical distributions over the two data
spaces and allows both sampling and likelihood evaluation for both data sets.
We compare our method (parOT) to related optimal transport approaches in the
context of domain adaptation and domain translation on benchmark data sets.
Finally, to illustrate the impact of our work on applied problems, we apply
parOT to a real scientific application: spectral calibration for
high-dimensional measurements from two vastly different environments
