  The need to reason about uncertainty in large, complex, and multi-modal
datasets has become increasingly common across modern scientific environments.
The ability to transform samples from one distribution $P$ to another
distribution $Q$ enables the solution to many problems in machine learning
(e.g. Bayesian inference, generative modeling) and has been actively pursued
from theoretical, computational, and application perspectives across the fields
of information theory, computer science, and biology. Performing such
transformations, in general, still leads to computational difficulties,
especially in high dimensions. Here, we consider the problem of computing such
"measure transport maps" with efficient and parallelizable methods. Under the
mild assumptions that $P$ need not be known but can be sampled from, and that
the density of $Q$ is known up to a proportionality constant, and that $Q$ is
log-concave, we provide in this work a convex optimization problem pertaining
to relative entropy minimization. We show how an empirical minimization
formulation and polynomial chaos map parameterization can allow for learning a
transport map between $P$ and $Q$ with distributed and scalable methods. We
also leverage findings from nonequilibrium thermodynamics to represent the
transport map as a composition of simpler maps, each of which is learned
sequentially with a transport cost regularized version of the aforementioned
problem formulation. We provide examples of our framework within the context of
Bayesian inference for the Boston housing dataset and generative modeling for
handwritten digit images from the MNIST dataset.
