  Transportation of measure provides a versatile approach for modeling complex
probability distributions, with applications in density estimation, Bayesian
inference, generative modeling, and beyond. Monotone triangular transport
maps$\unicode{x2014}$approximations of the Knothe$\unicode{x2013}$Rosenblatt
(KR) rearrangement$\unicode{x2014}$are a canonical choice for these tasks. Yet
the representation and parameterization of such maps have a significant impact
on their generality and expressiveness, and on properties of the optimization
problem that arises in learning a map from data (e.g., via maximum likelihood
estimation). We present a general framework for representing monotone
triangular maps via invertible transformations of smooth functions. We
establish conditions on the transformation such that the associated
infinite-dimensional minimization problem has no spurious local minima, i.e.,
all local minima are global minima; and we show for target distributions
satisfying certain tail conditions that the unique global minimizer corresponds
to the KR map. Given a sample from the target, we then propose an adaptive
algorithm that estimates a sparse semi-parametric approximation of the
underlying KR map. We demonstrate how this framework can be applied to joint
and conditional density estimation, likelihood-free inference, and structure
learning of directed graphical models, with stable generalization performance
across a range of sample sizes.
