  We propose a framework for solving high-dimensional Bayesian inference
problems using \emph{structure-exploiting} low-dimensional transport maps or
flows. These maps are confined to a low-dimensional subspace (hence, lazy), and
the subspace is identified by minimizing an upper bound on the
Kullback--Leibler divergence (hence, structured). Our framework provides a
principled way of identifying and exploiting low-dimensional structure in an
inference problem. It focuses the expressiveness of a transport map along the
directions of most significant discrepancy from the posterior, and can be used
to build deep compositions of lazy maps, where low-dimensional projections of
the parameters are iteratively transformed to match the posterior. We prove
weak convergence of the generated sequence of distributions to the posterior,
and we demonstrate the benefits of the framework on challenging inference
problems in machine learning and differential equations, using inverse
autoregressive flows and polynomial maps as examples of the underlying density
estimators.
