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.

Additional Feedback: ### POST AUTHOR FEEDBACK ### I am raising my score as the authors have done a good job of addressing my feedback and the other reviews were favourable. I like the idea of intelligently reducing a higher-dimensional problem to a series of lower-dimensional problems, the adaptive error bounds on the approximation, and the map-learning procedure which involves more than just defining a loss function and blindly optimizing. However, I also have some comments / questions which, if addressed, would very much solidify this paper's contribution to the field of machine learning in my opinion. As mentioned a few times already, I would like some clarity on Proposition 3. Specifically: (### POST FEEDBACK NOTE - I misunderstood on first read, thank you for clarifying in your response.) I guess this could be considered a good thing for weak convergence, but then why even include this condition?

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.