Greedy inference with structure-exploiting lazy maps
This provides a principled method for exploiting low-dimensional structure in inference problems, which is incremental as it builds on existing transport map techniques.
The paper tackles high-dimensional Bayesian inference by introducing structure-exploiting low-dimensional transport maps that focus on directions of significant discrepancy from the posterior, proving weak convergence and demonstrating benefits on challenging problems in machine learning and differential equations.
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.