On the representation and learning of triangular transport maps

Transport maps characterize probability distributions by coupling random variables using a deterministic transformation. While bijective transport maps enable sampling and density evaluations of the transformed random variable, learning the parameters of such transformations in high dimensions is challenging given few samples from an unknown target distribution. Moreover, structural choices for these transformations can have a significant impact on their performance and the optimization procedure for finding these maps. In this talk, we will present a framework for representing and learning monotone triangular maps via invertible transformations of smooth functions, and will first demonstrate that the associated optimization problem has no spurious local minima, i.e., all local minima are global minima. Second, we propose a sample-efficient adaptive algorithm that construct a sparse approximation for the map. We demonstrate how this framework can be applied for 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.