TITLE: Generative models for statistical inference

Abstract: We consider the problem of learning a continuous probability density function from data, a fundamental problem in statistics known as density estimation. It also arises in distributionally robust optimization (DRO), where the goal is to find the worst-case distribution to represent scenario departure from observations. Such a problem is known to be hard in high dimensions and incurs a significant computational challenge. In this talk, I will present a new approach to tackle these challenges, leveraging recent advances in neural-networks-based generative models, a type of machine learning model that aims to generate new samples following the underlying distributions of the observations. We develop a new flow-based generative model with guarantees, inspired by the celebrated Jordan-Kinderleherer-Otto (JKO) scheme, to represent the data distribution using a composition of a sequence of optimal transport maps applying to a source distribution (such as Gaussian). Our method can greatly reduce computational costs when achieving competitive performance over existing generative models. The connection of our JKO-flow method with proximal gradient descent in the Wasserstein-2 space enables us to prove a density learning guarantee with an exponential convergence rate. Besides density estimation, we also demonstrate that the JKO-flow generative model can be used in various applications, including adversarial learning, robust hypothesis testing, and data-driven differential privacy.