Collaborative Research: Statistical Optimal Transport: Foundation, Computation and Applications

Comparing probability models is a fundamental task in almost every data-enabled problem, and Optimal Transport (OT) offers a powerful and versatile framework to do so. Recent years have witnessed a rapid development of computational OT, which has expanded applications of OT to statistics, including clustering, generative modeling, domain adaptation, distribution-to-distribution regression, dimension reduction, and sampling. Still, understanding the fundamental strengths and limitations of OT as a statistical tool is much to be desired. This research project aims to fill this important gap by advancing statistical analysis (estimation and inference) and practical approximation of two fundamental notions (average and quantiles) in statistics and machine learning, demonstrated through modern applications for measure-valued data. The project also provides research training opportunities for graduate students. The award contains three main research projects. The first project will develop a new regularized formulation of the Wasserstein barycenter based on the multi-marginal OT and conduct an in-depth statistical analysis, encompassing sample complexity, limiting distributions, and bootstrap consistency. The second project will establish asymptotic distribution and bootstrap consistency results for linear functionals of OT maps and will study sharp asymptotics for entropically regularized OT maps when regularization parameters tend to zero. Building on the first two projects, the third project explores applications of the OT methodology to two important statistical tasks: dimension reduction and vector quantile regression. The research agenda will develop a novel and computationally efficient principal component method for measure-valued data and a statistically valid duality-based estimator for quantile regression with multivariate responses. The three projects will produce novel technical tools integrated from OT theory, empirical process theory, and partial differential equations, which are essential for OT-based inferential methods and will inspire new applications of OT to measure-valued and multivariate data. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.