Xiaohui Chen, USC

Title: Recent progress in statistical and computational optimal transport barycenters

Abstract: The Wasserstein barycenter plays a fundamental role in averaging measure-valued data under the framework of optimal transport (OT). However, there are tremendous challenges in computing and estimating the Wasserstein barycenter for high-dimensional distributions. In this talk, we will discuss some recent progress in advancing the statistical and computational frontiers of optimal transport barycenters. We first introduce a multimarginal Schrödinger barycenter (MSB) based on the entropy regularized multimarginal optimal transport problem that admits general-purpose fast algorithms for computation. By recognizing a proper dual geometry, we derive sharp non-asymptotic rates of convergence for estimating several key MSB quantities (cost functional, Schrödinger coupling and barycenter) from point clouds randomly sampled from the input marginal distributions. We will also consider the computation exact (i.e., unregularized) Wasserstein barycenter, which can be recast into a nonconvex-concave minimax optimization. By alternating between the primal Wasserstein and dual potential Sobolev optimization geometries, we introduce a linear-time and linear-space Wasserstein-Descent H-Ascent (WDHA) algorithm and prove its algorithmic convergence to a stationary point.