We propose a new method to estimate Wasserstein distances and optimal transport plans between two probability distributions from samples in high dimension. Unlike plug-in rules that simply replace the true distributions by their empirical counterparts, our method promotes couplings with low transport rank, a new structural assumption that is similar to the nonnegative rank of a matrix. Regularizing based on this assumption leads to drastic improvements on high-dimensional data for various tasks, including domain adaptation in single-cell RNA sequencing data. These findings are supported by a theoretical analysis that indicates that the transport rank is key in overcoming the curse of dimensionality inherent to data-driven optimal transport.
GAN Estimation of Lipschitz Optimal Transport Maps
Institut de Mathématiques de Toulouse, Université Paul Sabatier（图卢兹数学研究所，保罗·萨巴捷大学）；Institut de Recherche en Informatique de Toulouse, Université Paul Sabatier（图卢兹计算机科学研究所，保罗·萨巴捷大学）
Statistical bounds for entropic optimal transport: sample complexity and the central limit theorem
Harvard University（哈佛大学）；Massachusetts Institute of Technology（麻省理工学院）
我们证明了几个基本的统计界熵OT的平方欧几里德成本之间的亚高斯概率测度在任意维。首先，通过一个新的样本复杂度的结果，我们建立了收敛速率的熵OT的经验措施。我们的分析在Genevay et al.（2019）的边界上呈指数级改进，并将他们的工作扩展到无界测量。其次，我们基于Del Barrio和Loubes（2019）开发的技术建立了熵OT的中心极限定理。以前，这样的结果只在有限度量空间中才知道。作为我们的结果的应用，我们开发和分析了一种新的技术，用于估计被高斯噪声破坏的随机变量的熵。
Statistical Inference for Optimal Transport Maps: Recent Advances and Perspectives
Department of Statistics and Data Science（统计与数据科学系）；Machine Learning Department（机器学习系）；Carnegie Mellon University（卡内基梅隆大学）；Statistics and Data Science Center（统计与数据科学中心）；Massachusetts Institute of Technology（麻省理工学院）
Wasserstein Parallel Transport for Predicting the Dynamics of Statistical Systems
Department of Statistics and Data Science, Carnegie Mellon University（卡内基梅隆大学统计与数据科学系）；Machine Learning Department, Carnegie Mellon University（卡内基梅隆大学机器学习系）；Department of Economics, Emory University（埃默里大学经济系）
Minimax Optimal Estimation of Transport-Growth Pairs in Unbalanced Optimal Transport
Chiang Mai University（清迈大学）；RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能研究中心）；Graduate School of Arts and Science, The University of Tokyo（东京大学人文社会系研究科）；Graduate School of Science, Kyoto University（京都大学理学研究科）
