Collaborative Research: Statistical Optimal Transport in High Dimensional Mixtures

该项目研究高维混合模型，一类统计模型，可用于分析语言学，计算生物学和粒子物理学中出现的数据。该研究项目旨在定义一种新的分布之间的距离度量，该度量可以测量它们与混合模型的相似性，并提供一种比较，转换和分析高维数据集的原则性方法。作为该项目的一部分，研究人员将开发用于估计该距离的快速算法和允许该距离用于统计推断的理论保证。具体而言，该项目定义了草图Wasserstein距离（SWD），并将开发其计算和统计特性。的主要目的是建立对偶关系，这个距离，开发计算可行的估计为SWD使用原始和对偶配方，并研究新的估计的收敛速度。此外，本研究的目的是开发下界，以建立这些估计量的速率最优性，并建立分布限制，以允许建设渐近有效的置信区间。这些工具将应用于文本分析、系统生物学和高能物理学中的数据。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project studies high-dimensional mixture models, a class of statistical models that can be used to analyze data arising in linguistics, computational biology, and particle physics. This research project aims to define a new measure of distance between distributions that measures their similarity with respect to a mixture model and offers a principled way to compare, transform, and analyze high-dimensional data sets. As part of this project, the investigators will develop fast algorithms for estimating this distance and theoretical guarantees allowing this distance to be used for statistical inference.Specifically, this project defines a sketched Wasserstein distance (SWD) and will develop its computational and statistical properties. The primary aims are to establish duality relations for this distance, develop computationally feasible estimators for SWD using both primal and dual formulations, and to study the rates of convergence of the new estimators. In addition, the research aims to develop lower bounds to establish the rate optimality of these estimators and establish distributional limits to allow for the construction of asymptotically valid confidence intervals. These tools will be applied to data in text analysis, systems biology, and high-energy physics.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.