Collaborative Research: Statistical Optimal Transport in High Dimensional Mixtures

这个项目研究高维混合模型，这是一类统计模型，可用于分析语言学、计算生物学和粒子物理学中产生的数据。这项研究项目旨在定义一种新的分布之间距离的度量，该度量度量它们相对于混合模型的相似性，并提供一种原则性的方法来比较、转换和分析高维数据集。作为这个项目的一部分，研究人员将开发快速算法来估计这个距离，并从理论上保证这个距离可以用于统计推断。具体地说，这个项目定义了一个草图沃瑟斯坦距离(SWD)，并将开发它的计算和统计特性。主要目的是建立此距离的对偶关系，使用原始公式和对偶公式开发SWD在计算上可行的估计量，并研究新估计量的收敛速度。此外，研究的目的是开发下界以建立这些估计量的速率最优性，并建立分布极限以允许构造渐近有效的置信度区间。这些工具将应用于文本分析、系统生物学和高能物理中的数据。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.