Statistical Inference for Optimal Transport

本研究项目涉及最佳运输，这是一种将一种概率分布转换为另一种概率分布的数学方法。 最佳传输被用于将数据从一个科学领域转移到另一个领域，从而使科学家能够将不同来源的数据联合收割机结合起来。它还被用来确保算法不会对人口统计群体产生意想不到的偏见。该项目的重点是为最佳运输开发严格的统计方法，这些方法允许精确评估由于我们只能访问有限数据集而导致的不确定性。这些方法将与粒子物理学的合作者一起使用，以解决粒子加速器获得的数据中出现的数据分析问题。研究生将通过将他们纳入研究来进行培训。物理系为该奖项提供共同资助。这个项目有三个重点。第一个是为运输地图开发统计推断。 目的是证明这些估计映射的中心极限定理，然后利用这些定理来构造置信区间。研究人员还将把推论的想法扩展到运输的鲁棒版本和Gromov-Wasserstein距离，它将运输的想法扩展到不同空间的测量。 然后，他们将考虑半参数理论（双鲁棒性），高阶推理和最优假设检验。 第二个重点是开发新的交通地图。 通过偏离原始定义，他们可以导出效率稍低的映射，这些映射更容易估计，并且仍然具有良好的属性。 第三个重点是将这些方法应用于物理科学。 这包括使用最优传输来估计粒子物理学中的背景分布，用于基于模拟器的推理，以量化粒子展开中的系统不确定性，以及将信号分类器与受保护变量解相关。该奖项反映了NSF的法定使命，并通过使用基金会的智力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 research project concerns optimal transport, which is a mathematical method for transforming one probability distribution into another probability distribution. Optimal transport has been used to transfer data from one scientific domain to another, thus enabling scientists to combine data from different sources. It has also been used to ensure that algorithms do not create unintended biases against demographic groups. The focus of this project is to develop rigorous statistical methods for optimal transport that permit precise assessment of uncertainty due to the fact that we only have access to finite datasets. The methods will be used with collaborators in particle physics to address data analysis problems that arise in data obtained for particle accelerators. Graduate students will be trained by including them in the research. Division of Physics provides cofunding for this award. This project has three thrusts. The first is to develop statistical inference for transport maps. The aim is to prove central limit theorems for these estimated maps and then use these theorems to construct confidence intervals. The investigators will also extend inferential ideas to robust versions of transport and to the Gromov-Wasserstein distance, which extends the idea of transport to measures on different spaces. THey will then consider semiparametric theory (double robustness), higher-order inference, and optimal hypothesis testing. The second thrust is the development of new transport maps. By departing from the original definition, they can derive slightly less efficient maps that are easier to estimate and that still have good properties. The third thrust is to apply the methods to the physical sciences. This includes using optimal transport for estimating background distributions in particle physics, for simulator-based inference, to quantify the systematic uncertainty in particle unfolding, and to decorrelate signal classifiers from protected variables. They will also develop missing data transport for use when data from a signal region is not available.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.