Statistical Inference for Optimal Transport

这个研究项目涉及最优运输，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。最优传输被用来将数据从一个科学领域转移到另一个科学领域，从而使科学家能够合并来自不同来源的数据。它还被用来确保算法不会对人口统计群体产生无意的偏见。该项目的重点是为最佳运输制定严格的统计方法，以便能够准确地评估由于我们只能获得有限的数据集这一事实而产生的不确定性。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。这个项目有三个推力。第一个是开发运输地图的统计推断。其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度区间。研究人员还将把推论概念扩展到交通的稳健版本，以及格罗莫夫-瓦瑟斯坦距离，该距离将交通的概念扩展到不同空间上的测量。然后，他们将考虑半参数理论(双重稳健性)、高阶推理和最优假设检验。第二个推动力是开发新的运输图。通过背离最初的定义，他们可以得出效率略低的地图，这些地图更容易估计，而且仍然具有良好的性质。第三个推动力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中的背景分布，用于基于模拟器的推理，量化粒子展开中的系统不确定性，以及解除信号分类器与受保护变量的关联。他们还将开发丢失的数据传输，以便在来自信号区的数据不可用时使用。这一裁决反映了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 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.