Statistical Inference for Optimal Transport

本课题研究的是最优运输，它是一种将一种概率分布转化为另一种概率分布的数学方法。最优传输用于将数据从一个科学领域传输到另一个科学领域，从而使科学家能够组合来自不同来源的数据。它还被用来确保算法不会对人口群体产生意想不到的偏见。该项目的重点是开发严格的统计方法，以实现最佳运输，从而精确评估由于我们只能访问有限的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。研究生将接受培训，包括他们的研究。物理系为该奖项提供联合资助。这个项目有三个重点。首先是发展交通地图的统计推断。目的是证明这些估计映射的中心极限定理，然后利用这些定理构造置信区间。研究人员还将把推理思想扩展到传输的鲁棒版本和Gromov-Wasserstein距离，后者将传输的思想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。通过偏离最初的定义，他们可以推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性质。第三个重点是将这些方法应用于物理科学。这包括在粒子物理中使用最优传输来估计背景分布，用于基于模拟器的推理，量化粒子展开中的系统不确定性，以及从受保护变量中去相关信号分类器。他们还将开发丢失数据传输，以便在无法获得来自信号区域的数据时使用。该奖项反映了美国国家科学基金会的法定使命，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 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.