New Challenges in Statistical Inference with Regularized Optimal Transport

在数据丰富和计算进步的推动下，统计推断的应用领域不断增长。面向人类的技术，如自动驾驶汽车或机器人辅助手术，需要有严格性能保证的原则性推理方法。由于许多推理任务减少到比较概率分布，最优传输理论-它提供了一个强大的框架，这样做-已经成为一个工具的选择，设计和分析推理方法。然而，统计最优传输受到维数灾难的影响，定量结果要么随维数呈指数级恶化（例如，估计率），要么在很大程度上不可用（例如，极限分布，再分配等）。为了克服这一僵局，该项目将探索最佳运输距离的现代正则化技术，并开发一个全面的统计理论，以促进高维的原则性推理。这一创新预计将对统计推断在工业、商业、科学和社会中的广泛应用领域产生强烈影响，通过促进理论保证支持的大规模原则性实施。同时，教育部分将为学生提供严格的培训和多样化的招聘机会，沿着一个深思熟虑的计划，以促进统计和工程社区之间在最佳运输和相关领域的合作。这个项目将探索三个突出的最佳运输正则化方法：（1）通过选择核的卷积平滑;（2）通过低维投影切片;（3）通过低维投影切片;（4）通过低维投影切片。（3）通过熵罚的凸化。这些技术保留了经典最优运输的良性结构，但降低了其复杂性，这为可扩展的统计理论打开了大门。研究议程将解决有关统计推断与正则化最佳运输距离的关键理论挑战，包括经验错误率，极限分布，半参数效率，恢复方法，Berry-Esseen类型界限和计算统计差距。所开发的理论将被用来解决各种推理应用，包括生成建模，测试，向量分位数回归和内在维度估计。该项目将在最优运输理论的基础上，为大规模的推理方法提供理论基础，为实际实施提供指导和见解。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。

Driven by the abundance of data and computational advances, the application domain of statistical inference is ever-growing. Human-facing technologies, such as autonomous vehicles or robotic-assisted surgery, demand principled inference methods subject to rigorous performance guarantees. As many inference tasks reduce to comparing probability distributions, optimal transport theory — which provides a powerful framework for doing so — has emerged as a tool of choice for designing and analyzing inference methods. However, statistical optimal transport is bottlenecked by the curse of dimensionality, whereby quantitative results either deteriorate exponentially with dimension (for example, estimation rates) or are largely unavailable (for example, limit distributions, resampling, and more). To overcome this impasse, this project will explore modern regularization techniques for optimal transport distances and develop a comprehensive statistical theory to facilitate principled inference in high dimensions. This innovation is expected to have a strong impact on the broad application domain of statistical inference in industry, commerce, science, and society, by promoting principled implementations at scale backed by theoretical assurances. In conjunction, the educational component will provide rigorous training and diverse recruitment opportunities for students, along with a deliberate plan to promote collaborations between statistics and engineering communities working on optimal transport and related fields.This project will explore three prominent optimal transport regularization methods: (1) smoothing via convolution with a chosen kernel; (2) slicing via lower-dimensional projections; and (3) convexification via an entropic penalty. These techniques preserve the virtuous structure of classic optimal transport but reduce its complexity, which opens the door to a scalable statistical theory. The research agenda will tackle key theoretical challenges concerning statistical inference with regularized optimal transport distances, encompassing empirical error rates, limit distributions, semiparametric efficiency, resampling methods, Berry-Esseen type bounds, and computational-statistical gaps. The developed theory will be leveraged to address various inference applications, including generative modeling, testing, vector quantile regression, and intrinsic dimension estimation. The project will result in the theoretical underpinnings of inference methods at scale based on optimal transport theory, providing guidance and insight for practical implementations.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.