CAREER: Geometric and Combinatorial Methods for Distribution-Free Inference and Dependent Network Data

现代统计应用经常涉及违反独立抽样和易处理的参数形式的方便假设的多变量数据。例如，参数化方法通常不足以分析基因组学、流行病学和生物信息学中的复杂高维数据。这就需要开发的程序是不可知的数据的分布，计算效率高，但统计功能强大的大型非参数类。类似地，在由社交网络产生的组合数据集中，经典的独立性假设经常被违反，这使得开发用于对高维分布中的结构和依赖进行建模的现实的和数学上易于处理的方法变得越来越重要。该项目利用最优传输理论，随机几何图和统计物理学的最新发展的思想，以更深入地了解（1）多变量分布自由推理和（2）相关网络数据。该项目的教育和推广部分旨在通过课程开发、指导阅读小组和暑期项目，促进本科生研究，并为研究生提供指导。 该项目的第一部分将研究基于新兴的多元秩理论的非参数、无分布双样本检验的效率特性，其中包括著名的能量距离检验的秩模拟。该项目还将探讨基于最佳匹配的检验的渐近性质及其在观察性研究中检测平衡的应用。该项目的第二部分将侧重于使用Ising模型和更一般的高阶（张量）马尔可夫随机场对复杂关系数据中的依赖性进行建模。这里的目标是建立一个框架，同时建模的网络依赖（由邻域相互作用引起）和单个节点效应，并利用随机张量和统计物理学工具的最新进展，在这些模型中建立一个参数估计的整体理论。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准。

Modern statistical applications often involve multivariate data that violate the convenient assumptions of independent sampling and tractable parametric forms. For instance, parametric methods are often inadequate in the analysis of complex high-dimensional data arising from genomics, epidemiology, and bioinformatics. This necessitates the development of procedures that are agnostic to the distribution of the data, computationally efficient, and yet statistically powerful for large nonparametric classes. Similarly, the classical assumption of independence is routinely violated in combinatorial datasets arising from social networks, making it increasingly important to develop realistic and mathematically tractable methods for modeling structure and dependence in high-dimensional distributions. This project leverages ideas from recent developments in optimal transport theory, random geometric graphs, and statistical physics to gain a deeper understanding of (1) multivariate distribution-free inference and (2) dependent network data. The educational and outreach component of this project will aim to foster undergraduate research and prepare graduate students in mentoring, through curriculum development, directed reading groups, and summer programs. The first component of this project will study the efficiency properties of nonparametric, distribution-free two-sample tests based on the emerging theory of multivariate ranks, which include, among others, the rank analogue of the celebrated energy distance test. The project will also explore the asymptotic properties of tests based on optimal matchings and their applications to detecting balance in observational studies. The second component of this project will focus on modeling dependence in complex relational data, using the Ising model and, more generally, higher-order (tensor) Markov random fields. The goal here is to build a framework for simultaneously modeling the network dependency (arising from neighborhood interactions) and the individual node effects, and to develop a holistic theory of parameter estimation in these models using recent advances on random tensors and tools from statistical 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.