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

现代统计应用经常涉及多元数据，这些数据违背了独立抽样和易于处理的参数形式的方便假设。例如，参数化方法在分析基因组学、流行病学和生物信息学中产生的复杂高维数据时往往不足。这就需要开发与数据分布无关、计算效率高、对大型非参数类具有强大统计功能的过程。同样，经典的独立性假设在来自社交网络的组合数据集中经常被违反，这使得开发现实的和数学上易于处理的方法来模拟高维分布中的结构和依赖性变得越来越重要。该项目利用了最优传输理论、随机几何图和统计物理的最新发展思想，以更深入地理解(1)多元无分布推断和(2)依赖网络数据。该项目的教育和推广部分旨在通过课程开发、定向阅读小组和暑期项目，促进本科生的研究，并为研究生的指导做好准备。该项目的第一个组成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。该项目还将探索基于最佳匹配的检验的渐近性质及其在观测研究中检测平衡的应用。该项目的第二个组成部分将侧重于复杂关系数据中的依赖建模，使用伊辛模型和更普遍的高阶（张量）马尔可夫随机场。这里的目标是建立一个框架，用于同时建模网络依赖性（由邻里相互作用产生）和单个节点效应，并利用随机张量和统计物理工具的最新进展，在这些模型中开发参数估计的整体理论。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。

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.