CAREER: Inference with graphs: density skeleton and Markov missing graph

该项目介绍了使用图分析复杂数据集的新框架。图形的这些新应用使研究人员能够研究感兴趣的量之间的复杂关系。新开发的方法将为研究星系的生长和演化提供新的方向。PI还计划开发方法来处理国家阿尔茨海默病协调中心数据库中复杂的数据缺失问题。该项目强调了抽象的数学对象，如图形，如何为天文学和痴呆症研究等不同领域的问题提供统一的处理方法。PI还将启动几个新的教育项目，并以各种方式让研究生和本科生参与研究。PI计划研究将图形应用于统计问题的两个新的研究方向。在第一个方向上，PI开发了一种称为密度骨架的新图形方法，这是一种总结协变量分布形状的无向图。PI将研究如何将密度骨架应用于各种统计学习问题，包括回归，算法公平性和拓扑数据分析。在项目的第二部分，PI开发了一种新的基于图的方法，称为马尔可夫缺失图，以处理缺失数据问题。马尔可夫缺失图定义了一个识别假设来恢复缺失条目的分布。PI旨在研究建模、计算和效率理论如何与图形几何相互作用。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。

This project introduces novel frameworks for using graphs in analyzing complex datasets. These new applications of graphs allow researchers to investigate the intricate relation among quantities of interest. The newly developed methods will offer novel directions for studying the growth and evolution of a galaxy. The PI also plans to develop methodologies to handle complex missing data problems in the National Alzheimer's Coordinating Center's database. The project highlights how abstract mathematical objects like graphs offer a unified treatment on problems arising from different fields such as astronomy and dementia studies. The PI will also initiate several new educational programs and engage both graduate and undergraduate students in research in various ways. The PI plans to investigate two novel research directions of applying graphs to statistical problems. In the first direction, the PI develops a novel graphical approach called density skeleton, an undirected graph summarizing the shape of the covariate distribution. The PI will study how to apply density skeleton to various statistical learning problems, including regression, algorithmic fairness, and topological data analysis. In the second part of the project, the PI develops a new graph-based method called Markov missing graph to handle missing data problems. The Markov missing graph defines an identifying assumption to recover the missing entries' distribution. The PI intends to study how the modeling, computation, and efficiency theory interacts with graph geometry.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.