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

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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.