Inference for Geometric and Topological Data Analysis

Geometric and topological data analysis comprises a large collection of useful tools and methods in the modern landscape of data science with numerous applications in a broad variety of fields. Relevant concepts have evolved into versatile tools that are reshaping the manner in which we handle, scrutinize, and make sense of intricate data structures. This project aims at tightening the connections between geometric and topological data analysis on the one hand, and statistics and machine learning on the other. This goes along with the training of both graduate and undergraduate students, which is another integral part of the project. More specifically, the project will focus on theory and methodology based on core concepts in the field: the Euler characteristic, the graph Laplacians, and the heat kernel. Among others, this includes the development of large sample distribution theory for an appropriately weighted Euler characteristic process, and a novel two-sample testing procedure for isometry of manifolds based on the heat kernel signature. Crucially, the contributions of the project will result in novel insights catalyzing further developments, which will also benefit practitioners. The theoretical tools used in this project are a combination of tools from geometric probability theory, the analysis on Riemannian manifolds, and differential geometry. While these advanced topics lend themselves naturally to the training of graduate students in statistics, the overarching geometric nature of the project also allows for an exciting and meaningful involvement of advanced undergraduates. 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.