OODA: Statistics with geometric sample spaces

University Of North Carolina At Chapel Hill, Chapel Hill NC

Investigators

Abstract

This project develops mathematical foundations and statistical methods for data when each data object or the set of all possible data points has complex geometric structure. Data of this sort arises in an increasing multitude of societally important contexts. In forensic science, for example, fingerprints are not described by single numbers or even lists of numbers. In geoscience, patterns in thin slices of rock or ice serve as signatures of chemical composition or history. In medicine, imaging commonly produces high-resolution 3D scans that capture shapes, or even 4D scans (videos) that capture shapes evolving in time. The shapes themselves can be complicated geometric objects: branching trees of blood vessels, airways, or nerve cell dendrites; folded or kinked surfaces of brains or teeth; segmented or irregular blobs that vary subtly from patient to patient. Often, the geometry is not smooth: even from very close up, a branch point or a kink, for instance, does not look like a flat line, plane, or Euclidean space of higher dimension. These sorts of resolutely non-Euclidean phenomena pose fundamental challenges for data analysis, whose goal is to identify trends, search for anomalies, or classify. Even making these tasks precise in non-Euclidean geometric settings requires the mathematical foundations and statistical techniques targeted by this project. These scientific pursuits serve as a platform to mentor a group of interdisciplinary junior researchers, from high school to postdoctoral, in a vertically integrated scientific research lab environment, and enhance their professional development through direct research funding and travel support. The supported research will develop statistical methods to handle data sampled from non-Euclidean geometric spaces such as manifolds, algebraic varieties, simplices, and more general singular spaces, drawing on methods from probability, notably surrounding geometric central limit theorems on stratified spaces, as well as from algebraic, differential, and convex geometry. The project aims to produce specific contributions to (i) fundamental geometric statistics in non-smooth settings, including confidence regions and hypothesis testing for singularities as well as (ii) exploratory data analysis by deformation and slicing to find modes of variation in geometric sample spaces such as polyspheres, probability simplices (for compositional data), and more general semialgebraic varieties. This research meets the critical need for new methods in the emerging and important area of statistics and data science of complex data. 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.