Title: Manifold Learning on Fibre Bundles

Abstract:

Spectral geometry has played an important role in modern geometric data analysis, where the technique is widely known as Laplacian eigenmaps or diffusion maps. In this talk, we present a geometric framework that studies graph representations of complex datasets, where each edge of the graph is equipped with a non-scalar transformation or correspondence. This new framework models such a dataset as a fibre bundle with a connection, and interprets the collection of pairwise functional relations as defining a horizontal diffusion process on the bundle driven by its projection on the base. The eigenstates of this horizontal diffusion process encode the “consistency” among objects in the dataset, and provide a lens through which the geometry of the dataset can be revealed. We demonstrate an application of this geometric framework on evolutionary anthropology.

Tingran Gao is a William H. Kruskal Instructor in the Department of Statistics and Committee on Computational and Applied Mathematics at the University of Chicago. He obtained his PhD in Mathematics at Duke University in 2015 and was a Visiting Assistant Professor there from 2015 to 2017. Prior to that, he received BS in Mathematics at Tsinghua University in Beijing, China. His research centers around the geometry and topology of massive, high-dimensional datasets, with interests ranging from signal and image processing to geometry modeling, optimization, and statistical learning.