Speaker: Tingran Gao (U. Chicago)
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.