Peter Bubenik: Distances for Topological Data Analysis

Abstract

Topological data analysis starts with encoding data as a diagram of topological spaces. To this diagram we apply standard topological, algebraic, and combinatorial tools to produce a summary of our data. Now we would like to be able perform a quantitative analysis. I will show how a family of distances, called Wasserstein distances, arise in a natural way, and how an extension of these ideas produces a summary for which we have not only distances but also angles (in fact, a Hilbert space) allowing us to apply statistics and machine learning.