Studying the Space of Persistence Diagrams Using Optimal Partial Transport - Part I

Explore the intersection of topological data analysis and optimal transport in this comprehensive lecture on persistence diagrams. Delve into the theoretical background of studying persistence diagrams as measure spaces and their representation using optimal partial transport problems. Discover how this approach leads to a generalization of persistence diagrams as Radon measures supported on the upper half plane. Examine the topological properties of this new space and its implications for the closed subspace of persistence diagrams. Learn about the characterization of convergence in persistence diagram spaces and its applications in machine learning pipelines. Gain insights into the manipulation of limit objects like expected persistence diagrams and their role in proving convergence rates and stability results in random settings. Based on the work "Understanding the topology and the geometry of the persistence diagram space via optimal partial transport," this lecture provides a deep dive into advanced concepts in applied algebraic topology.