Persistent homology filtering of signals over graphs | BIMSA Topology Seminar

Abstract

Persistent homology provides essential topological insights into datasets, representing eachtopological feature with an interval whose length, known as the feature's lifetime, represents itspersistence. Features with short lifetimes are typically regarded as noise, while those with longellifetimes are considered meaningful characteristics of the dataset. We introduce a novel filteringmethod in graph signal processing, named the Low Persistence Filter. This technigue filters out lowpersistence classes in the persistence modules of the sublevel filtration of a signal over a graph,resulting in a topologically simplified version of the signal. Our method introduces a new structurecalled the Basin Hierarchy Tree. This structure encodes information about the persistence modulesof the sublevel filtration and details how different intervals in the persistence diagram arecorrelated, which is crucial for defining the Low Persistence Filter. Finally, we showcase severalapplications of the Low Persistence Filter using the open-source Python implementation wedeveloped.

This is a joint work with S. E. Graiff Zurita, and S. Kaji.