Inés García-Redondo: Stability for Inference with Persistent Homology Rank Functions

Speaker
Inés García-Redondo,  London School of Geometry and Number Theory

Abstract
Persistent homology (PH) barcodes and diagrams are a cornerstone of topological data analysis. Widely used in many real data settings, they relate variation in topological information with variation in data, however, they are challenging to use in statistical settings due to their complex geometric structure. In this talk, we will revisit persistent rank functions and rank invariants as alternative representations of the PH output, easily integrable in Machine Learning and inferential tasks. Due to their functional nature, these invariants are amenable to Functional Data Analysis (FDA), a well-stablished branch of statistics dealing with data coming in the form of functions. I will present stability results for rank functions and rank invariants under a metric suitable for FDA applications, to then showcase the effectiveness of such approach through applications of rank functions and biparameter rank invariants to real and simulated data. This is joint work with Qiquan Wang, Pierre Faugère, Anthea Monod and Gregory Henselman-Petrusek.