Inés García-Redondo (11/29/23): Computable Stability for Persistence Rank Function Machine Learning
Title: Computable Stability for Persistence Rank Function Machine Learning
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 (as measured by cellular homology) 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, a well-established branch of statistics dealing with data precisely coming in the form of functions. I will present and showcase the effectiveness of such approach through three applications of rank functions and biparameter rank invariants to real and simulated data. After that, stability results for rank functions and rank invariants under $L^p$ metrics, the metric space needed for the applications above, will be discussed. This is joint work with Qiquan Wang, Pierre Faugère, Anthea Monod and Gregory Henselman-Petrusek.