New families of simplicial filtration functors

TL;DR

This paper introduces a new theoretical framework for stable simplicial filtration functors in topological data analysis, extending classical filtrations like Čech and Vietoris-Rips with novel, feature-diverse variants.

Contribution

It develops a valuation-induced framework for constructing stable filtration functors, including new types beyond classical ones, and classifies these functors using basepoint concepts.

Findings

Framework encompasses classical and new filtration functors

Ensures stability with respect to Gromov-Hausdorff distance

Provides a classification of filtration functors

Abstract

The so called \v{C}ech and Vietoris-Rips simplicial filtrations are designed to capture information about the topological structure of metric datasets. These filtrations are two of the workhorses in the field of topological data analysis. They enjoy stability with respect to the Gromov-Hausdorff (GH) distance, and this stability property allows us to estimate the GH distance between finite metric space representations of the underlying datasets. Via the concept of Gromov's curvature sets we construct a rich theoretical framework of valuation-induced stable filtration functors. This framework includes the \v{C}ech and Vietoris-Rips filtration functors as well as many novel filtration functors that capture diverse features present in datasets. We further explore the concept of basepoint filtrations functors and use it to provide a classification of the filtration functors that we identify.