New families of stable simplicial filtration functors

Abstract

When quantifying the topological properties of a metric dataset using persistent homology, the first step is to produce a simplicial filtration on the data. The Čech and Vietoris-Rips filtrations are two of the workhorses of persistent homology, but over time, various other filtrations which capture different properties of data or have different computational burdens have been introduced. Towards a program of characterizing all the possible simplicial filtrations on a metric dataset, we introduce and develop the framework of valuation-induced stable filtration functors. This framework is based on the concept of curvature sets due to Gromov, and encapsulates the Vietoris-Rips and various other filtrations while simultaneously providing a model for generating families of novel filtration functors that capture diverse features present in datasets. We further extend this foundation by incorporating the notion of basepoint-dependent filtration functors and proving the associated functoriality and stability properties. This rich theoretical framework provides a unifying language for various extant simplicial filtrations, and is also a mechanism for generating arbitrarily large families of novel filtration functors with control over basepoint dependence/independence as well as the locality of the filtration. We exemplify our constructions on both toy datasets and on 3D shapes from a publicly available shape database. Our paper is accompanied by a Matlab software package incorporating an interactive platform for visualizing and testing new filtrations on datasets.