Please note different day: Tuesday, May 19th.
Two central concepts from topological data analysis (TDA) are persistence and the Mapper construction. Persistence employs a sequence of objects built on data called a filtration. A Mapper produces insightful summaries of data, and has found widespread applications in diverse areas.
We define a new filtration called the cover filtration built from a single cover based on a generalized Jaccard distance. We prove a stability result: the cover filtrations of two covers are a/m interleaved, where a is a bound on bottleneck distance between covers and m is the size of smallest set in either cover. We also show our construction is equivalent to the Cech filtration under certain settings, and the Vietoris-Rips filtration completely determines the cover filtration in all cases. We then develop a theory for stable paths within this filtration.
We demonstrate how our framework can be employed in a variety of applications where a metric is not obvious but a cover is readily available. First we present a new model for recommendation systems using cover filtration. Stable paths identified on a movies data set represent gentle transitions from one genre to another. As a second application in explainable machine learning, we apply the Mapper for model induction, and provide explanations of a predictive model in the form of paths between subpopulations of images.
A preprint is available at https://arxiv.org/abs/1906.08256.