Cover Filtration and Stable Paths in the Mapper

The contributions of this paper are two-fold. We define a new filtration called the cover filtration built from a single cover based on a generalized Steinhaus distance, which is a generalization of Jaccard distance. We then develop a language and theory for stable paths within this filtration, inspired by ideas of persistent homology. This framework can be used to develop several new learning representations in applications where an obvious metric may not be defined but a cover is readily available. We demonstrate the utility of our framework as applied to recommendation systems and explainable machine learning. We demonstrate a new perspective for modeling recommendation system data sets that does not require manufacturing a bespoke metric. As a direct application, we find that the stable paths identified by our framework in a movies data set represent a sequence of movies constituting a gentle transition and ordering from one genre to another. For explainable machine learning, we apply the Mapper for model induction, providing explanations in the form of paths between subpopulations. Our framework provides an alternative way of building a filtration from a single mapper that is then used to explore stable paths. As a direct illustration, we build a mapper from a supervised machine learning model trained on the FashionMNIST data set. We show that the stable paths in the cover filtration provide improved explanations of relationships between subpopulations of images.