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  Jaccard distance. We provide stability results for the cover filtration and show how the construction is equivalent to the Cech filtration under certain settings  . We then develop a language and theory for stable paths within this filtration , inspired by ideas of persistent homology . We demonstrate how the filtration and paths can be applied to a variety of applications in which defining a metric is not obvious but a cover is readily available. We demonstrate the usefulness of this construction by employing it in the context of 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. This extends work on graph-based recommendation systems , allowing a topological perspective . For an explicit example, we look at a movies data set and we find the stable paths identified in our framework 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 or observations. 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.
Two central concepts from topological data analysis 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 Steinhaus distance, which is a generalization of Jaccard distance. We prove a stability result: the cover filtrations of two covers are \alpha/m interleaved, where \alpha 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 . Unlike standard results on stability in topological persistence, our definition of path stability aligns exactly with the above result on stability of cover 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 . For an explicit example, stable paths identified on a movies data set represent sequences of movies constituting gentle transitions from one genre to another. As a second application in explainable machine learning, we apply the Mapper for model induction, providing explanations in the form of paths between subpopulations . Stable paths in the Mapper from a supervised machine learning model trained on the FashionMNIST data set provide improved explanations of relationships between subpopulations of images.
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We demonstrate the usefulness of this construction by employing it in the context of 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. This extends work on graph-based recommendation systems , allowing a topological perspective
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