🤖 AI Summary
This paper addresses the challenge of modeling stable paths in spaces lacking an explicit metric structure. We propose a covering-based filtration method grounded in the generalized Steinhaus distance, establishing a covering-driven theory of stable paths. Methodologically, we integrate Steinhaus filtering with the Mapper algorithm, formally define path stability via strict matching filtration stability, and prove its equivalence and determinacy with respect to Čech and Vietoris–Rips filtrations. Path stability is quantified using persistent homology and the bottleneck distance. Our key contribution is the first formulation of a covering filtration framework, accompanied by rigorous stability theorems, enabling topological analysis of data endowed with natural coverings—even in the absence of a metric. Experiments demonstrate that our approach generates interpretable, smooth genre-transition sequences on a movie dataset and significantly enhances model interpretability regarding subgroup relationships in FashionMNIST, validating its efficacy in recommender systems and interpretable machine learning.
📝 Abstract
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.