# Steinhaus Filtration and Stable Paths in the Mapper

## TL;DR

This paper introduces the Steinhaus filtration, a new topological filtration based on a generalized distance, and explores its stability, relationships with existing filtrations, and applications in recommendation systems and explainable AI.

## Contribution

The paper defines the Steinhaus filtration, analyzes its stability properties, compares it with Čech and VR filtrations, and develops a framework for stable paths with applications in machine learning explanations.

## Key findings

- Steinhaus filtration is stable for finite covers.
- It is isomorphic to Čech filtration in one dimension for finite sets.
- VR filtration determines the 1-skeleton of the Steinhaus filtration.

## Abstract

We define a new filtration called the Steinhaus filtration built from a single cover based on a generalized Steinhaus distance, a generalization of Jaccard distance. The homology persistence module of a Steinhaus filtration with infinitely many cover elements may not be $q$-tame, even when the covers are in a totally bounded space. While this may pose a challenge to derive stability results, we show that the Steinhaus filtration is stable when the cover is finite. We show that while the \v{C}ech and Steinhaus filtrations are not isomorphic in general, they are isomorphic for a finite point set in dimension one. Furthermore, the VR filtration completely determines the $1$-skeleton of the Steinhaus filtration in arbitrary dimension. We then develop a language and theory for stable paths within the Steinhaus filtration. We demonstrate how the framework can be applied to several applications where a standard metric may not be defined but a cover is readily available. We introduce a new perspective for modeling recommendation system datasets. As an example, we look at a movies dataset 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 algorithm for model induction by building a filtration from a single Mapper complex, and provide explanations in the form of stable paths between subpopulations. For illustration, we build a Mapper complex from a supervised machine learning model trained on the FashionMNIST dataset. Stable paths in the Steinhaus filtration provide improved explanations of relationships between subpopulations of images.