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 Čech 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.

Using autonomous mobile robots to dynamically track oil plume propagation in ocean environments is challenging. Based on a model of advection-diffusion equation that describes point-source pollution propagation in marine environments, we have previously proposed a model-based estimator-controller design for autonomous robots to dynamically track plume front propagation. In this paper we study the robustness of the controller in a robot simulator, and evaluate it's performance in a realistic environmental model setup. A probabilistic Lagrangian environmental model is used that can capture both the time-averaged, idealized structure and the instantaneous, realistic structure of a dynamic plume. The controller is implemented on the Field Robotics Lab Vehicle Software, which uses Lightweight Communication and Marshalling library to facilitate process communication and for easy transit to field testing. Simulation results are shown with discussions and lessons learned to guide future field experiments.