Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Steinhaus Filtration and Stable Paths in the Mapper   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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The categories of this paper are: q-bio
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The categories of this paper are: funct-an
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The categories of this paper are: chao-dyn
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The categories of this paper are: supr-con
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The categories of this paper are: hep-lat
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The categories of this paper are: eess
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The categories of this paper are: hep-ex
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The categories of this paper are: comp-gas
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The categories of this paper are: nlin
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The categories of this paper are: chem-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: astro-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: patt-sol
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The categories of this paper are: cond-mat