Persistent interaction topology in data analysis
arxiv(2024)
摘要
Topological data analysis, as a tool for extracting topological features and
characterizing geometric shapes, has experienced significant development across
diverse fields. Its key mathematical techniques include persistent homology and
the recently developed persistent Laplacians. However, classic mathematical
models like simplicial complexes often struggle to provide a localized
topological description for interactions or individual elements within a
complex system involving a specific set of elements. In this work, we introduce
persistent interaction homology and persistent interaction Laplacian that
emphasize individual interacting elements in the system. We demonstrate the
stability of persistent interaction homology as a persistent module.
Furthermore, for a finite discrete set of points in the Euclidean space, we
provide the construction of persistent interaction Vietoris-Rips complexes and
compute their interaction homology and interaction Laplacians. The proposed
methods hold significant promise for analyzing heterogeneously interactive data
and emphasizing specific elements in data. Their utility for data science is
demonstrated with applications to molecules.
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