Ephemeral Persistence Features and the Stability of Filtered Chain Complexes

Authors Facundo Mémoli, Ling Zhou

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Facundo Mémoli
  • Department of Mathematics and Department of Computer Science and Engineering, The Ohio State University, Columbus, OH, USA
Ling Zhou
  • Department of Mathematics, The Ohio State University, Columbus, OH, USA

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Facundo Mémoli and Ling Zhou. Ephemeral Persistence Features and the Stability of Filtered Chain Complexes. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 51:1-51:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


We strengthen the usual stability theorem for Vietoris-Rips (VR) persistent homology of finite metric spaces by building upon constructions due to Usher and Zhang in the context of filtered chain complexes. The information present at the level of filtered chain complexes includes ephemeral points, i.e. points with zero persistence, which provide additional information to that present at homology level. The resulting invariant, called verbose barcode, which has a stronger discriminating power than the usual barcode, is proved to be stable under certain metrics which are sensitive to these ephemeral points. In some situations, we provide ways to compute such metrics between verbose barcodes. We also exhibit several examples of finite metric spaces with identical (standard) VR barcodes yet with different verbose VR barcodes thus confirming that these ephemeral points strengthen the discriminating power of the standard VR barcode.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Algebraic topology
  • Mathematics of computing → Topology
  • filtered chain complexes
  • Vietoris-Rips complexes
  • barcode
  • bottleneck distance
  • matching distance
  • Gromov-Hausdorff distance


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