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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 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 that 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 standard VR barcode.
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Maximum Subbarcode Matching and Subbarcode Distance
We investigate the maximum subbarcode matching problem which arises from the study of persistent homology and introduce the subbarcode distance on barcodes. A barcode is a set of intervals which correspond to topological features in data and is the output of a persistent homology computation. A barcode A has a subbarcode matching to B if each interval in A matches to an interval in B which contains it. We present an algorithm which takes two barcodes, A and B, and returns a maximum subbarcode matching.
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- Award ID(s):
- 2017980
- PAR ID:
- 10422657
- Editor(s):
- Bahoo, Yeganeh; Georgiou, Konstantinos
- Date Published:
- Journal Name:
- Canadian Conference in Computational Geometry
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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