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On interleaving distance as a complete metric for bifiltrations in Rn
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DOI：10.1080/27684830.2025.2573572.png)
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Abstract
En 中文
In this study, we investigate the completeness of the interleaving distance dI on the space of bifiltrations constructed from finite subsets of a metric space X subset of Rn. In persistent homology and topological data analysis, bifiltrations arise naturally as understanding of metric properties is essential for comparison and stability. The interleaving distance dI is a key tool for comparing bifiltrated structures. While the interleaving distance has been widely studied in one parameter persistence, its completeness in the setting of bifiltrations has not been formally established. We provide a constructive proof that every Cauchy sequence of bifiltrations converges to a well defined limit bifiltration under dI. This result establishes dI as a complete extended pseudometric enabling reliable comparison of bifiltrated data. We present a detailed proof of completeness by showing that an arbitrary Cauchy sequence {Bkr(m)} of bifiltrations in X converges to a unique bifiltration in X.
Keywords:
Extended pseudometrics
completeness
topological data analysis (TDA)
bifiltrations
interleaving distances
persistent homology
Journal
R
IF:
1.1
Papers:
72
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0
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Cited Papers