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The Universal ℓp-Metric on Merge Trees

arXiv 2112.12165 ↗First posted 22 Dec 2021cs.CG cs.LG math.ATcited by 0

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Abstract (arXiv)

Adapting a definition given by Bjerkevik and Lesnick for multiparameter persistence modules, we introduce an ℓp-type extension of the interleaving distance on merge trees. We show that our distance is a metric, and that it upper-bounds the p-Wasserstein distance between the associated barcodes. For each p∈[1,∞], we prove that this distance is stable with respect to cellular sublevel filtrations and that it is the universal (i.e., largest) distance satisfying this stability property. In the p=∞ case, this gives a novel proof of universality for the interleaving distance on merge trees.

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References · 20

  1. 2021Wasserstein Distances, Geodesics and Barycenters of Merge Treescited by 16
  2. 2013Hypothesis Testing for Topological Data Analysiscited by 9
  3. 2013Measuring Distance between Reeb Graphscited by 6
  4. 2015Computing the Gromov-Hausdorff Distance for Metric Treescited by 4
  5. 2013Metrics for generalized persistence modulescited by 3
  6. 2016Persistence Diagrams as Diagrams: A Categorification of the Stability Theoremcited by 3
  7. 2021Reeb Graph Metrics from the Ground Upcited by 3
  8. 2021Comparative Analysis of Merge Trees using Local Tree Edit Distancecited by 3
  9. 2014The edit distance for Reeb graphs of surfacescited by 2
  10. 2015Categorified Reeb Graphscited by 2
  11. 2018The Reeb Graph Edit Distance is Universalcited by 2
  12. 2018FPT-algorithms for computing Gromov-Hausdorff and interleaving distances between treescited by 2
8 more
  1. 2020On the persistent homology of almost surely C0 stochastic processescited by 2
  2. 2014Strong Equivalence of the Interleaving and Functional Distortion Metrics for Reeb Graphscited by 1
  3. 2016Generalized Cluster Trees and Singular Measurescited by 1
  4. 2017Universality of the Homotopy Interleaving Distancecited by 1
  5. 2018The ℓ^∞-Cophenetic Metric for Phylogenetic Trees as an Interleaving Distancecited by 1
  6. 2020A family of metrics from the truncated smoothing of Reeb graphscited by 1
  7. 2020On C0-persistent homology and treescited by 1
  8. 2021Tight quasi-universality of Reeb graph distancescited by 1

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