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The Universal 𝓁^p-Metric on Merge Trees

Authors Robert Cardona, Justin Curry , Tung Lam, Michael Lesnick

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  • 20 pages

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Author Details

Robert Cardona
  • University at Albany, State University of New York (SUNY), NY, USA
Justin Curry
  • University at Albany, State University of New York (SUNY), NY, USA
Tung Lam
  • University at Albany, State University of New York (SUNY), NY, USA
Michael Lesnick
  • University at Albany, State University of New York (SUNY), NY, USA


While Håvard Bjerkevik was not directly involved in this project, he has had a major influence on it, via his collaboration with ML on presentation distances for multiparameter persistence modules [Bjerkevik and Lesnick, 2021]. In particular, Håvard kindly agreed to share an early draft of [Bjerkevik and Lesnick, 2021] with our group in July 2020, which inspired many of the ideas in our paper.

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Robert Cardona, Justin Curry, Tung Lam, and Michael Lesnick. The Universal 𝓁^p-Metric on Merge Trees. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 24:1-24:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


Adapting a definition given by Bjerkevik and Lesnick for multiparameter persistence modules, we introduce an 𝓁^p-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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Algebraic topology
  • Theory of computation → Unsupervised learning and clustering
  • Theory of computation → Computational geometry
  • merge trees
  • hierarchical clustering
  • persistent homology
  • Wasserstein distances
  • interleavings


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