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DOI: 10.4230/LIPIcs.SoCG.2024.44

Stability and Approximations for Decorated Reeb Spaces

Authors: Justin Curry, Washington Mio, Tom Needham, Osman Berat Okutan, and Florian Russold

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Abstract

Given a map f:X → M from a topological space X to a metric space M, a decorated Reeb space consists of the Reeb space, together with an attribution function whose values recover geometric information lost during the construction of the Reeb space. For example, when M = ℝ is the real line, the Reeb space is the well-known Reeb graph, and the attributions may consist of persistence diagrams summarizing the level set topology of f. In this paper, we introduce decorated Reeb spaces in various flavors and prove that our constructions are Gromov-Hausdorff stable. We also provide results on approximating decorated Reeb spaces from finite samples and leverage these to develop a computational framework for applying these constructions to point cloud data.

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Justin Curry, Washington Mio, Tom Needham, Osman Berat Okutan, and Florian Russold. Stability and Approximations for Decorated Reeb Spaces. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 44:1-44:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{curry_et_al:LIPIcs.SoCG.2024.44,
  author =	{Curry, Justin and Mio, Washington and Needham, Tom and Okutan, Osman Berat and Russold, Florian},
  title =	{{Stability and Approximations for Decorated Reeb Spaces}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{44:1--44:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.44},
  URN =		{urn:nbn:de:0030-drops-199891},
  doi =		{10.4230/LIPIcs.SoCG.2024.44},
  annote =	{Keywords: Reeb spaces, Gromov-Hausdorff distance, Persistent homology}
}

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DOI: 10.4230/LIPIcs.SoCG.2022.24

The Universal 𝓁^p-Metric on Merge Trees

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

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Abstract

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.

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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)

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@InProceedings{cardona_et_al:LIPIcs.SoCG.2022.24,
  author =	{Cardona, Robert and Curry, Justin and Lam, Tung and Lesnick, Michael},
  title =	{{The Universal 𝓁^p-Metric on Merge Trees}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{24:1--24:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.24},
  URN =		{urn:nbn:de:0030-drops-160325},
  doi =		{10.4230/LIPIcs.SoCG.2022.24},
  annote =	{Keywords: merge trees, hierarchical clustering, persistent homology, Wasserstein distances, interleavings}
}

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Documents authored by Curry, Justin

Stability and Approximations for Decorated Reeb Spaces

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The Universal 𝓁^p-Metric on Merge Trees

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