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Stability and Approximations for Decorated Reeb Spaces

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

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Author Details

Justin Curry
  • University at Albany, State University of New York, NY, USA
Washington Mio
  • Florida State University, Tallahassee, FL, USA
Tom Needham
  • Florida State University, Tallahassee, FL, USA
Osman Berat Okutan
  • Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany
Florian Russold
  • Graz University of Technology, Austria

Funding

  • Curry, Justin: Supported by NSF CCF-1850052 and NASA 80GRC020C0016.
  • Mio, Washington: Supported by NSF DMS-1722995.
  • Needham, Tom: Supported by NSF DMS-2107808 and DMS-2324962.
  • Russold, Florian: Supported by the Austrian Science Fund (FWF): W1230.

Cite AsGet BibTex

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)
https://doi.org/10.4230/LIPIcs.SoCG.2024.44

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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Algebraic topology
  • Theory of computation → Computational geometry
Keywords
  • Reeb spaces
  • Gromov-Hausdorff distance
  • Persistent homology

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