License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2020.56
URN: urn:nbn:de:0030-drops-122145
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12214/
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Maria, Clément ; Oudot, Steve ; Solomon, Elchanan

Intrinsic Topological Transforms via the Distance Kernel Embedding

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LIPIcs-SoCG-2020-56.pdf (0.6 MB)

Abstract

Topological transforms are parametrized families of topological invariants, which, by analogy with transforms in signal processing, are much more discriminative than single measurements. The first two topological transforms to be defined were the Persistent Homology Transform (PHT) and Euler Characteristic Transform (ECT), both of which apply to shapes embedded in Euclidean space. The contribution of this paper is to define topological transforms for abstract metric measure spaces. Our proposed pipeline is to pre-compose the PHT or ECT with a Euclidean embedding derived from the eigenfunctions and eigenvalues of an integral operator. To that end, we define and study an integral operator called the distance kernel operator, and demonstrate that it gives rise to stable and quasi-injective topological transforms. We conclude with some numerical experiments, wherein we compute and compare the eigenfunctions and eigenvalues of our operator across a range of standard 2- and 3-manifolds.

BibTeX - Entry

@InProceedings{maria_et_al:LIPIcs:2020:12214,
  author =	{Cl{\'e}ment Maria and Steve Oudot and Elchanan Solomon},
  title =	{{Intrinsic Topological Transforms via the Distance Kernel Embedding}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{56:1--56:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Sergio Cabello and Danny Z. Chen},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12214},
  URN =		{urn:nbn:de:0030-drops-122145},
  doi =		{10.4230/LIPIcs.SoCG.2020.56},
  annote =	{Keywords: Topological Transforms, Persistent Homology, Inverse Problems, Spectral Geometry, Algebraic Topology, Topological Data Analysis}
}

Keywords: Topological Transforms, Persistent Homology, Inverse Problems, Spectral Geometry, Algebraic Topology, Topological Data Analysis
Collection: 36th International Symposium on Computational Geometry (SoCG 2020)
Issue Date: 2020
Date of publication: 08.06.2020

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