Intrinsic Topological Transforms via the Distance Kernel Embedding

Authors Clément Maria, Steve Oudot, Elchanan Solomon

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Author Details

Clément Maria
  • INRIA Sophia Antipolis-Méditerranée, Valbonne, France
Steve Oudot
  • INRIA Saclay, Palaiseau, France
Elchanan Solomon
  • Department of Mathematics, Duke University, Durham, NC USA

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Clément Maria, Steve Oudot, and Elchanan Solomon. Intrinsic Topological Transforms via the Distance Kernel Embedding. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 56:1-56:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Algebraic topology
  • Topological Transforms
  • Persistent Homology
  • Inverse Problems
  • Spectral Geometry
  • Algebraic Topology
  • Topological Data Analysis


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