Document Open Access Logo

Intrinsic Topological Transforms via the Distance Kernel Embedding

Authors Clément Maria, Steve Oudot, Elchanan Solomon

Thumbnail PDF


  • Filesize: 0.55 MB
  • 15 pages

Document Identifiers

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

Cite AsGet BibTex

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


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Jonathan Bates. The embedding dimension of Laplacian eigenfunction maps. Applied and Computational Harmonic Analysis, 37(3):516-530, 2014. Google Scholar
  2. Mikhail Belkin and Partha Niyogi. Laplacian eigenmaps and spectral techniques for embedding and clustering. In Advances in neural information processing systems, pages 585-591, 2002. Google Scholar
  3. Mikhail Belkin and Partha Niyogi. Laplacian eigenmaps for dimensionality reduction and data representation. Neural computation, 15(6):1373-1396, 2003. Google Scholar
  4. Mikhail Belkin and Partha Niyogi. Convergence of laplacian eigenmaps. In Advances in Neural Information Processing Systems, pages 129-136, 2007. Google Scholar
  5. Robin Lynne Belton, Brittany Terese Fasy, Rostik Mertz, Samuel Micka, David L Millman, Daniel Salinas, Anna Schenfisch, Jordan Schupbach, and Lucia Williams. Learning simplicial complexes from persistence diagrams. arXiv preprint, 2018. URL:
  6. Dmitri Burago, Sergei Ivanov, and Yaroslav Kurylev. A graph discretization of the laplace-beltrami operator. arXiv preprint, 2013. URL:
  7. Ronald R Coifman and Stéphane Lafon. Diffusion maps. Applied and computational harmonic analysis, 21(1):5-30, 2006. Google Scholar
  8. Lorin Crawford, Anthea Monod, Andrew X Chen, Sayan Mukherjee, and Raúl Rabadán. Topological summaries of tumor images improve prediction of disease free survival in glioblastoma multiforme. arXiv preprint, 2016. URL:
  9. Justin Curry, Sayan Mukherjee, and Katharine Turner. How many directions determine a shape and other sufficiency results for two topological transforms. arXiv preprint, 2018. URL:
  10. Tamal K Dey, Dayu Shi, and Yusu Wang. Comparing graphs via persistence distortion. arXiv preprint, 2015. URL:
  11. Gerald B Folland. A Guide to Advanced Real Analysis. Number 37 in Dolciani Mathematical Expositions. MAA, 2009. Google Scholar
  12. Robert Ghrist, Rachel Levanger, and Huy Mai. Persistent homology and euler integral transforms. Journal of Applied and Computational Topology, 2(1-2):55-60, 2018. Google Scholar
  13. Christopher Heil. Compact and hilbert-schmidt operators. In A Basis Theory Primer, pages 481-490. Springer, 2011. Google Scholar
  14. Fritz John et al. The ultrahyperbolic differential equation with four independent variables. Duke Mathematical Journal, 4(2):300-322, 1938. Google Scholar
  15. Clément Maria, Steve Oudot, and Elchanan Solomon. Intrinsic topological transforms via the distance kernel embedding. arXiv preprint, 2019. URL:
  16. Steve Oudot and Elchanan Solomon. Barcode embeddings for metric graphs. arXiv preprint, 2017. URL:
  17. Steve Oudot and Elchanan Solomon. Inverse problems in topological persistence. arXiv preprint, 2018. URL:
  18. Iosif Polterovich, Leonid Polterovich, and Vukašin Stojisavljević. Persistence barcodes and laplace eigenfunctions on surfaces. Geometriae Dedicata, 201(1):111-138, 2019. Google Scholar
  19. Joshua B Tenenbaum, Vin De Silva, and John C Langford. A global geometric framework for nonlinear dimensionality reduction. science, 290(5500):2319-2323, 2000. Google Scholar
  20. Katharine Turner, Sayan Mukherjee, and Doug M Boyer. Persistent homology transform for modeling shapes and surfaces. Information and Inference: A Journal of the IMA, 3(4):310-344, 2014. Google Scholar
  21. Sathamangalam R Srinivasa Varadhan. On the behavior of the fundamental solution of the heat equation with variable coefficients. Communications on Pure and Applied Mathematics, 20(2):431-455, 1967. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail