On the Metric Distortion of Embedding Persistence Diagrams into Separable Hilbert Spaces

Authors Mathieu Carrière, Ulrich Bauer

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Mathieu Carrière
  • Department of Systems Biology, Columbia University, New York, USA
Ulrich Bauer
  • Department of Mathematics, Technical University of Munich (TUM), Germany

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Mathieu Carrière and Ulrich Bauer. On the Metric Distortion of Embedding Persistence Diagrams into Separable Hilbert Spaces. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 21:1-21:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


Persistence diagrams are important descriptors in Topological Data Analysis. Due to the nonlinearity of the space of persistence diagrams equipped with their diagram distances, most of the recent attempts at using persistence diagrams in machine learning have been done through kernel methods, i.e., embeddings of persistence diagrams into Reproducing Kernel Hilbert Spaces, in which all computations can be performed easily. Since persistence diagrams enjoy theoretical stability guarantees for the diagram distances, the metric properties of the feature map, i.e., the relationship between the Hilbert distance and the diagram distances, are of central interest for understanding if the persistence diagram guarantees carry over to the embedding. In this article, we study the possibility of embedding persistence diagrams into separable Hilbert spaces with bi-Lipschitz maps. In particular, we show that for several stable embeddings into infinite-dimensional Hilbert spaces defined in the literature, any lower bound must depend on the cardinalities of the persistence diagrams, and that when the Hilbert space is finite dimensional, finding a bi-Lipschitz embedding is impossible, even when restricting the persistence diagrams to have bounded cardinalities.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Algebraic topology
  • Topological Data Analysis
  • Persistence Diagrams
  • Hilbert space embedding


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