Exact Computation of the Matching Distance on 2-Parameter Persistence Modules

Authors Michael Kerber , Michael Lesnick , Steve Oudot

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Michael Kerber
  • Graz University of Technology, Graz, Austria
Michael Lesnick
  • University at Albany, SUNY, United States
Steve Oudot
  • Inria Saclay - Île-de-France, Palaiseau, France


This work was initiated at the BIRS workshop "Multiparameter Persistent Homology" (18w55140) in Oaxaca, Mexico (Aug. 2018). We thank Jan Reininghaus and the other members of the discussion group on this topic for fruitful initial exchanges. We thank Matthew Wright for helpful discussions about line arrangements, slices, and the computational aspects of 2-parameter persistence.

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Michael Kerber, Michael Lesnick, and Steve Oudot. Exact Computation of the Matching Distance on 2-Parameter Persistence Modules. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 46:1-46:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


The matching distance is a pseudometric on multi-parameter persistence modules, defined in terms of the weighted bottleneck distance on the restriction of the modules to affine lines. It is known that this distance is stable in a reasonable sense, and can be efficiently approximated, which makes it a promising tool for practical applications. In this work, we show that in the 2-parameter setting, the matching distance can be computed exactly in polynomial time. Our approach subdivides the space of affine lines into regions, via a line arrangement. In each region, the matching distance restricts to a simple analytic function, whose maximum is easily computed. As a byproduct, our analysis establishes that the matching distance is a rational number, if the bigrades of the input modules are rational.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Algebraic topology
  • Mathematics of computing → Mathematical optimization
  • Topological Data Analysis
  • Multi-Parameter Persistence
  • Line arrangements


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