Discrete & Computational Geometry

, Volume 42, Issue 1, pp 71–93 | Cite as

The Theory of Multidimensional Persistence

  • Gunnar Carlsson
  • Afra ZomorodianEmail author


Persistent homology captures the topology of a filtration—a one-parameter family of increasing spaces—in terms of a complete discrete invariant. This invariant is a multiset of intervals that denote the lifetimes of the topological entities within the filtration. In many applications of topology, we need to study a multifiltration: a family of spaces parameterized along multiple geometric dimensions. In this paper, we show that no similar complete discrete invariant exists for multidimensional persistence. Instead, we propose the rank invariant, a discrete invariant for the robust estimation of Betti numbers in a multifiltration, and prove its completeness in one dimension.


Computational topology Multidimensional analysis Persistent homology Persistence 


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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of MathematicsStanford UniversityStanfordUSA
  2. 2.Department of Computer ScienceDartmouth CollegeHanoverUSA

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