The Theory of the Interleaving Distance on Multidimensional Persistence Modules

2011/06/30 by Michael Lesnick · 220 citations
Computer Science · Mathematics · #Characterization (materials science) #Combinatorics #Computer science #Discrete mathematics #Homotopy and Cohomology in Algebraic Topology #Interleaving #Isomorphism (crystallography) #Mathematics #Persistence (discontinuity) #Physics #Topological and Geometric Data Analysis #Universality (dynamical systems) #cs.CG #math.AT

paper · pdf · doi:10.1007/s10208-015-9255-y

published in Foundations of Computational Mathematics 15(3), 613-650 (Springer Science+Business Media) · Major revision; exposition improved throughout. To appear in Foundations of Computational Mathematics. 36 pages

arxiv created 2015/02/02 · openalex publication_date 2015/03/23 · arxiv updated 2015/05/22 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

In 2009, Chazal et al. introduced ε-interleavings of persistence modules. ε-interleavings induce a pseudometric d_I on (isomorphism classes of) persistence modules, the interleaving distance. The definitions of ε-interleavings and d_I generalize readily to multidimensional persistence modules. In this paper, we develop the theory of multidimensional interleavings, with a view towards applications to topological data analysis. We present four main results. First, we show that on 1-D persistence modules, d_I is equal to the bottleneck distance d_B. This result, which first appeared in an earlier preprint of this paper, has since appeared in several other places, and is now known as the isometry theorem. Second, we present a characterization of the ε-interleaving relation on multidimensional persistence modules. This expresses transparently the sense in which two ε-interleaved modules are algebraically similar. Third, using this characterization, we show that when we define our persistence modules over a prime field, d_I satisfies a universality property. This universality result is the central result of the paper. It says that d_I satisfies a stability property generalizing one which d_B is known to satisfy, and that in addition, if d is any other pseudometric on multidimensional persistence modules satisfying the same stability property, then d≤ d_I. We also show that a variant of this universality result holds for d_B, over arbitrary fields. Finally, we show that d_I restricts to a metric on isomorphism classes of finitely presented multidimensional persistence modules.

Citations

• A Course in Metric Geometry
• Sheaves, Cosheaves and Applications
• Multidimensional Interleavings and Applications to Topological Inference
• Unzerlegbare Darstellungen I
• Stability of Persistence Diagrams
• Computing Persistent Homology
• Algebra
• The structure and stability of persistence modules
• Stable comparison of multidimensional persistent homology groups with torsion

Cited by

• Local Versus Global Distances for Zigzag Persistence Modules
• Bottleneck stability for generalized persistence diagrams
• Geometry in the space of persistence modules
• Topological data analysis [wikipedia]