Interleaving and Gromov-Hausdorff distance

Abstract

One of the central notions to emerge from the study of persistent homology is that of interleaving distance. It has found recent applications in symplectic and contact geometry, sheaf theory, computational geometry, and phylogenetics. Here we present a general study of this topic. We define interleaving of functors with common codomain as solutions to an extension problem. In order to define interleaving distance in this setting we are led to categorical generalizations of Hausdorff distance, Gromov-Hausdorff distance, and the space of metric spaces. We obtain comparisons with previous notions of interleaving via the study of future equivalences. As an application we recover a definition of shift equivalences of discrete dynamical systems.

35 pages, v3: changed title and added references to uses of interleaving (Section 1.3)

References in corpus (3)

• The ${\ell }^{\infty }$-Cophenetic Metric for Phylogenetic Trees as an Interleaving Distance
• Topological Analysis of Nerves, Reeb Spaces, Mappers, and Multiscale Mappers
• Categories, norms and weights

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• A family of metrics from the truncated smoothing of Reeb graphs
• Multiplicative persistent distances
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• Nonparametric Risk Assessment and Density Estimation for Persistence Landscapes
• Reeb Graph Metrics from the Ground Up
• Finite Mixture Model of Nonparametric Density Estimation using Sampling Importance Resampling for Persistence Landscape