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Bottleneck Profiles and Discrete Prokhorov Metrics for Persistence Diagrams

Paweł Dłotko, Niklas Hellmer

TL;DR

A notion of discrete Prokhorov metrics for PDs as a generalization of the bottleneck distance is proposed to satisfy a stability result and can be used to bound Wasserstein metrics from above and from below.

Abstract

In topological data analysis (TDA), persistence diagrams have been a succesful tool. To compare them, Wasserstein and Bottleneck distances are commonly used. We address the shortcomings of these metrics and show a way to investigate them in a systematic way by introducing bottleneck profiles. This leads to a notion of discrete Prokhorov metrics for persistence diagrams as a generalization of the Bottleneck distance. They satisfy a stability result and bounds with respect to Wasserstein metrics. We provide algorithms to compute the newly introduced quantities and end with an discussion about experiments.

Bottleneck Profiles and Discrete Prokhorov Metrics for Persistence Diagrams

TL;DR

A notion of discrete Prokhorov metrics for PDs as a generalization of the bottleneck distance is proposed to satisfy a stability result and can be used to bound Wasserstein metrics from above and from below.

Abstract

In topological data analysis (TDA), persistence diagrams have been a succesful tool. To compare them, Wasserstein and Bottleneck distances are commonly used. We address the shortcomings of these metrics and show a way to investigate them in a systematic way by introducing bottleneck profiles. This leads to a notion of discrete Prokhorov metrics for persistence diagrams as a generalization of the Bottleneck distance. They satisfy a stability result and bounds with respect to Wasserstein metrics. We provide algorithms to compute the newly introduced quantities and end with an discussion about experiments.

Paper Structure

This paper contains 16 sections, 27 theorems, 66 equations, 11 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Measure Theory
  4. Persistent Homology
  5. Bottleneck Profiles
  6. Relation to Wasserstein distances
  7. Algorithms
  8. Discrete Prokhorov Metrics for Persistence Diagrams
  9. Comparison with Wasserstein
  10. Metric and Topological Properties
  11. Algorithms
  12. Experiments
  13. Highlighting Geometric Intuition
  14. Classification Experiments
  15. Discussion
  16. ...and 1 more sections

Key Result

Lemma 2.1

Let $(X,\Sigma, \mu)$ be a measure space and let $f\colon X \to \mathbb{R}$ be a measurable function. Then for any $p>0$ and $t>0$,

Figures (11)

  • Figure 1: Illustration of two Prokhorov-close measures which are not Wasserstein-close.
  • Figure 2: Illustration of two measures $\mu$ and $\nu$ and a coupling $\gamma$ of them.
  • Figure 3: Four bottlenecks on the left, a single bottleneck on the right, realizing almost the same bottleneck distance.
  • Figure 4: The PD $X$ has bottleneck distance $3$ to each of the PDs $Y,Z, W$ (first three images). However, it is attained with different multiplicities, which one can read off from the bottleneck profile (right-most image)
  • Figure 5: The situation in the proof of Lemma \ref{['lem:TriangleD']}
  • ...and 6 more figures

Theorems & Definitions (66)

  • Lemma 2.1: Chebychev's inequality
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6: Crawley-Boevey2015Decomposition, Theorem 1.1
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 56 more