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Generic families of finite metric spaces with identical or trivial 1-dimensional persistence

Philip Smith, Vitaliy Kurlin

TL;DR

This paper describes generic families of point sets in metric spaces that have identical or even trivial one-dimensional persistence and motivates stronger invariants to distinguish finite point sets up to isometry.

Abstract

Persistent homology is a popular and useful tool for analysing finite metric spaces, revealing features that can be used to distinguish sets of unlabeled points and as input into machine learning pipelines. The famous stability theorem of persistent homology provides an upper bound for the change of persistence in the bottleneck distance under perturbations of points, but without giving a lower bound. This paper clarifies the possible limitations persistent homology may have in distinguishing finite metric spaces, which is evident for non-isometric point sets with identical persistence. We describe generic families of point sets in metric spaces that have identical or even trivial one-dimensional persistence. The results motivate stronger invariants to distinguish finite point sets up to isometry.

Generic families of finite metric spaces with identical or trivial 1-dimensional persistence

TL;DR

This paper describes generic families of point sets in metric spaces that have identical or even trivial one-dimensional persistence and motivates stronger invariants to distinguish finite point sets up to isometry.

Abstract

Persistent homology is a popular and useful tool for analysing finite metric spaces, revealing features that can be used to distinguish sets of unlabeled points and as input into machine learning pipelines. The famous stability theorem of persistent homology provides an upper bound for the change of persistence in the bottleneck distance under perturbations of points, but without giving a lower bound. This paper clarifies the possible limitations persistent homology may have in distinguishing finite metric spaces, which is evident for non-isometric point sets with identical persistence. We describe generic families of point sets in metric spaces that have identical or even trivial one-dimensional persistence. The results motivate stronger invariants to distinguish finite point sets up to isometry.
Paper Structure (6 sections, 8 theorems, 3 equations, 7 figures)

This paper contains 6 sections, 8 theorems, 3 equations, 7 figures.

Table of Contents

  1. Motivations, problems, and outline of results
  2. Edges that are important for 1D persistence
  3. Tails without medium edges in a metric space
  4. Persistence for long wedges and with tails
  5. Experiments on persistence of random sets
  6. Conclusions and discussion of other invariants

Key Result

Lemma 2.3

For any finite set $A$ and a filtration $\{C(A;\alpha)\}$ from Definition dfn:complexes, all edges are split into disjoint classes: short, medium, long. $\blacksquare$

Figures (7)

  • Figure 1: Many non-isometric sets have the same 0D persistence and trivial 1D persistence. Theorem \ref{['thm:tail']} extends these examples to generic families of sets by adding 'tails' at red corners.
  • Figure 2: The set $A$ of 10 points in the centre is extended by four tails going out from red points. All such sets have trivial 1D persistence by Corollary \ref{['cor:PD=0']}, but all such sets in general position are not isometric to each other. The black edges form a Minimum Spanning Tree.
  • Figure 3: Left: an edge $[p,q]$ opposite to a non-acute angle in a 2-simplex $\triangle pqw$, see the proof of Proposition \ref{['prop:long_edges']}(d). Middle and Right: classes of edges by Definition \ref{['dfn:edges']} in Example \ref{['exa:edges']}.
  • Figure 4: A tail $T$ around a ray $R$ with vertex $v$ in ${\mathbb R}^2$, see Definitions \ref{['dfn:ang_deviation']} and \ref{['dfn:thickness']}. Left: all angles are not greater than the angular deviation $\omega(T;R)$. Right: the angular thickness $\theta(T;R)$ can be smaller than the angular deviation $\omega(T;R)$.
  • Figure 5: Left: the cloud $A$ in Theorem \ref{['thm:tail']} can be a single red point extendable by tails of blue points along straight rays that form non-acute angles. Then all Delaunay triangles are obtuse, circumscribed by orange circles, meaning that $\mathrm{PD}_1\{\mathrm{Del}(C;\alpha)\}=\emptyset$. Right: a tail $T$ can be generically perturbed under conditions of Theorem \ref{['thm:tail']} without changing $\mathrm{PD}_1$.
  • ...and 2 more figures

Theorems & Definitions (26)

  • Definition 1.1: A filtration of complexes $\{C(A;\alpha)\}$
  • Definition 1.2: 1D persistence and barcode
  • Definition 2.1: Complexes $\mathrm{VR}$, $\mathrm{\check{C}ech}$, and $\mathrm{Del}$
  • Definition 2.2: Short, medium, long edges in a filtration
  • Lemma 2.3: Classes of edges
  • proof
  • Proposition 2.4: Long edges in $\mathrm{VR}$, $\mathrm{\check{C}ech}$, $\mathrm{Del}$
  • proof
  • Example 2.5: Classes of edges on 3 and 4 points
  • Proposition 3.1: No medium edges $\Rightarrow$ trivial $H_1$
  • ...and 16 more