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Stability of Bipath Persistence Diagrams

Shunsuke Tada

Abstract

Recently, bipath persistent homology has been proposed as an extension of standard persistent homology, along with its visualization (bipath persistence diagram) and computational methods. In the setting of standard persistent homology, the stability theorem with respect to real-valued functions on a topological space is one of the fundamental results, which gives a mathematical justification for using persistent homology to noisy data. In proving the stability theorem, the algebraic stability theorem/the isometry theorem for persistence modules plays a central role. In this point of view, the stability property for bipath persistent homology is desired for analyzing data. In this paper, we prove the stability theorem of bipath persistent homology with respect to bipath functions on a topological space. This theorem suggests a stability of bipath persistence diagrams: small changes in a bipath function (except at their ends) result in only small changes in the bipath persistence diagram. Similar to the stability theorem of standard persistent homology, we deduce the stability theorem of bipath persistent homology by using the algebraic stability theorem/the isometry theorem of bipath persistence modules.

Stability of Bipath Persistence Diagrams

Abstract

Recently, bipath persistent homology has been proposed as an extension of standard persistent homology, along with its visualization (bipath persistence diagram) and computational methods. In the setting of standard persistent homology, the stability theorem with respect to real-valued functions on a topological space is one of the fundamental results, which gives a mathematical justification for using persistent homology to noisy data. In proving the stability theorem, the algebraic stability theorem/the isometry theorem for persistence modules plays a central role. In this point of view, the stability property for bipath persistent homology is desired for analyzing data. In this paper, we prove the stability theorem of bipath persistent homology with respect to bipath functions on a topological space. This theorem suggests a stability of bipath persistence diagrams: small changes in a bipath function (except at their ends) result in only small changes in the bipath persistence diagram. Similar to the stability theorem of standard persistent homology, we deduce the stability theorem of bipath persistent homology by using the algebraic stability theorem/the isometry theorem of bipath persistence modules.

Paper Structure

This paper contains 14 sections, 34 theorems, 66 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Persistence modules
  4. Distances
  5. Translations
  6. Additional observations of the bottleneck interleaving and the bottleneck distances
  7. Relation to graph matching
  8. Bipath persistence
  9. Finite bipath posets
  10. The bipath poset
  11. Stability theorem of bipath persistent homology
  12. Interleaving and partial matching for bipath persistence modules
  13. Proof of the algebraic stability theorem of bipath persistence modules
  14. Proof of the isometry theorem of bipath persistence modules

Key Result

Theorem 2.1

Let $P$ be a poset. Any pfd $P$-persistence module is uniquely decomposed into indecomposable pfd $P$-persistence modules up to isomorphism and permutations of terms.

Figures (3)

  • Figure 1: A correspondence of intervals of $B_{n,m}$ and points in the plane. For an interval $I=\langle s,t \rangle \in \mathbb{I}(B_{n,m})\setminus\mathcal{B}(B_{n,m})$, we plot a point at $(s,t)$. For the interval $B_{n,m}$, we plot a point at the upper left region in the plane.
  • Figure 2: A geometric realization of the $B_{2,2}$-filtration $S$.
  • Figure 3: A visualization of the $0$th and $1$st biapth persistence diagrams of the $B_{2,2}$-filtration $S$.

Theorems & Definitions (71)

  • Theorem 2.1: cf. botnan2020decomposition, Azumaya_1950
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Definition 2.5
  • Lemma 2.6: blanchette2021homological
  • Lemma 2.7
  • Definition 2.8: $\Lambda_\epsilon$-interleaving and the interleaving distance
  • Definition 2.9: bottleneck $\Lambda_\epsilon$-interleaving and the bottleneck interleaving distance
  • Proposition 2.10: oudot2024differential
  • ...and 61 more