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Computing the Bottleneck Distance between Persistent Homology Transforms

Michael Kerber, Elena Xinyi Wang

TL;DR

The paper tackles the problem of comparing Persistent Homology Transforms by computing the maximum bottleneck distance across all directions, providing the first algorithms with provable bounds in both 2D and 3D. It introduces an event-driven, randomized band-refinement framework that navigates a combinatorial set of candidate extrema, updating bottleneck matchings through diagram changes with vineyard updates and augmenting-path searches. In 2D, it achieves {O(n^3)} time for the max distance and {O(n^5)} for the integral variant, while in 3D it attains {O(n^5)} time for the max distance, leveraging an arrangement-based approach on the sphere. As a byproduct, the integral distance in 2D is improved to {O(n^5)}. These results enable efficient, exact PHT comparisons on moderate-scale data, with potential extensions to related directional transforms in topological data analysis.

Abstract

The Persistent Homology Transform (PHT) summarizes a shape in $\R^m$ by collecting persistence diagrams obtained from linear height filtrations in all directions on $\mathbb{S}^{m-1}$. It enjoys strong theoretical guarantees, including continuity, stability, and injectivity on broad classes of shapes. A natural way to compare two PHTs is to use the bottleneck distance between their diagrams as the direction varies. Prior work has either compared PHTs by sampling directions or, in 2D, computed the exact \textit{integral} of bottleneck distance over all angles via a kinetic data structure. We improve the integral objective to $\tilde O(n^5)$ in place of earlier $\tilde O(n^6)$ bound. For the \textit{max} objective, we give a $\tilde O(n^3)$ algorithm in $\mathbb{R}^2$ and a $\tilde O(n^5)$ algorithm in $\mathbb{R}^3$.

Computing the Bottleneck Distance between Persistent Homology Transforms

TL;DR

The paper tackles the problem of comparing Persistent Homology Transforms by computing the maximum bottleneck distance across all directions, providing the first algorithms with provable bounds in both 2D and 3D. It introduces an event-driven, randomized band-refinement framework that navigates a combinatorial set of candidate extrema, updating bottleneck matchings through diagram changes with vineyard updates and augmenting-path searches. In 2D, it achieves {O(n^3)} time for the max distance and {O(n^5)} for the integral variant, while in 3D it attains {O(n^5)} time for the max distance, leveraging an arrangement-based approach on the sphere. As a byproduct, the integral distance in 2D is improved to {O(n^5)}. These results enable efficient, exact PHT comparisons on moderate-scale data, with potential extensions to related directional transforms in topological data analysis.

Abstract

The Persistent Homology Transform (PHT) summarizes a shape in by collecting persistence diagrams obtained from linear height filtrations in all directions on . It enjoys strong theoretical guarantees, including continuity, stability, and injectivity on broad classes of shapes. A natural way to compare two PHTs is to use the bottleneck distance between their diagrams as the direction varies. Prior work has either compared PHTs by sampling directions or, in 2D, computed the exact \textit{integral} of bottleneck distance over all angles via a kinetic data structure. We improve the integral objective to in place of earlier bound. For the \textit{max} objective, we give a algorithm in and a algorithm in .

Paper Structure

This paper contains 30 sections, 23 theorems, 24 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Persistent Homology
  4. Persistent Homology Transform
  5. Maximum Bottleneck Distance
  6. Computing $d_{\infty}$ in $\mathbb{R}^2$
  7. Events
  8. Overview of the algorithm
  9. Decision Problem
  10. Candidate Pruning via Band Refinement
  11. Predicate 1: Existence Test
  12. Predicate 2: Band Refinement
  13. Randomized Analysis
  14. Computing $d_1$ in $\mathbb{R}^2$
  15. Computing $d_{\infty}$ in $\mathbb{R}^3$
  16. ...and 15 more sections

Key Result

Lemma 2.1

For the above construction of $G$, $d_B(G) = d_B(X, Y)$.

Figures (2)

  • Figure 1: Construction of the bipartite graph $G$ based on the persistence diagrams $X$ and $Y$.
  • Figure 2: On the left are two simplices with vertices $v_{0, 1, 2}$ and $w_{0, 1, 2}$ respectively. On the right is $\mathbb{S}^1$ partitioned into regions with the same active vertex pairs $(v_i, w_j)$.

Theorems & Definitions (47)

  • Lemma 2.1: Reduction Lemma EdelsHarer2010
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • Theorem 3.2: Events for $d_\infty$ in $\mathbb{R}^2$
  • proof
  • Remark 3.3
  • Corollary 3.4
  • proof
  • Remark 3.5
  • ...and 37 more