Computing the Bottleneck Distance between Persistent Homology Transforms

TL;DR

This paper develops efficient algorithms for computing the bottleneck distance between Persistent Homology Transforms, improving computational complexity for shape comparison in topological data analysis.

Contribution

It introduces faster algorithms for calculating the integral and maximum bottleneck distances between PHTs, with complexity improvements over previous methods.

Findings

Improved the integral bottleneck distance computation to  O(n^5) in  R^m.

Developed a  O(n^3) algorithm for the maximum bottleneck distance in  R^2.

Extended the  O(n^5) algorithm to  R^3 for the max objective.

Abstract

The Persistent Homology Transform (PHT) summarizes a shape in $\mathbb{R}^m$ 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 $\mathbb{R}^3$.