Computing the Bottleneck Distance between Persistent Homology Transforms

2025-11-30

AI Summary

This work addresses the problem of efficiently computing the bottleneck distance between persistence diagrams induced by persistent homology transforms (PHTs). Existing algorithms suffer from high computational complexity, hindering their application to topological comparison of high-dimensional shapes. To overcome this, we propose a novel computational framework based on linear height filtrations under directional changes, integrating kinetic data structures with adaptive directional sampling. Our approach yields the first exact algorithms for computing the maximum bottleneck distance in two and three dimensions, achieving $ ilde{O}(n^3)$ and $ ilde{O}(n^5)$ time complexity, respectively. Furthermore, we improve the time bound for the associated integral optimization objective to $ ilde{O}(n^5)$. The method significantly enhances both the efficiency and accuracy of PHT comparisons, delivering a theoretical breakthrough in asymptotic complexity. It provides a scalable new tool for shape analysis at the intersection of topological data analysis and computational geometry.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Web Mining and Content Analysis: Web data integration and cleaning

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 extit{integral} of bottleneck distance over all angles via a kinetic data structure. We improve the integral objective to $ ilde O(n^5)$ in place of earlier $ ilde O(n^6)$ bound. For the extit{max} objective, we give a $ ilde O(n^3)$ algorithm in $mathbb{R}^2$ and a $ ilde O(n^5)$ algorithm in $mathbb{R}^3$.

Problem

• Improves computational efficiency of bottleneck distance in Persistent Homology Transforms.
• Develops faster algorithms for integral and max objectives in shape comparison.
• Enhances methods for comparing shapes via persistent homology across directions.

Innovation

• Improved integral objective to O(n^5) from O(n^6)
• Developed O(n^3) max objective algorithm in R^2
• Proposed O(n^5) max objective algorithm in R^3