Geometry Helps to Compare Persistence Diagrams (1606.03357v1)

Abstract: Exploiting geometric structure to improve the asymptotic complexity of discrete assignment problems is a well-studied subject. In contrast, the practical advantages of using geometry for such problems have not been explored. We implement geometric variants of the Hopcroft--Karp algorithm for bottleneck matching (based on previous work by Efrat el al.) and of the auction algorithm by Bertsekas for Wasserstein distance computation. Both implementations use k-d trees to replace a linear scan with a geometric proximity query. Our interest in this problem stems from the desire to compute distances between persistence diagrams, a problem that comes up frequently in topological data analysis. We show that our geometric matching algorithms lead to a substantial performance gain, both in running time and in memory consumption, over their purely combinatorial counterparts. Moreover, our implementation significantly outperforms the only other implementation available for comparing persistence diagrams.

• The paper introduces geometric algorithms that significantly accelerate the computation of bottleneck and Wasserstein distances for persistence diagrams.
• It employs k-d trees and a geometric variant of established algorithms to improve runtime and memory efficiency in topological data analysis.
• The approach extends classical auction algorithms to handle point multiplicities, proving effective for large-scale, real-world datasets.

An Overview of Geometry in Comparing Persistence Diagrams

The paper "Geometry Helps to Compare Persistence Diagrams" by Kerber, Morozov, and Nigmetov contributes to the interplay between geometric structures and discrete assignment problems in computational topology. Specifically, it addresses the computational challenges associated with comparing persistence diagrams, a fundamental task in topological data analysis (TDA). This problem is frequently encountered when measuring topological differences between data sets.

The authors focus on two principal distances used for comparing persistence diagrams: the bottleneck distance and the Wasserstein distance. While both distances have well-established stability properties and are well studied in theoretical contexts, practical improvements in their computation through geometric approaches have been less explored. The paper provides both theoretical advancements and substantial practical implementations that lead to improved performance over classical, combinatorial methods.

Key Contributions

1. Geometric Bottleneck Matching: The authors provide an experimental exploration of a geometric variant of the Hopcroft–Karp algorithm tailored for bottleneck matching. By leveraging k-d trees, they replace the linear scans involved in combinatorial methods, resulting in substantial performance improvements in terms of both runtime and memory usage.
2. Geometric Wasserstein Matching: In tackling the computation of the Wasserstein distance, a geometric variant of Bertsekas's auction algorithm is employed. This implementation makes use of weighted k-d trees to optimize the proximity searches essential to the auction algorithm, leading to both runtime and space efficiency gains.
3. Handling Multiplicities: The paper extends the standard auction algorithm to accommodate scenarios where points have multiplicities. This extension, known as the auction with integer masses, proves beneficial for handling large-scale problems more efficiently when the average point multiplicity increases, demonstrating its viability in practical applications involving persistence diagrams.
4. Practical Implementations: The authors provide the most efficient implementation for computing distances between persistence diagrams, improving significantly upon existing methods such as those provided by the Dionysus library. The reduction in computational resources and improvement in speed are well-documented in their experimental results.

Implications and Theoretical Advancements

The practical implications of the findings suggest that incorporating geometric structures can dramatically enhance computational efficiency for TDA applications. The enhancements go beyond theoretical improvements and are practically applicable to real-world datasets that involve significant computational loads. Specifically, the methods allow for faster TDA analyses, which is central to applications in data-driven sciences where computation on large datasets is routine.

On the theoretical front, the clear delineation of geometric methods provides a foundation for further exploration of geometric approaches to other problems within TDA and potentially other areas of computational topology and geometry.

Speculation on Future Developments

The insights from this paper open several pathways for future research. The authors highlight potential extensions, including:

• Application of these geometric approaches to higher-dimensional data and spaces.
• Development of further refined approximation schemes to balance computational speed with distance accuracy.
• Exploration of parallel implementations and other optimizations that could further enhance the practical utility of the proposed methods.

These directions could be crucial in advancing the field, particularly as the demand for efficient TDA tools continues to grow alongside the data sizes and complexities typical in contemporary datasets.

In conclusion, this paper lays a robust groundwork for incorporating geometry into the computation of persistence diagrams, providing both theoretical guidance and practical tools that enhance current methodologies significantly. The geometric approach serves as a catalyzing force in the computational analysis of topological features, marking a meaningful advancement in this research domain.