Approximation algorithms for 1-Wasserstein distance between persistence diagrams

TL;DR

This paper introduces two near-linear time algorithms for approximating the 1-Wasserstein distance between persistence diagrams, significantly improving efficiency over previous methods and enabling scalable topological data analysis.

Contribution

The authors adapt quadtree-based approximation algorithms for optimal transport to efficiently compute the 1-Wasserstein distance between persistence diagrams, providing a practical and scalable solution.

Findings

Algorithms run in near-linear time, outperforming previous methods.

Extensive experiments demonstrate high efficiency and accuracy.

Code is publicly available for reproducibility.

Abstract

Recent years have witnessed a tremendous growth using topological summaries, especially the persistence diagrams (encoding the so-called persistent homology) for analyzing complex shapes. Intuitively, persistent homology maps a potentially complex input object (be it a graph, an image, or a point set and so on) to a unified type of feature summary, called the persistence diagrams. One can then carry out downstream data analysis tasks using such persistence diagram representations. A key problem is to compute the distance between two persistence diagrams efficiently. In particular, a persistence diagram is essentially a multiset of points in the plane, and one popular distance is the so-called 1-Wasserstein distance between persistence diagrams. In this paper, we present two algorithms to approximate the 1-Wasserstein distance for persistence diagrams in near-linear time. These algorithms primarily follow the same ideas as two existing algorithms to approximate optimal transport between two finite point-sets in Euclidean spaces via randomly shifted quadtrees. We show how these algorithms can be effectively adapted for the case of persistence diagrams. Our algorithms are much more efficient than previous exact and approximate algorithms, both in theory and in practice, and we demonstrate its efficiency via extensive experiments. They are conceptually simple and easy to implement, and the code is publicly available in github.