Geometry Helps to Compare Persistence Diagrams

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.

Topological signatures of interstellar magnetic fields – I. Betti numbers and persistence diagrams

“…Details of algorithms for computation of the bottleneck distance can be found in Kerber et al (2016). We used the software package dipha at https://github.com/DIPHA/dipha.…”

Geometry Helps to Compare Persistence Diagrams

“…If several bidders want the same object, it is assigned to the bidder who offers the highest price increment, ∆p ij + ε. The Jacobi auction, which was used in the ALENEX version of this paper [21], has a drawback if many objects provide the same value to many bidders. In that case, it may happen that all of these bidders bid for the same object in one iteration, and all but one of them remain unassigned.…”

“…A conference version of this article appeared in ALENEX 2016 [21]. The major novelty of the present version is the discussion of the auction with integer masses in Section 5.…”

“…Moreover, we employed a different variant in the (standard) auction algorithm, which improved the running time of the geometric version by more than a magnitude. Technical explanations and updated experimental evaluation compared to [21] are discussed in Section 4.…”

A User’s Guide to Topological Data Analysis

“…Dionysus (Morozov, 2015) is one of the most versatile packages available, with functionality for not only persistent homology, but also persistent cohomology, zigzag persistence, and the bottleneck and Wasserstein distances. Kerber, Morozov, and Nigmetov (2016) also have software for computing the bottleneck and Wasserstein distances. Perseus (Nanda, 2015;Mischaikow & Nanda, 2013) utilizes discrete Morse theory to speed up persistence computation, and provides easy access to persistence for point clouds as well as for images.…”