Approximation algorithms for 1-Wasserstein distance between persistence diagrams

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.

Approximating 1-Wasserstein Distance between Persistence Diagrams by Graph Sparsification

“…This means finding the nearest PD from a set of PDs for a given query PD with respect to the W 1 metric. Following [5,22], define recall@1 for a given algorithm as the percentage of nearest neighbor queries that are correct when using that algorithm for distance computation. We also use the phrase "prediction accuracy" synonymously with recall@1.…”

“…Although at s = 1 there are no approximation guarantees, PDoptFlow still obtains high recall@1; see Figure 7 and Table 6 for a demonstration of the low empirical error from our experiments. Other approximation algorithms [5,22,52] are incomparable in prediction accuracy though they run much faster.…”

“…This suggests most of PDoptFlow's speedup comes from the approximation algorithm design irrespective of the parallelism. The Software from [22] is not in Table 2 since its theoretical relative error (2× (height of its quadtree)-1) [5,22] is not comparable to values (0.5 and 0.2) from Table 2. In fact, it has theoretical relative errors of 75, 41, 61, 41 for bh, AB, mri, rips respectively.…”

“…In fact, it has theoretical relative errors of 75, 41, 61, 41 for bh, AB, mri, rips respectively. flowtree [22] is much faster than PDoptFlow and is less accurate empirically. On these datasets, there is a 10.1x, 2.6x, 18.8x and 90.3x speedup against PDoptFlow(s = 18) at ε = 1.…”

“…The O(n 2 ) memory requirement is demanding for large n, especially on GPU. It was shown in [22] that the quadtree [47] and flowtree [5] algorithms can be adapted to achieve a O(log ∆) approximation in O(n log ∆) memory and time where ∆ is the ratio of the largest pairwise distance between PD points divided by their closest pairwise distance. One has no control over the error with this approach and in practice the approximation factor is large.…”