Approximating 1-Wasserstein Distance between Persistence Diagrams by Graph Sparsification

Abstract

Persistence diagrams (PD)s play a central role in topological data analysis. This analysis requires computing distances among such diagrams such as the $1$-Wasserstein distance. Accurate computation of these PD distances for large data sets that render large diagrams may not scale appropriately with the existing methods. The main source of difficulty ensues from the size of the bipartite graph on which a matching needs to be computed for determining these PD distances. We address this problem by making several algorithmic and computational observations in order to obtain, in theory, a near-linear fully polynomial-time approximation scheme. This is theoretically optimal assuming the $(1+\epsilon)$-approximate EMD conjecture in constant dimension, which is that the EMD problem on the plane cannot be approximated by a PTAS in time $O(\frac{1}{\epsilon^2}n)$ up to polylog factors. In our implementation, first, taking advantage of the distribution of PD points, we \emph{condense} them thereby decreasing the number of nodes in the graph for computation. The increase in point multiplicities is addressed by reducing the matching problem to a min-cost flow problem on a transshipment network. Second, we use Well Separated Pair Decomposition to sparsify the graph to a size that is linear in the number of points. Both node and arc sparsifications contribute to the approximation factor where we leverage a lower bound given by the Relaxed Word Mover's distance. Third, we eliminate bottlenecks during the sparsification procedure by introducing parallelism. Fourth, we develop an open source software called PDoptFlow based on our algorithm, exploiting parallelism by GPU and multicore. We perform extensive experiments and show that the actual empirical error is very low. We also show that we can achieve high performance at low guaranteed relative errors, improving upon the state of the arts.