The DOPE Distance is SIC: A Stable, Informative, and Computable Metric on Time Series And Ordered Merge Trees

Metrics for merge trees that are simultaneously stable, informative, and efficiently computable have so far eluded researchers. We show in this work that it is possible to devise such a metric when restricting merge trees to ordered domains such as the interval and the circle. We present the ``dynamic ordered persistence editing'' (DOPE) distance, which we prove is stable and informative while satisfying metric properties. We then devise a simple $O(N^2)$ dynamic programming algorithm to compute it on the interval and an $O(N^3)$ algorithm to compute it on the circle. Surprisingly, we accomplish this by ignoring all of the hierarchical information of the merge tree and simply focusing on a sequence of ordered critical points, which can be interpreted as a time series. Thus our algorithm is more similar to string edit distance and dynamic time warping than it is to more conventional merge tree comparison algorithms. In the context of time series with the interval as a domain, we show empirically on the UCR time series classification dataset that DOPE performs better than bottleneck/Wasserstein distances between persistence diagrams.