A Persistence-Driven Edit Distance for Trees with Abstract Weights

TL;DR

This paper introduces a new tree edit distance metric based on persistence and topological data analysis, enabling more meaningful comparisons of trees with abstract weights, with applications in stability and merge tree estimation.

Contribution

It proposes a novel persistence-driven tree edit distance metric and a dynamic programming method for its computation, advancing topological data analysis techniques.

Findings

The metric effectively captures topological features of trees.

It can be computed efficiently using binary linear programming.

Applications include stability analysis and merge tree estimation.

Abstract

In this work we define a novel edit distance for trees considered with some abstract weights on the edges. The metric is driven by the idea of considering trees as topological summaries in the context of persistence and topological data analysis. Several examples related to persistent sets are presented. The metric can be computed with a dynamical binary linear programming approach. This framework is applied and further studied in other works focused on merge trees, where the problems of stability and merge trees estimation are also assessed.