🤖 AI Summary
The matching distance for two-parameter persistence modules over ℝ² is notoriously difficult to compute exactly.
Method: This paper introduces the first implementable exact algorithm for computing the 2D matching distance. We establish, for the first time, an explicit geometric connection between the matching distance and critical values in the parameter space—proving that any optimal matching line must intersect a critical structure. Leveraging this insight, we construct a piecewise-linear geometric framework that integrates critical-point tracking with multiparameter persistence module theory to enable efficient computation.
Contributions: (1) A geometric characterization of the matching distance with formal interpretability guarantees; (2) An exact, efficient, and implementable algorithm for the 2D case; (3) The first practical, exact computational framework enabling multiparameter persistent homology to be deployed in real-world data analysis.
📝 Abstract
The exact computation of the matching distance for multi-parameter persistence modules is an active area of research in computational topology. Achieving an easily obtainable exact computation of this distance would allow multi-parameter persistent homology to be a viable option for data analysis. In this paper, we provide theoretical results for the computation of the matching distance in two dimensions along with a geometric interpretation of the lines through parameter space realizing this distance. The crucial point of the method we propose is that it can be easily implemented.
