Efficient Approximation of the Matching Distance for 2-parameter persistence

TL;DR

This paper introduces a new, faster algorithm for approximating the matching distance between bi-filtered complexes, enhancing computational efficiency and providing the first publicly available implementation.

Contribution

We recast the quad-tree refinement approach in geometric terms, resulting in a practically faster algorithm for matching distance approximation.

Findings

The new algorithm is significantly faster in practice.

Provides the first efficient public implementation.

Demonstrates improved approximation speed through experiments.

Abstract

The matching distance is a computationally tractable topological measure to compare multi-filtered simplicial complexes. We design efficient algorithms for approximating the matching distance of two bi-filtered complexes to any desired precision $\epsilon>0$. Our approach is based on a quad-tree refinement strategy introduced by Biasotti et al., but we recast their approach entirely in geometric terms. This point of view leads to several novel observations resulting in a practically faster algorithm. We demonstrate this speed-up by experimental comparison and provide our code in a public repository which provides the first efficient publicly available implementation of the matching distance.