Local Versus Global Distances for Zigzag and Multi-Parameter Persistence Modules

In this paper, we establish explicit and broadly applicable relationships between persistence-based distances computed locally and globally. In particular, we show that the bottleneck distance and the Wasserstein distance between two zigzag persistence modules restricted to an interval is always bounded above by the distance between the unrestricted versions. While this result is not surprising, it could have potential practical implications. We give two related applications for metric graph distances, as well as an extension for the matching distance between multiparameter persistence modules. We also prove a similar restriction inequality for the interleaving distance between two multi-parameter persistence modules.

在本文中，我们在本地和全球计算的基于持久性的距离之间建立了明确且广泛的关系。特别是，我们表明瓶颈距离和限制在间隔的两个锯齿形持久模块之间的瓦斯汀距离始终在不受限制的版本之间的距离上方界定。尽管此结果不足为奇，但它可能具有潜在的实际含义。我们给出了两个相关的度量图距离应用程序，以及多组分持久模块之间匹配距离的扩展。对于两个多参数持久模块之间的交织距离，我们还证明了类似的限制不平等。