ℓ^p-Distances on Multiparameter Persistence Modules

2021/06/25 by Bjerkevik, Håvard Bakke, Lesnick, Michael
#Algebraic Topology (math.AT) #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics

paper · doi:10.48550/arxiv.2106.13589

Abstract

Motivated both by theoretical and practical considerations in topological data analysis, we generalize the p-Wasserstein distance on barcodes to multiparameter persistence modules. For each p∈ [1,∞], we in fact introduce two such generalizations d_{\mathcal I}^p and d_{\mathcal M}^p, such that d_{\mathcal I}^∞ equals the interleaving distance and d_{\mathcal M}^∞ equals the matching distance. We show that on 1- or 2-parameter persistence modules over prime fields, d_{\mathcal I}^p is the universal (i.e., largest) metric satisfying a natural stability property; this extends a stability theorem of Skraba and Turner for the p-Wasserstein distance on barcodes in the 1-parameter case, and is also a close analogue of a universality property for the interleaving distance given by the second author. We also show that d_{\mathcal M}^p≤ d_{\mathcal I}^p for all p∈ [1,∞], extending an observation of Landi in the p=∞ case. We observe that on 2-parameter persistence modules, d_{\mathcal M}^p can be efficiently approximated. In a forthcoming companion paper, we apply some of these results to study the stability of (2-parameter) multicover persistent homology.

Related

• The Universal ℓ^p-Metric on Merge Trees
• Wasserstein Stability of Persistence Landscapes and Barcodes
• Local Equivalence of Metrics for Multiparameter Persistence Modules
• Stable resolutions of multi-parameter persistence modules
• Projected distances for multi-parameter persistence modules