2023/12/07 by Patrizio Frosini, Eloy Mósig García, Frosini, Patrizio +5 · 2 citations
Biochemistry, Genetics and Molecular Biology · Computer Science · #Algebraic Topology (math.AT) #Anomaly Detection Techniques and Applications #FOS: Mathematics #Metabolomics and Mass Spectrometry Studies #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2312.04201
openalex publication_date 2023/12/07 · openalex created_date 2023/12/09 · openalex updated_date 2026/07/28
One of the most animated themes of multidimensional persistence is the comparison between invariants. The matching distance between persistent Betti numbers functions (or rank invariants), is among the most studied metrics in this context, particularly in 2-parameter persistence. The main reason for this interest is that, in the 2-parameter case, the foliation method allows for an effective computation of the matching distance, based on filtering the space along lines of positive slope. Our work provides a qualitative analysis, based on a construction called extended Pareto grid, of the filtering lines that actually contribute to the computation of the matching distance. Under certain genericity assumptions, we show that these lines must either be horizontal, vertical, of slope 1 or belong to a finite collection of special lines associated with discontinuity phenomena.