2016/03/12 by Andrea Cerri, Cerri, Andrea, Marc Ethier +3 · 1 citation
Computer Science · Mathematics · Medicine · #65D18 #Advanced Neuroimaging Techniques and Applications #Algebraic Topology (math.AT) #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #I.3.5 #I.4.7 #Primary 55N35 -- Secondary 68U05 #Topological and Geometric Data Analysis #acm:55N35 #acm:65D18 #acm:68U05 #cs.CG #math.AT #msc:55N35 #msc:65D18 #msc:68U05
paper · pdf · doi:10.48550/arxiv.1603.03886
11 pages, 1 figure
arxiv created 2016/03/12 · openalex publication_date 2016/03/12 · arxiv updated 2016/03/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Comparison between multidimensional persistent Betti numbers is often based on the multidimensional matching distance. While this metric is rather simple to define and compute by considering a suitable family of filtering functions associated with lines having a positive slope, it has two main drawbacks. First, it forgets the natural link between the homological properties of filtrations associated with lines that are close to each other. As a consequence, part of the interesting homological information is lost. Second, its intrinsically discontinuous definition makes it difficult to study its properties. In this paper we introduce a new matching distance for 2D persistent Betti numbers, called coherent matching distance and based on matchings that change coherently with the filtrations we take into account. Its definition is not trivial, as it must face the presence of monodromy in multidimensional persistence, i.e. the fact that different paths in the space parameterizing the above filtrations can induce different matchings between the associated persistent diagrams. In our paper we prove that the coherent 2D matching distance is well-defined and stable.