On the geometrical properties of the coherent matching distance in 2D persistent homology

open3siIn this paper we study a new metric for comparing Betti numbers functions in bidimensional persistent homology, based on coherent matchings, i.e. families of matchings that vary in a continuous way. We prove some new results about this metric, including its stability. In particular, we show that the computation of this distance is strongly related to suitable filtering functions associated with lines of slope 1, so underlining the key role of these lines in the study of bidimensional persistence. In order to prove these results, we introduce and study the concepts of extended Pareto grid for a normal filtering function as well as of transport of a matching. As a by-product, we obtain a theoretical framework for managing the phenomenon of monodromy in 2D persistent homology.Work carried out under the auspices of INdAM-GNSAGA. M.E. has been partially supported by the Toposys project FP7-ICT-318493-STREP, as well as an ESF Short Visit Grant under the Applied and Computational Algebraic Topology networking programme. He would like to acknowledge his former employers, the Université de Saint-Boniface in Winnipeg, Canada, and Champlain–Saint-Lawrence College in Québec, Canada, for work on this paper done during his employment. A.C. has been partially supported by the FP7 Integrated Project IQmulus, FP7-ICT-2011–318787, and the H2020 Project Gravitate, H2020-REFLECTIVE-7-2014-665155.openCerri, Andrea; Ethier, Marc; Frosini, PatrizioCerri, Andrea; Ethier, Marc; Frosini, Patrizi