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

Abstract

In 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.

39 pages, 15 figures. Corrected the definition of multiplicity of points in the extended Pareto grid and the definition of normal function. Removed Rem. 3.3. Added Ex. 3.9, Fig. 11, Fig. 12, Rem. 5.3 and Fig. 15. Changed Rem. 4.9 into regular text. Reformulated statements of Theorems 5.1, 5.2, 5.4. Some changes in their proofs. Added references. Some small changes in the text and in the figures