A Study of Monodromy in the Computation of Multidimensional Persistence

摘要

The computation of multidimensional persistent Betti numbers for a sublevel filtration on a suitable topological space equipped with a R~n-valued continuous filtering function can be reduced to the problem of computing persistent Betti numbers for a parameterized family of one-dimensional filtering functions. A notion of continuity for points in persistence diagrams exists over this parameter space excluding a discrete number of so-called singular parameter values. We have identified instances of nontrivial monodromy over loops in nonsingular parameter space. In other words, following cornerpoints of the persistence diagrams along nontrivial loops can result in them switching places. This has an important incidence, e.g., in computer-assisted shape recognition, as we believe that new, improved distances between shape signatures can be defined by considering continuous families of matchings between cornerpoints along paths in nonsingular parameter space. Considering that nonhomotopic paths may yield different matchings will therefore be necessary. In this contribution we will discuss theoretical properties of the monodromy in question and give an example of a filtration in which it can be shown to be nontrivial.