Necessary conditions for discontinuities of multidimensional persistent Betti numbers

Topological persistence has proven to be a promising framework for dealing with problems concerning the analysis of data. In this context, it was originally introduced by taking into account 1-dimensional properties of data, modeled by real-valued functions. More recently, topological persistence has been generalized to consider multidimensional properties of data, coded by vector-valued functions. This extension enables the study of multidimensional persistent Betti numbers, which provide a representation of data based on the properties under examination. In this contribution, we establish a new link between multidimensional topological persistence and Pareto optimality, proving that discontinuities of multidimensional persistent Betti numbers are necessarily pseudocritical or special values of the considered functions.