Stable comparison of multidimensional persistent homology groups with torsion

The present lack of a stable method to compare persistent homology groups with torsion is a relevant problem in current research about Persistent Homology and its applications in Pattern Recognition. In this paper we introduce a pseudo-distance $d_T$ that represents a possible solution to this problem. Indeed, $d_T$ is a pseudo-distance between multidimensional persistent homology groups with coefficients in an Abelian group, hence possibly having torsion. Our main theorem proves the stability of the new pseudo-distance with respect to the change of the filtering function, expressed both with respect to the max-norm and to the natural pseudo-distance between topological spaces endowed with $R^n$-valued filtering functions. Furthermore, we prove a result showing the relationship between $d_T$ and the matching distance in the 1-dimensional case, when the homology coefficients are taken in a field and hence the comparison can be made.