  Finite metric spaces are the object of study in many data analysis problems.
We examine the concept of weak isometry between finite metric spaces, in order
to analyse properties of the spaces that are invariant under strictly
increasing rescaling of the distance functions. In this paper, we analyse some
of the possible complete and incomplete invariants for weak isometry and we
introduce a dissimilarity measure that asses how far two spaces are from being
weakly isometric. Furthermore, we compare these ideas with the theory of
persistent homology, to study how the two are related.
