  The use of persistent homology in applications is justified by the validity
of certain stability results. At the core of such results is a notion of
distance between the invariants that one associates with data sets. Here we
introduce a general framework to compare distances and invariants in
multiparameter persistence, where there is no natural choice of invariants and
distances between them. We define amplitudes, monotone, and subadditive
invariants that arise from assigning a non-negative real number to objects of
an abelian category. We then present different ways to associate distances to
such invariants, and we provide a classification of classes of amplitudes
relevant to topological data analysis. In addition, we study the the
relationships as well as the discriminitative power of such amplitude distances
arising in topological data analysis scenarios.
