The formalism of metric measure spaces permits inducing a number of different distributional invariants or signatures of datasets/shapes which enable fast estimations of the pairwise Gromov-Wasserstein distance. A question of clear importance arising from these constructions is that of understanding, for a given signature S, what is the largest possible class of shapes with the property that any two shapes X and Y in this class are isomorphic if and only if S(X) = S(Y). This talk we will overvie…