  In this paper, we introduce the notion of DTM-signature, a measure on R +
that can be associated to any metric-measure space. This signature is based on
the distance to a measure (DTM) introduced by Chazal, Cohen-Steiner and
M\'erigot. It leads to a pseudo-metric between metric-measure spaces,
upper-bounded by the Gromov-Wasserstein distance. Under some geometric
assumptions, we derive lower bounds for this pseudo-metric. Given two
N-samples, we also build an asymptotic statistical test based on the
DTM-signature, to reject the hypothesis of equality of the two underlying
metric-measure spaces, up to a measure-preserving isometry. We give strong
theoretical justifications for this test and propose an algorithm for its
implementation.
