In this talk, we introduce the distance-to-measure signature, a probability measure on R+ that is associated to a metric measure space (mm-space). This signature is based on the distance-to-measure function, introduced in [1]. This signature provides a lower bound for the Gromov-Wasserstein distance [2], on the space of mm-spaces. Moreover, given two samples from two mm-spaces, we derive an asymptotic statistical test, to test equality on the mm-spaces up to isometries that preserve measures. This work is available in [3].

[1] F. Chazal, D. Cohen-Steiner, Q. Mérigot, Geometric Inference for Probability Measure, Foundations of Computational Mathematics, 2011, 11 : 733-51

[2] F. Mémoli, Gromov–Wasserstein Distances and the Metric Approach to Object Matching, Found Comput Math, 2011, 11:417-487

[3] C. Brécheteau, A statistical test of isomorphism between metric-measure spaces using the distance-to-a-measure signature. Electronic Journal of Statistics 2019, Vol. 13, 1:795-849