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The Gromov-Wasserstein (GW) distance is a generalization of the standard Wasserstein distance between two probability measures on a given ambient metric space. The GW distance assumes that these two probability measures might live on different ambient spaces and therefore implements an actual comparison of pairs of metric measure spaces. Metric-measure spaces are triples (X,dX,muX) where (X,dX) is a metric space and muX is a Borel probability measure over X and serve as a model for datasets.

In practical applications, this distance is estimated either directly via gradient based optimization approaches, or through the computation of lower bounds which arise from distributional invariants of metric-measure spaces. One particular such invariant is the so called ‘global distance distribution’ of pairwise distances.

This talk will overview the construction of the GW distance, the stability of distribution based invariants, and will discuss some results regarding the injectivity of the global distribution of distances for smooth planar curves, hypersurfaces, and metric trees.