The Gromov-Wasserstein distance and distributional invariants of datasets

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 metric spaces and therefore implements an actual comparison of pairs of metric measure spaces. A metric-measure space is a triple (X,dX,muX) where (X,dX) is a metric space and muX is a Borel probability measure over X.

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’ which precisely encodes the distribution of pairwise distances between points in a given metric measure space. This invariant has been used in many applications yet its classificatory power is not yet well understood.

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