Gromov-Wasserstein Averaging in a Riemannian Framework

Abstract

We introduce a theoretical framework for performing statistical tasks-including, but not limited to, averaging and principal component analysis-on the space of (possibly asymmetric) matrices with arbitrary entries and sizes. This is carried out under the lens of the Gromov-Wasserstein (GW) distance, and our methods translate the Riemannian framework of GW distances developed by Sturm into practical, implementable tools for network data analysis. Our methods are illustrated on datasets of asymmetric stochastic blockmodel networks and planar shapes viewed as metric spaces. On the theoretical front, we supplement the work of Sturm by producing additional results on the tangent structure of this "space of spaces", as well as on the gradient flow of the Fréchet functional on this space.

Gromov-Wasserstein Distance based Object Matching: Asymptotic Inference

“…Notable exceptions are very recent. We mention Liebscher (2018), who suggested a poly-time algorithm for a Gromov-Hausdorff type metric on the space of phylogenetic trees, Chowdhury and Needham (2019), who applied the Gromov-Kantorovich distance to develop new tools for network analysis, and Gellert et al (2019), who used and empirically compared several lower bounds for the Gromov-Kantorovich distance for clustering of various redoxins, including our lower bound in (5). In fact, to reduce the computational complexity they employed a bootstrap scheme related to the one investigated in this paper and reported empirically good results.…”