Gromov-Wasserstein Averaging in a Riemannian Framework

TL;DR

This paper develops a Riemannian framework for statistical analysis of matrices and networks using Gromov-Wasserstein distances, enabling averaging and PCA on complex data structures.

Contribution

It introduces a practical Riemannian approach to Gromov-Wasserstein distances for network and shape analysis, extending Sturm's theoretical work.

Findings

Effective methods for averaging and PCA on network data.

Applications demonstrated on letter graphs, stochastic blockmodels, and shape spaces.

Theoretical insights into the tangent structure and gradient flows in the space of metric spaces.

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 letter graphs, 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 Fr\'{e}chet functional on this space.