Gromov-Wasserstein Averaging in a Riemannian Framework
This provides a practical tool for network data analysis, but it is incremental as it builds on existing theoretical work by Sturm.
The paper tackles the problem of performing statistical tasks like averaging and principal component analysis on matrices of arbitrary sizes and entries, using the Gromov-Wasserstein distance in a Riemannian framework, and demonstrates applications on datasets such as letter graphs and asymmetric networks.
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 Fréchet functional on this space.