Gromov-Wasserstein Learning in a Riemannian Framework

• Activity Number: 47 - Geometric and Topological Information in Data Analysis
• Type: Topic-Contributed
• Date/Time: Sunday, August 8, 2021 : 3:30 PM to 5:20 PM
• Sponsor: IMS
• Author(s): Samir Chowdhury*
• Companies: Stanford University
• Keywords: Gromov-Wasserstein; Riemannian statistics; Heat kernel; Fréchet means; Quadratic programming

Abstract

Geometric and topological data analysis methods are increasingly being used to derive insights from data arising in the empirical sciences. We start with a use case where such techniques are applied to human neuroimaging data to obtain graphs which can then yield insights connecting neurobiology to human task performance. Reproducing such insights across populations requires statistical learning techniques such as averaging and PCA across graphs without known node correspondences. We formulate this problem using the Gromov-Wasserstein (GW) distance and present a recently-developed Riemannian framework for GW-averaging and tangent PCA. Beyond graph adjacency matrices, this framework permits consuming derived network representations such as distance or kernel matrices, and such choices lead to additional structure on the GW problem that can be exploited for theoretical and computational advantages. We show how replacing the adjacency matrix representation with a spectral representation leads to theoretical guarantees allowing efficient use of the Riemannian framework as well as state of the art accuracy and runtime in graph learning tasks such as matching and partitioning.