SPEAKER: Daniel McBride (UTK)

TITLE: Metric properties of partial and robust Gromov-Wasserstein distances
ABSTRACT: The Gromov-Wasserstein (GW) distances define a family of metrics, based on ideas from optimal transport, which enable comparisons between probability measures defined on distinct metric spaces. Although GW distances have proven useful for various applications in the recent machine learning literature, it has been observed that they are inherently sensitive to outlier noise and cannot accommodate partial matching. This has been addressed by various constructions building on the GW framework; in this talk, I focus specifically on a natural relaxation of the GW optimization problem, introduced by Chapel et al., which is aimed at addressing exactly these shortcomings. The goal is to present the properties of this relaxed optimization problem from the viewpoint of metric geometry. While the relaxed problem fails to induce a metric, in particular the axioms of non-degeneracy and triangle inequality are not satisfied, I will present a novel family of distances that define true metrics. These new distances, whose construction is inspired by the Prokhorov and Ky Fan distances, as well as by the recent work of Raghvendra et al. on robust versions of classical Wasserstein distance, are not only true metrics, but they also induce the same topology as the GW distances and enjoy additional robustness to perturbations.