Applied Math Seminar- New Paradigm in Optimal Transport: Calculus and Riemannian Structure of Gromov-Wasserstein Distance

title: New Paradigm in Optimal Transport: Calculus and Riemannian Structure of Gromov-Wasserstein Distance
Date and Time: Friday, September 20, 1:30pm-2:30pm

Place: Zoom link: https://gwu-edu.zoom.us/j/2957364221
 

abstract: The Gromov-Wasserstein (GW) distance, rooted in optimal transport (OT) theory, quantifies dissimilarity between heterogeneous datasets and provides a natural framework for aligning them. As such, GW distance enables applications including object matching, single-cell genomics, and matching language models. While computational aspects of the GW distance have been studied heuristically, most of the mathematical theories remained elusive. Indeed, initial theories on GW's duality and Brenier map were only established in recent years, and its geometry and variational differentiation have remained largely obscure, while all aforementioned aspects are prevalent in the well-established OT theory. This work closes these gaps for the inner product GW (IGW) distance on the same ambient space. We present (i) a thorough investigation of the Jordan-Kinderlehrer-Otto (JKO) scheme for the gradient flow of IGW, and (ii) a dynamical formulation of IGW, which generalizes the Benamou-Brenier formula for the Wasserstein distance. Central to both contributions is a Riemannian structure on the (tangent) space of probability distributions, based on which we also propose novel numerical schemes for data evolution and deformation.