The Gromov-Wasserstein (GW) distance, rooted in optimal transport (OT) theory, quantifies dissimilarity between metric measure spaces 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 about GW duality, Brenier maps, geometry, etc., remained elusive, despite the rapid progress these aspects have seen under the classical OT paradigm in recent decades. This talk will cover recent progress on closing these gaps for the GW. We present (i) sharp statistical estimation rates through duality, (ii) a thorough investigation of the Jordan-Kinderlehrer-Otto (JKO) scheme for the gradient flow of inner product GW (IGW) distance, and (iii) a dynamical formulation of IGW, which generalizes the Benamou-Brenier formula for the Wasserstein distance. Central to (ii) and (iii) is a Riemannian structure on the space of probability distributions, based on which we also propose novel numerical schemes for measure evolution and deformation. [Joint work with Zhengxin Zhang (Cornell), Ziv Goldfeld (Cornell), Kristjan Greenewald (IBM Research), and Youssef Mroueh (IBM Research)]
