Abstract: The Gromov-Wasserstein (GW) distance quantifies dissimilarity between metric measure (mm) spaces and provides a natural correspondence between them. As such, it serves as a figure of merit for applications involving alignment of heterogeneous datasets, including object matching, single-cell genomics, and matching language models. While various heuristic methods for approximately evaluating the GW distance from data have been developed, formal guarantees for such approaches---both statistical and computational---remained elusive. This work closes these gaps for the quadratic GW distance between Euclidean mm spaces of different dimensions. At the core of our proofs is a novel dual representation of the GW problem as an infimum of certain optimal transportation problems. The dual form enables deriving, for the first time, sharp empirical convergence rates for the GW distance by providing matching upper and lower bounds. For computational tractability, we consider the entropically regularized GW distance. We derive bounds on the entropic approximation gap, establish sufficient conditions for convexity of the objective, and devise efficient algorithms with global convergence guarantees. These advancements facilitate principled estimation and inference methods for GW alignment problems, that are efficiently computable via the said algorithms.
Bio: Ziv Goldfeld is an Assistant Professor in the School of Electrical and Computer Engineering, and a graduate field member in the Center of Applied Mathematics, Statistics and Data Science, Computer Science, and Operations Research and Information Engineering, at Cornell University. Before joining Cornell, he was a postdoctoral research fellow in LIDS at MIT. Ziv graduated with a B.Sc., M.Sc., and Ph.D. (all summa cum laude) in Electrical and Computer Engineering from Ben Gurion University, Israel. Ziv’s research interests include optimal transport theory, statistical learning theory, information theory, and mathematical statistics. Honors include the NSF CAREER Award, the IBM University Award, the Rothschild Fellowship, and the Michael Tien ’72 Excellence in Teaching Award