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A lecture by Cornell University's Ziv Goldfeld exploring the Gromov-Wasserstein (GW) problem in optimal transport theory, focusing on the statistical, computational, and geometric aspects of aligning metric measure spaces. Discover how GW alignment minimizes pairwise distance distortion to match internal structures of heterogeneous datasets. The presentation addresses previously unresolved challenges by introducing a novel variational representation of the GW problem as an infimum of optimal transport problems, enabling sharp empirical convergence rates through matching upper and lower bounds. Learn about the entropically regularized GW distance, including bounds on entropic approximation gaps, conditions for objective convexity, and efficient algorithms with global convergence guarantees. The lecture also covers recent advances in gradient flows and interpolation schemes in GW geometry, presenting new structure-preserving evolution dynamics for probability distributions. Recorded at IPAM's Statistical and Numerical Methods for Non-commutative Optimal Transport Workshop at UCLA.
