Recorded 20 May 2025. Ziv Goldfeld of Cornell University presents "Gromov-Wasserstein Alignment: Statistics, Computation, and Geometry" at IPAM's Statistical and Numerical Methods for Non-commutative Optimal Transport Workshop. Abstract: The Gromov-Wasserstein (GW) problem, rooted in optimal transport theory, considers optimal alignment of metric measure (mm) spaces that minimizes pairwise distance distortion. As such, it matches up the mm spaces’ internal structures and offers a powerful framework for aligning heterogeneous datasets. While various heuristic methods for approximately computing the GW cost and plan from data have been developed, principled approaches with formal guarantees—both statistical and computational—have 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 variational representation of the GW problem as an infimum of a certain class of optimal transport problems. This connection enables deriving, for the first time, sharp empirical convergence rates for the GW distance via matching upper and lower bounds. For computational tractability, we consider the entropically regularized GW distance. We establish bounds on the entropic approximation gap, provide sufficient conditions for convexity of the objective, and propose efficient algorithms with global convergence guarantees. If time permits, we will also discuss recent progress on gradient flows and interpolation schemes in GW geometry, leading to new structure-preserving evolution dynamics for probability distributions. Learn more online at: https://www.ipam.ucla.edu/programs/workshops/workshop-iii-statistical-and-numerical-methods-for-non-commutative-optimal-transport/