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Explore the convergence of entropically regularized optimal transport (EOT) to classical Monge optimal transport in this 57-minute lecture by Promit Ghosal from the University of Chicago. Delve into the challenges of understanding small-regularization limits, particularly in dimensions greater than one. Learn how concepts from optimal transport, geometry, and large deviations theory can quantify local exponential convergence of entropic optimizers and characterize large deviation rate functions using Kantorovich potential. The lecture introduces a novel selection principle for Monge transport plans with Euclidean distance cost and absolutely continuous marginals, demonstrating how the limiting plan is supported on transport rays and uniquely determined by minimizing a relative entropy functional. This mathematical exploration resolves an open problem in higher dimensions through a complete variational characterization of limiting transport plans.
