Regularity of Entropic Potentials, Stability of Entropic Optimal Transport

Explore the mathematical foundations of entropic optimal transport through a comprehensive examination of regularity theory and stability analysis in this 37-minute conference lecture. Delve into the fundamental relationship between regularity of optimal solutions and stability of optimizers when problem parameters or constraints are perturbed, and discover how this stability enables theoretical convergence guarantees for numerical schemes. Learn about a probabilistic approach to establishing monotonicity estimates for gradients of Schrödinger potentials, understanding the mathematical techniques used to analyze these complex functions. Investigate how weak semiconcavity properties of potentials lead to stability of optimal plans with respect to marginal inputs, providing crucial insights into the behavior of optimal transport solutions under perturbations. Examine the theoretical underpinnings of Sinkhorn's algorithm and understand how stability results serve as key ingredients in proving exponential convergence rates for this widely-used computational method. Gain insights into the intersection of probability theory, optimization, and numerical analysis as applied to optimal transport problems, with particular focus on the entropic regularization approach that makes these problems computationally tractable while maintaining important theoretical properties.