Entropic Optimal Transport: Convergence Rates and ODE Characterization

Explore recent advancements in entropic (multi-marginal) optimal transport through this comprehensive conference talk. Begin with a crash introduction to the topic before delving into convergence rates and a novel numerical method based on an ODE characterization. Examine lower and upper bounds on the difference with the unregularized cost, including explicit dimensional constants dependent on marginals and ground cost. Investigate upper bounds for Lipschitz and semi-concave costs, as well as lower bounds for C² costs with specific signature conditions. Learn about matching bounds in situations where the optimal plan is deterministic, such as Wasserstein barycenters. Discover a new numerical method for solving multi-marginal optimal transport problems, introducing a one-parameter family of cost functions that interpolates between the original and a special cost function. Understand how to recover the solution to the original problem by solving an ordinary differential equation, and explore simulations using explicit Euler and higher-order Runge-Kutta schemes. Finally, examine the extension of this approach to more general entropic optimal transport problems with linear constraints.