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Luca Nenna: On (Multi-Marginal) Entropic Optimal Transport: from convergence rate to an ODE characterisation #ICBS2024
In this talk we review recent advances concerning entropic (multi-marginal) optimal transport starting by a crash introduction to the topic. We then focus on convergence rate and a novel numerical method based on an ODE characterisation. As for the former we establish lower and upper bounds on the difference with the unregularized cost of the form Cεlog(1/ε)+O(ε) for some explicit dimensional constants C depending on the marginals and on the ground cost, but not on the optimal transport plans themselves. Upper bounds are obtained for Lipschitz costs or semi-concave costs (for a finer estimate), and lower bounds for C² costs satisfying some signature condition on the mixed second derivatives that may include degenerate costs, thus generalizing results previously obtained by Carlier, Pegon and Tamanini, and by Eckstein and Nutz. We obtain in particular matching bounds in some typical situations where the optimal plan is deterministic, like in the case of Wasserstein barycenters. Concerning the ODE characterisation we will introduce a new numerical method to solve multi-marginal optimal transport problems. Notice that the complexity of multi-marginal optimal transport generally scales exponentially in the number of marginals $m$. We introduce a one parameter family of cost functions that interpolates between the original and a special cost function for which the problem's complexity scales linearly in $m$. We then show that the solution to the original problem can be recovered by solving an ordinary differential equation in the parameter $\eta$, whose initial condition corresponds to the solution for the special cost function mentioned above; we then present some simulations, using both explicit Euler and explicit higher order Runge-Kutta schemes to compute solutions to the ODE, and, as a result, the multi-marginal optimal transport problem. Finally, we extend this approach to more general entropic optimal transport problems with linear constraints.