Convergence rate of entropy-regularized multi-marginal optimal transport costs

Abstract

We investigate the convergence rate of multi-marginal optimal transport costs that are regularized with the Boltzmann-Shannon entropy, as the noise parameter $\epsilon$ tends to $0$. We establish lower and upper bounds on the difference with the unregularized cost of the form $C\epsilon \log (1/\epsilon )+O(\epsilon )$ 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 locally semi-concave costs for a finer estimate, and lower bounds for ${\mathcal{C}}^2$ costs satisfying some signature condition on the mixed second derivatives that may include degenerate costs, thus generalizing results previously in the two marginals case and for non-degenerate costs. We obtain in particular matching bounds in some typical situations where the optimal plan is deterministic.

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