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

Nenna, Luca  ; Pegon, Paul

Canadian journal of mathematics, Tome 77 (2025) no. 3, pp. 1072-1092

Résumé

We investigate the convergence rate of multi-marginal optimal transport costs that are regularized with the Boltzmann–Shannon entropy, as the noise parameter $\varepsilon $ tends to $0$. We establish lower and upper bounds on the difference with the unregularized cost of the form $C\varepsilon \log (1/\varepsilon )+O(\varepsilon )$ 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 semiconcave costs for a finer estimate, and lower bounds for $\mathscr {C}^2$ 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 nondegenerate costs. We obtain in particular matching bounds in some typical situations where the optimal plan is deterministic.

Détail

DOI : 10.4153/S0008414X24000257

Mots-clés : Optimal transport, multi-marginal optimal transport, entropic regularization, Schrödinger problem, convex analysis