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Article Dans Une Revue Canadian Journal of Mathematics = Journal Canadien de Mathématiques Année : 2024

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

Résumé

We investigate the convergence rate of multi-marginal optimal transport costs that are regularized with the Boltzmann-Shannon entropy, as the noise parameter ε tends to 0. We establish lower and upper bounds on the difference with the unregularized cost of the form Cε log(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 locally semi-concave 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 non-degenerate costs. We obtain in particular matching bounds in some typical situations where the optimal plan is deterministic.
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Licence : CC BY - Paternité

Dates et versions

hal-04154453 , version 1 (06-07-2023)
hal-04154453 , version 2 (01-09-2023)
hal-04154453 , version 3 (09-04-2024)

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Luca Nenna, Paul Pegon. Convergence rate of entropy-regularized multi-marginal optimal transport costs. Canadian Journal of Mathematics = Journal Canadien de Mathématiques, 2024, ⟨10.4153/S0008414X24000257⟩. ⟨hal-04154453v3⟩
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