Published Paper

Inserted: 6 jul 2023
Last Updated: 9 apr 2024

Journal: Canadian Journal of Mathematics
Pages: pp. 1-21
Year: 2024
Doi: 10.4153/S0008414X24000257

Abstract:

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 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.

Keywords: Optimal transport, Entropic regularization, Multi-marginal optimal transport, Schrödinger problem, convex analysis, Rényi dimension

Download:

• convergence_rate_emot.pdf