Asymptotics for semi-discrete entropic optimal transport

Abstract

We compute exact second-order asymptotics for the cost of an optimal solution to the entropic optimal transport problem in the continuous-to-discrete, or semi-discrete, setting. In contrast to the discrete-discrete or continuous-continuous case, we show that the first-order term in this expansion vanishes but the second-order term does not, so that in the semi-discrete setting the difference in cost between the unregularized and regularized solution is quadratic in the inverse regularization parameter, with a leading constant that depends explicitly on the value of the density at the points of discontinuity of the optimal unregularized map between the measures. We develop these results by proving new pointwise convergence rates of the solutions to the dual problem, which may be of independent interest.

References in corpus (2)

• Central Limit Theorems for General Transportation Costs
• A Study of Performance of Optimal Transport

Cited by in corpus (5)

• Entropic estimation of optimal transport maps
• Convergence Rates of the Regularized Optimal Transport : Disentangling Suboptimality and Entropy
• Nearly Tight Convergence Bounds for Semi-discrete Entropic Optimal Transport
• Scaling positive random matrices: concentration and asymptotic convergence
• Large deviations for dynamical Schrödinger problems