Asymptotics for semi-discrete entropic optimal transport

TL;DR

This paper derives precise second-order asymptotics for entropic optimal transport costs in semi-discrete settings, revealing a quadratic relationship with the inverse regularization parameter and dependence on measure discontinuities.

Contribution

It provides the first detailed second-order asymptotic analysis for semi-discrete entropic optimal transport, highlighting differences from fully discrete or continuous cases.

Findings

Second-order term in cost difference is quadratic in inverse regularization parameter.

Leading constant depends explicitly on density at discontinuities.

New convergence rates for dual problem solutions are established.

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.