Geometry and Convergence of Quadratically Regularized Optimal Transport II

Abstract: We study quadratically regularized optimal transport with quadratic cost in the regime of small regularization $\varepsilon$, where the support of the optimal coupling is sparse. For smooth marginal densities in $\mathbb{R}<sup>d$, we show that the support contains the graph of the Brenier map. After centering by the Brenier image and rescaling by $\ell=\varepsilon<sup>{1/(d+2)}$, the sections of the support have explicit ellipsoidal limits for $\varepsilon\to0$, except at boundary points, where the limits are half-space profiles. The optimal dual potentials admit expansions at order $\ell<sup>2$, uniformly up to the boundary. We identify the leading coefficients, which consist of a common local profile and opposite global corrections determined by a linear Neumann problem. Finally, we analyze two approximations to the Brenier map, namely the gradient of the dual potential and the conditional mean of the coupling. We obtain the sharp $L<sup>p$ rate $\ell<sup>{1+1/p}$ with exact leading constants and further identify the leading interior and boundary biases. Taken together, our results illustrate that quadratic regularization induces an accurate sparse approximation of classical optimal transport.