Sparsity of Quadratically Regularized Optimal Transport: Bounds on Concentration and Bias

We study the quadratically regularized optimal transport (QOT) problem for quadratic cost and compactly supported marginals μ and v. It has been empirically observed that the optimal coupling (formula presenetd) for the QOT problem has sparse support for small regularization parameter (formula presenetd). In this article, we provide the first quantitative description of this phenomenon in the general dimension: we derive bounds on the size and the location of the support of (formula presenetd) compared to those of the Monge coupling. Our analysis is based on pointwise bounds on the density of (formula presenetd) together with Minty’s trick, which provides a quadratic detachment from the optimal transport duality gap. In the self-transport setting μ =v, we obtain optimal rates of order (formula presenetd).