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 nu. It has been empirically observed that the optimal coupling πε for the QOT problem has sparse support for small regularization parameter ε>0. In this article we provide the first quantitative description of this phenomenon in general dimension: we derive bounds on the size and on the location of the support of πε compared to the Monge coupling. Our analysis is based on pointwise bounds on the density of πε together with Minty's trick, which provides a quadratic detachment from the optimal transport duality gap. In the self-transport setting μ=nu we obtain optimal rates of order ε^frac12+d.