Sparsity of Quadratically Regularized Optimal Transport: Scalar Case

The quadratically regularized optimal transport problem is empirically known to have sparse solutions: its optimal coupling πvarepsilon has sparse support for small regularization parameter varepsilon, in contrast to entropic regularization whose solutions have full support for any varepsilon>0. Focusing on continuous and scalar marginals, we provide the first precise description of this sparsity. Namely, we show that the support of πvarepsilon shrinks to the Monge graph at the sharp rate varepsilon1/3. This result is based on a detailed analysis of the dual potential fvarepsilon for small varepsilon. In particular, we prove that fvarepsilon is twice differentiable a.s. and bound the second derivative uniformly in varepsilon, showing that fvarepsilon is uniformly strongly convex. Convergence rates for fvarepsilon and its derivative are also obtained.