Stability of Quadratically Regularized Optimal Transport

Quadratically regularized optimal transport (QOT) is a sparse alternative to entropic optimal transport. We develop a quantitative stability theory for QOT under perturbations of the marginals, the transport cost function, and the regularization parameter. The centerpiece is an $L^\infty$-stability result for the dual potentials. Starting from an error bound in an $L^2$-space that varies with the marginals, we use a self-bound for the potentials to derive a local $L^\infty$-Lipschitz bound that is uniform over marginals. This bound also yields stability of the optimal coupling and of its support. In particular, we show that for the quadratic transport cost, the support of the optimal coupling is locally Lipschitz in Hausdorff distance under perturbations of the marginals. To the best of our knowledge, this is the first stability result for the optimal support in regularized optimal transport.