Quadratically Regularized Optimal Transport: Localization Bounds and Affine Case Analysis

Quadratic regularization has emerged as a potential alternative to the popular entropic regularization in computational optimal transport, offering the theoretical advantage of producing sparse couplings through its hinge density structure. Despite recent progress in one-dimensional settings and general upper bounds, fundamental questions about the localization rate of QOT optimizers around the Monge coupling have remained open. In this work, we establish a general lower bound showing that the support of the QOT optimizer cannot concentrate around the Monge graph faster than order varepsilon^frac1d+2 in the directed Hausdorff distance, matching the conjectured optimal exponent under standard regularity assumptions in citetwiesel2025sparsity. We also show that the QOT value gap controls the mean-squared deviation mathbb Eπ_varepsilon|y-T(x)|2 by the scale of varepsilon^frac2d+2. As a corollary, in the affine Brenier regime, which includes Gaussian-to-Gaussian transport, we derive a sharp pointwise tube bound of order varepsilon^frac1d+2 by reducing the problem to self-transport and applying recent self-transport sparsity results. Finally, we validate our theoretical bound with a synthetic experiment in high-dimensional settings.