Quadratically Regularized Optimal Transport: Localization Bounds and Affine Case Analysis

2026-05-23

AI Summary

This work investigates the localization rate of solutions to quadratically regularized optimal transport (QOT) near Monge couplings, establishing the first theoretical lower bound on the convergence rate of their support sets. By deriving a general lower bound on the directed Hausdorff distance between the support of QOT optimizers and the Monge map, and leveraging tools from optimal transport theory, regularization analysis, and self-transport sparsity, the authors prove that the localization rate in general settings scales as ε^{1/(d+2)}, which is further linked to the mean squared deviation via the duality gap. In the affine Brenier setting—such as Gaussian-to-Gaussian transport—they obtain sharp pointwise tubular bounds. High-dimensional synthetic experiments confirm the tightness and universality of the theoretical rates.

Abstract

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 $\varepsilon^{\frac{1}{d+2}}$ in the directed Hausdorff distance, matching the conjectured optimal exponent under standard regularity assumptions in \citet{wiesel2025sparsity}. We also show that the QOT value gap controls the mean-squared deviation $\mathbb E_{π_\varepsilon}\|y-T(x)\|^2$ by the scale of $\varepsilon^{\frac{2}{d+2}}$. As a corollary, in the affine Brenier regime, which includes Gaussian-to-Gaussian transport, we derive a sharp pointwise tube bound of order $\varepsilon^{\frac{1}{d+2}}$ 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.

Problem

• quadratic regularization
• optimal transport
• localization rate
• Monge coupling
• sparsity

Innovation

• quadratic regularization
• optimal transport
• localization rate
• sparsity
• Monge coupling