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 $\mathbb E_{\pi_\varepsilon}\|y-T(x)\|^2$ 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.
| Subjects: | Optimization and Control (math.OC); Machine Learning (stat.ML) |
| Cite as: | arXiv:2605.24644 [math.OC] |
| (or arXiv:2605.24644v1 [math.OC] for this version) | |
| https://doi.org/10.48550/arXiv.2605.24644 arXiv-issued DOI via DataCite (pending registration) |
|
| Journal reference: | Forty-Third International Conference on Machine Learning (ICML 2026) |
From: Long Nguyen-Chi [view email]
[v1]
Sat, 23 May 2026 16:21:03 UTC (68 KB)
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