发表机构
Columbia University（哥伦比亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究小正则化参数下二次正则化最优传输的几何结构，证明支撑集逼近Brenier映射，并给出收敛速率与精确常数，表明其可作为经典最优传输的稀疏近似。
AI 中文摘要
我们研究在小正则化参数 $\varepsilon$ 情形下具有二次代价的二次正则化最优传输问题，此时最优耦合的支撑集是稀疏的。对于 $\mathbb{R}^d$ 中的光滑边缘密度，我们证明支撑集包含 Brenier 映射的图。在以 Brenier 像为中心并按 $\ell=\varepsilon^{1/(d+2)}$ 缩放后，当 $\varepsilon\to0$ 时，支撑集的截面具有显式的椭球极限，但在边界点处极限为半空间剖面。最优对偶势允许在 $\ell^2$ 阶上展开，且一致到边界。我们识别出首项系数，其由公共局部剖面和由线性 Neumann 问题确定的相反全局修正组成。最后，我们分析 Brenier 映射的两种近似，即对偶势的梯度和耦合的条件均值。我们获得精确的 $L^p$ 阶 $\ell^{1+1/p}$ 及其精确首项常数，并进一步识别出主要的内部和边界偏差。综合来看，我们的结果表明二次正则化诱导了对经典最优传输的精确稀疏近似。
英文摘要
We study quadratically regularized optimal transport with quadratic cost in the regime of small regularization $\varepsilon$, where the support of the optimal coupling is sparse. For smooth marginal densities in $\mathbb{R}^d$, we show that the support contains the graph of the Brenier map. After centering by the Brenier image and rescaling by $\ell=\varepsilon^{1/(d+2)}$, the sections of the support have explicit ellipsoidal limits for $\varepsilon\to0$, except at boundary points, where the limits are half-space profiles. The optimal dual potentials admit expansions at order $\ell^2$, uniformly up to the boundary. We identify the leading coefficients, which consist of a common local profile and opposite global corrections determined by a linear Neumann problem. Finally, we analyze two approximations to the Brenier map, namely the gradient of the dual potential and the conditional mean of the coupling. We obtain the sharp $L^p$ rate $\ell^{1+1/p}$ with exact leading constants and further identify the leading interior and boundary biases. Taken together, our results illustrate that quadratic regularization induces an accurate sparse approximation of classical optimal transport.
