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
本文研究二次正则化最优传输在小正则化参数下的收敛速率，刻画支撑的稀疏几何，证明对偶势的一致 Hessian 界及收敛速率，并确定收敛失效的阈值。
AI 中文摘要
我们建立了在小正则化参数 $\varepsilon$ 下，具有二次代价的二次正则化最优传输的尖锐收敛速率。特别地，我们量化了正则化优化器支撑的稀疏性。对于 $\mathbb{R}^d$ 中的光滑边际密度，该支撑位于距 Brenier 图 $\varepsilon^{1/(d+2)}$ 阶的距离内，且支撑的每个截面都夹在半径为此阶的球之间。支撑的几何形状与对偶势密切相关。我们证明势满足一致的双侧 Hessian 界，并以 $\varepsilon^{2/(d+2)}$ 的速率一致收敛，而其梯度以 $\varepsilon^{1/(d+2)}$ 的速率收敛。这些速率是尖锐的，我们进一步确定了收敛失效的正则性阈值。
英文摘要
We establish sharp convergence rates for quadratically regularized optimal transport with quadratic cost in the regime of small regularization $\varepsilon$. In particular, we quantify the sparsity of the support of the regularized optimizer. For smooth marginal densities in $\mathbb{R}^d$, this support lies within a distance of order $\varepsilon^{1/(d+2)}$ from the Brenier graph, and every section of the support is sandwiched between balls with radii of that order. The geometry of the support is closely linked to the dual potentials. We show that the potentials satisfy uniform two-sided Hessian bounds and converge uniformly at rate $\varepsilon^{2/(d+2)}$, while their gradients converge at rate $\varepsilon^{1/(d+2)}$. The rates are sharp, and we further identify the regularity threshold where convergence breaks down.
Commentsadded reference to Part II
