发表机构
CEREMADE, Université Paris-Dauphine, Université PSL; CMAP, École polytechnique, Institut polytechnique de Paris（巴黎文理研究大学巴黎多芬大学CEREMADE研究所; 巴黎理工学院巴黎综合理工学院CMAP研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过正则化Brenier势能和对偶间隙误差估计，证明了二次最优传输映射的定量稳定性，并获得了依赖于源密度的Hölder指数及维度多项式依赖的常数。
AI 中文摘要
我们通过正则化Brenier势能并估计对偶间隙中的误差，证明了二次最优传输映射的定量稳定性估计。源测度具有相对于具有有界变差密度的概率测度的$L^p$密度，其中$p>1$。当一个目标测度位于固定球内时，我们获得$W_2$ Hölder指数$(p-1)/(4p-2)$。该常数通过三个显式的源量依赖于维度：方差、参考密度的全变差以及权重的$L^p$范数。证明使用了平均正则化Fenchel间隙的二阶界。通过限制势能的斜率，我们还处理了具有任意大于二阶的有界矩的目标测度。在这两个估计中，条件施加在一个目标上，而另一个目标可以是任何具有有限二阶矩的概率测度。对于高斯和各向同性对数凹源，常数随维度多项式增长。对于高斯测度的$L^p$变化，有界目标估计中的因子为$d^{(p-1)/(4p-2)}$。对于每个$p$，二维中的固定源证明了该源类中两个稳定性指数的最优性。高斯构造提供了维度依赖性的多项式下界。
英文摘要
We prove quantitative stability estimates for quadratic optimal transport maps by regularizing a Brenier potential and estimating the error in the duality gap. The source has an $L^p$ density, $p>1$, relative to a probability measure with a BV density. When one target lies in a fixed ball, we obtain the $W_2$ Hölder exponent $(p-1)/(4p-2)$. The constant depends on the dimension through three explicit source quantities: the variance, the total variation of the reference density and the $L^p$ norm of the weight. The proof uses a second-order bound on the averaged regularized Fenchel gap. By restricting the slopes of the potential, we also treat targets with a bounded moment of any order greater than two. In both estimates, the condition is imposed on one target, while the other can be any probability measure with finite second moment. For Gaussian and isotropic log-concave sources, the constants grow polynomially with the dimension. For $L^p$ changes of Gaussian measure, the factor in the bounded-target estimate is $d^{(p-1)/(4p-2)}$. For each $p$, a fixed source in two dimensions proves optimality of both stability exponents in this source class. A Gaussian construction provides polynomial lower bounds on the dimension dependence.
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