ON THE CONVERGENCE RATE OF POTENTIALS OF BRENIER MAPS
ON THE CONVERGENCE RATE OF POTENTIALS OF BRENIER MAPS
复制标题
论BRENIER图势的收敛率
DOI：
10.1017/s0266466621000037
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发表时间：
2021
影响因子：
0.8
通讯作者：
F. Gunsilius
中科院分区：
文献类型：
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作者：
F. Gunsilius
最优运输理论在许多经济研究领域，如最优匹配理论和计量经济识别中，受到了极大的关注。一个特别有价值的工具，由于它方便地表示为凸函数的梯度，是Brenier映射：以欧氏距离为代价函数作为Monge-Kantorovich最优运输问题的优化器获得的匹配。尽管它很受欢迎，但布雷尼尔图的统计特性尚未完全建立，这阻碍了它在估计和推断方面的实际应用。本文在这个方向上迈出了第一步，通过半对偶Monge-Kantorovich问题推导出Brenier映射势的简单插件估计量的收敛速率。根据光滑经验过程收敛的经典结果，证明了如果其中一个概率测度满足poincar<e:1>不等式，在核密度估计的极小极大收敛率下，该插入估计量以标准差收敛于其种群对应量。在势的归一化下，结果推广到收敛于
The theory of optimal transportation has experienced a sharp increase in interest in many areas of economic research such as optimal matching theory and econometric identification. A particularly valuable tool, due to its convenient representation as the gradient of a convex function, has been the Brenier map: the matching obtained as the optimizer of the Monge–Kantorovich optimal transportation problem with the euclidean distance as the cost function. Despite its popularity, the statistical properties of the Brenier map have yet to be fully established, which impedes its practical use for estimation and inference. This article takes a first step in this direction by deriving a convergence rate for the simple plug-in estimator of the potential of the Brenier map via the semi-dual Monge–Kantorovich problem. Relying on classical results for the convergence of smoothed empirical processes, it is shown that this plug-in estimator converges in standard deviation to its population counterpart under the minimax rate of convergence of kernel density estimators if one of the probability measures satisfies the Poincaré inequality. Under a normalization of the potential, the result extends to convergence in the
$L^2$
norm, while the Poincaré inequality is automatically satisfied. The main mathematical contribution of this article is an analysis of the second variation of the semi-dual Monge–Kantorovich problem, which is of independent interest.
独立非线性分量分析
影响因子：
3.7
作者：
Gunsilius, Florian;Schennach, Susanne
通讯作者：
Schennach, Susanne