论文标题
通过协方差不平等对Caffarelli收缩定理的熵概括
An entropic generalization of Caffarelli's contraction theorem via covariance inequalities
论文作者
论文摘要
正如Caffarelli的著名定理中首先观察到的，标准高斯度量与$α$ stronglongly log-concove概率度量之间的最佳传输图为$α^{ - 1/2} $ -IPSCHITZ。在本文中，我们应用了两种经典的协方差不平等（Brascamp-Lieb和Cramér-Rao不平等现象），以证明由熵正规化的最佳运输引起的地图的Lipschitz常数急剧结合。随着正则化的趋势趋于零，我们获得了咖啡雷利最初结果的优雅而简短的证明。我们还将Caffarelli的定理扩展到了措施对数密度的Hessians的设置，以任意的正定通勤矩阵界定。
The optimal transport map between the standard Gaussian measure and an $α$-strongly log-concave probability measure is $α^{-1/2}$-Lipschitz, as first observed in a celebrated theorem of Caffarelli. In this paper, we apply two classical covariance inequalities (the Brascamp-Lieb and Cramér-Rao inequalities) to prove a sharp bound on the Lipschitz constant of the map that arises from entropically regularized optimal transport. In the limit as the regularization tends to zero, we obtain an elegant and short proof of Caffarelli's original result. We also extend Caffarelli's theorem to the setting in which the Hessians of the log-densities of the measures are bounded by arbitrary positive definite commuting matrices.
