论文标题
稳定性和样本的差异正则最佳转运的样本复杂性
Stability and Sample Complexity of Divergence Regularized Optimal Transport
论文作者
论文摘要
我们研究差异正规化最佳转运（DOT）的稳定性和样品复杂性。首先，我们获得了在Wasserstein距离中测得的DOT优化者的定量稳定性结果，该距离适用于各种差异，并同时改善了熵最佳运输的已知结果。其次，我们研究了样品复杂性的情况，其中点问题是使用边缘的经验度量近似的。我们表明，与未注册的最佳运输相比，差异正则化可以提高相应的收敛速率。为此，我们证明了上限，可以利用成本函数的规律性和差异功能，以及边缘的内在维度。在此过程中，我们建立了DOT双重优化器的规律性特性，以及具有合适的测试功能类别的经验度量定理的一般限制定理。
We study stability and sample complexity properties of divergence regularized optimal transport (DOT). First, we obtain quantitative stability results for optimizers of DOT measured in Wasserstein distance, which are applicable to a wide class of divergences and simultaneously improve known results for entropic optimal transport. Second, we study the case of sample complexity, where the DOT problem is approximated using empirical measures of the marginals. We show that divergence regularization can improve the corresponding convergence rate compared to unregularized optimal transport. To this end, we prove upper bounds which exploit both the regularity of cost function and divergence functional, as well as the intrinsic dimension of the marginals. Along the way, we establish regularity properties of dual optimizers of DOT, as well as general limit theorems for empirical measures with suitable classes of test functions.
