Maximum Mean Discrepancy (MMD) has been widely used in the areas of machine learning and statistics to quantify the distance between two distributions in the $p$-dimensional Euclidean space. The asymptotic property of the sample MMD has been well studied when the dimension $p$ is fixed using the theory of U-statistic. As motivated by the frequent use of MMD test for data of moderate/high dimension, we propose to investigate the behavior of the sample MMD in a high-dimensional environment and develop a new studentized test statistic. Specifically, we obtain the central limit theorems for the studentized sample MMD as both the dimension $p$ and sample sizes $n,m$ diverge to infinity. Our results hold for a wide range of kernels, including popular Gaussian and Laplacian kernels, and also cover energy distance as a special case. We also derive the explicit rate of convergence under mild assumptions and our results suggest that the accuracy of normal approximation can improve with dimensionality. Additionally, we provide a general theory on the power analysis under the alternative hypothesis and show that our proposed test can detect difference between two distributions in the moderately high dimensional regime. Numerical simulations demonstrate the effectiveness of our proposed test statistic and normal approximation.

翻译：在机器学习和统计领域,广泛使用最大平均值差异(MMD),以量化以美元为单位的Euclidean空间中两个分布点之间的距离。当使用U-统计理论将维度固定为美元时,对MMD的无症状属性进行了充分研究。由于经常使用MMD测试中度/高度数据,我们提议调查抽样MMD在高度环境中的行为,并开发一个新的学生化测试统计。具体地说,我们获得了学生制样本MMD的中央限值,作为维度(美元)和样本大小(美元)的数值(美元)和无限的样本大小(美元)。在使用U-统计理论理论确定维度时,对样本MMD的无症状特性特性特性特性进行了充分的研究。我们的结果支持了广泛的内核核,包括流行的戈斯和拉普拉卡尼内核内核,并作为一个特殊的例子覆盖了能源距离。我们还从温度假设中得出明确的趋同率率,我们的结果表明正常近度的精确度可以随着维度而提高。此外,我们提供了一种在替代度假设下进行正常度度度的动力比值分析中度分析的一般动力分析的一般理论分析的一般理论,显示我们提议的试验。