Maximum mean discrepancy (MMD) has enjoyed a lot of success in many machine learning and statistical applications, including non-parametric hypothesis testing, because of its ability to handle non-Euclidean data. Recently, it has been demonstrated in Balasubramanian et al.(2021) that the goodness-of-fit test based on MMD is not minimax optimal while a Tikhonov regularized version of it is, for an appropriate choice of the regularization parameter. However, the results in Balasubramanian et al. (2021) are obtained under the restrictive assumptions of the mean element being zero, and the uniform boundedness condition on the eigenfunctions of the integral operator. Moreover, the test proposed in Balasubramanian et al. (2021) is not practical as it is not computable for many kernels. In this paper, we address these shortcomings and extend the results to general spectral regularizers that include Tikhonov regularization.

翻译：最大均值差异（MMD）因其处理非欧几里得数据的能力，在包括非参数假设检验在内的许多机器学习和统计应用中取得了巨大成功。近期，Balasubramanian等人（2021）的研究表明，基于MMD的拟合优度检验并非极小化最优，而其Tikhonov正则化版本在适当选择正则化参数时具有极小化最优性。然而，Balasubramanian等人（2021）的结果是在均值元素为零以及积分算子特征函数一致有界性等限制性假设下获得的。此外，该研究中提出的检验方法缺乏实用性，因为对于许多核函数而言该方法不可计算。本文针对这些不足，将结果推广到包含Tikhonov正则化的一般谱正则化器。