The kernel Maximum Mean Discrepancy~(MMD) is a popular multivariate distance metric between distributions that has found utility in two-sample testing. The usual kernel-MMD test statistic is a degenerate U-statistic under the null, and thus it has an intractable limiting distribution. Hence, to design a level-$\alpha$ test, one usually selects the rejection threshold as the $(1-\alpha)$-quantile of the permutation distribution. The resulting nonparametric test has finite-sample validity but suffers from large computational cost, since every permutation takes quadratic time. We propose the cross-MMD, a new quadratic-time MMD test statistic based on sample-splitting and studentization. We prove that under mild assumptions, the cross-MMD has a limiting standard Gaussian distribution under the null. Importantly, we also show that the resulting test is consistent against any fixed alternative, and when using the Gaussian kernel, it has minimax rate-optimal power against local alternatives. For large sample sizes, our new cross-MMD provides a significant speedup over the MMD, for only a slight loss in power.

翻译：核最大均值差异（MMD）是一种流行的分布间多元距离度量，在双样本检验中具有重要应用。通常的核MMD检验统计量在原假设下是退化的U统计量，因此其极限分布难以处理。为了设计水平为$\alpha$的检验，通常选择置换分布的$(1-\alpha)$分位数作为拒绝阈值。由此得到的非参数检验具有有限样本有效性，但计算成本高昂，因为每次置换都需要二次时间。我们提出交叉MMD，一种基于样本分割和学生化方法的新的二次时间MMD检验统计量。我们证明在温和假设下，交叉MMD在原假设下具有极限标准高斯分布。重要的是，我们还证明所得检验对任意固定备择假设具有一致性，且当使用高斯核时，对局部备择假设具有极小最大速率最优功效。对于大样本量，我们的新交叉MMD相比MMD能显著加速，仅轻微损失检验功效。