A Permutation-free Kernel Two-Sample Test
This work addresses the computational bottleneck in nonparametric two-sample testing for statisticians and machine learning practitioners, offering a more efficient alternative to permutation-based methods.
The authors tackled the computational inefficiency of the kernel Maximum Mean Discrepancy (MMD) two-sample test by proposing the cross-MMD, a new statistic based on sample-splitting and studentization, which achieves a significant speedup for large sample sizes with only a slight loss in power.
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-$α$ test, one usually selects the rejection threshold as the $(1-α)$-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.