  Motivated by the increasing use of kernel-based metrics for high-dimensional
and large-scale data, we study the asymptotic behavior of kernel two-sample
tests when the dimension and sample sizes both diverge to infinity. We focus on
the maximum mean discrepancy (MMD) using isotropic kernel, including MMD with
the Gaussian kernel and the Laplace kernel, and the energy distance as special
cases. We derive asymptotic expansions of the kernel two-sample statistics,
based on which we establish the central limit theorem (CLT) under both the null
hypothesis and the local and fixed alternatives. The new non-null CLT results
allow us to perform asymptotic exact power analysis, which reveals a delicate
interplay between the moment discrepancy that can be detected by the kernel
two-sample tests and the dimension-and-sample orders. The asymptotic theory is
further corroborated through numerical studies.
