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Nonparametric two sample testing deals with the question of consistentlydeciding if two distributions are different, given samples from both, withoutmaking any parametric assumptions about the form of the distributions. Thecurrent literature is split into two kinds of tests - those which areconsistent without any assumptions about how the distributions may differ(\textit{general} alternatives), and those which are designed to specificallytest easier alternatives, like a difference in means (\textit{mean-shift}alternatives).  The main contribution of this paper is to explicitly characterize the powerof a popular nonparametric two sample test, designed for general alternatives,under a mean-shift alternative in the high-dimensional setting. Specifically,we explicitly derive the power of the linear-time Maximum Mean Discrepancystatistic using the Gaussian kernel, where the dimension and sample size canboth tend to infinity at any rate, and the two distributions differ in theirmeans. As a corollary, we find that if the signal-to-noise ratio is heldconstant, then the test's power goes to one if the number of samples increasesfaster than the dimension increases. This is the first explicit powerderivation for a general nonparametric test in the high-dimensional setting,and also the first analysis of how tests designed for general alternativesperform when faced with easier ones.

On the High-dimensional Power of Linear-time Kernel Two-Sample Testing under Mean-difference Alternatives