  In this paper, we propose a test for the equality of multiple distributions
based on kernel mean embeddings. Our framework provides a flexible way to
handle multivariate or even high-dimensional data by virtue of kernel methods
and allows the number of distributions to increase with the sample size. This
is in contrast to previous studies that have been mostly restricted to
classical low-dimensional settings with a fixed number of distributions. By
building on Cramer-type moderate deviation for degenerate two-sample
V-statistics, we derive the limiting null distribution of the test statistic
and show that it converges to a Gumbel distribution. The limiting distribution,
however, depends on an infinite number of nuisance parameters, which makes it
infeasible for use in practice. To address this issue, the proposed test is
implemented via the permutation procedure and is shown to be minimax rate
optimal against sparse alternatives. During our analysis, an exponential
concentration inequality for the permuted test statistic is developed which may
be of independent interest.
