  We present a study of a kernel-based two-sample test statistic related to the
Maximum Mean Discrepancy (MMD) in the manifold data setting, assuming that
high-dimensional observations are close to a low-dimensional manifold. We
characterize the test level and power in relation to the kernel bandwidth, the
number of samples, and the intrinsic dimensionality of the manifold.
Specifically, when data densities $p$ and $q$ are supported on a
$d$-dimensional sub-manifold ${M}$ embedded in an $m$-dimensional space and are
H\"older with order $\beta$ (up to 2) on ${M}$, we prove a guarantee of the
test power for finite sample size $n$ that exceeds a threshold depending on
$d$, $\beta$, and $\Delta_2$ the squared $L^2$-divergence between $p$ and $q$
on the manifold, and with a properly chosen kernel bandwidth $\gamma$. For
small density departures, we show that with large $n$ they can be detected by
the kernel test when $\Delta_2$ is greater than $n^{- { 2 \beta/( d + 4 \beta )
}}$ up to a certain constant and $\gamma$ scales as $n^{-1/(d+4\beta)}$. The
analysis extends to cases where the manifold has a boundary and the data
samples contain high-dimensional additive noise. Our results indicate that the
kernel two-sample test has no curse-of-dimensionality when the data lie on or
near a low-dimensional manifold. We validate our theory and the properties of
the kernel test for manifold data through a series of numerical experiments.
