Kernel Two-Sample Tests for Manifold Data

Abstract: 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 $L<sup>2$-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 $n<sup>{-1/(d+4\beta)}$. 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.