Kernel Two-Sample Tests for Manifold Data

We present a study of kernel based two-sample test statistic, which is 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, we show that when data densities are supported on a $d$-dimensional sub-manifold $\mathcal{M}$ embedded in an $m$-dimensional space, the kernel two-sample test for data sampled from a pair of distributions $(p, q)$ that are H\"older with order $\beta$ is consistent and powerful when the number of samples $n$ is greater than $\delta_2(p,q)^{-2-d/\beta}$ up to certain constant, where $\delta_2$ is the squared $\ell_2$-divergence between two distributions on manifold. Moreover, to achieve testing consistency under this scaling of $n$, our theory suggests that the kernel bandwidth $\gamma$ scales with $n^{-1/(d+2\beta)}$. These results indicate that the kernel two-sample test does not have a curse-of-dimensionality when the data lie on a low-dimensional manifold. We demonstrate the validity of our theory and the property of the kernel test for manifold data using several numerical experiments.