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 $\mathcal{M}$ embedded in an m-dimensional space and are Hölder with order β (up to 2) on $\mathcal{M}$, we prove a guarantee of the test power for finite sample size n that exceeds a threshold depending on d, β, and ${\mathrm{\Delta }}_2$ the squared $L^2$-divergence between p and q on the manifold, and with a properly chosen kernel bandwidth γ. For small density departures, we show that with large n they can be detected by the kernel test when ${\mathrm{\Delta }}_2$ is greater than $n^{-2\mathit{\beta }∕(d+4\mathit{\beta })}$ up to a certain constant and γ scales as $n^{-1∕(d+4\mathit{\beta })}$. 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.