Kernel Two-Sample Tests for Manifold Data
This provides a theoretical foundation for statistical testing in high-dimensional settings like imaging or genomics, where data often have low intrinsic dimensionality, though it is incremental as it builds on existing MMD methods.
The paper tackles the problem of two-sample testing for high-dimensional data that lie on a low-dimensional manifold, proving that a kernel-based test can detect small density differences without suffering from the curse of dimensionality, with detection thresholds scaling as n^{-2β/(d+4β)} for large sample sizes.
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 Hölder with order $β$ (up to 2) on ${M}$, we prove a guarantee of the test power for finite sample size $n$ that exceeds a threshold depending on $d$, $β$, and $Δ_2$ 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 $Δ_2$ is greater than $n^{- { 2 β/( d + 4 β) }}$ up to a certain constant and $γ$ scales as $n^{-1/(d+4β)}$. 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.