Diffusion-Scale Martingale MMD Tests on Compact Riemannian Manifolds

Two-sample testing on a compact Riemannian manifold requires a kernel that respects geometry and remains sensitive to differences at unknown spatial scales. Diffusion kernels provide an intrinsic family, but smoothing attenuates both signal and noise, and neighboring scales can give redundant tests. We study this tradeoff within martingale maximum mean discrepancy testing. An exact spectral criterion shows that additional diffusion improves local asymptotic efficiency precisely when the average null-noise frequency exceeds the average signal frequency. For a fixed grid, thresholding the covariance pseudoinverse accommodates singularity and yields an asymptotic chi-square null limit with consistently estimated positive rank. Simulations on the circle, torus, and sphere support the predicted frequency dependence and conservative-to-near-nominal size. A matched torus comparison shows how the resampling-free calculation can improve detection by processing more available observations in similar time. These results connect intrinsic scale choice to both statistical power and computational cost.