The paper introduces a new kernel-based Maximum Mean Discrepancy (MMD) statistic for measuring the distance between two distributions given finitely-many multivariate samples. When the distributions are locally low-dimensional, the proposed test can be made more powerful to distinguish certain alternatives by incorporating local covariance matrices and constructing an anisotropic kernel. The kernel matrix is asymmetric; it computes the affinity between @math data points and a set of @math reference points, where @math can be drastically smaller than @math . While the proposed statistic can be viewed as a special class of Reproducing Kernel Hilbert Space MMD, the consistency of the test is proved, under mild assumptions of the kernel, as long as @math for any @math , based on a result of convergence in distribution of the test statistic. Applications to flow cytometry and diffusion MRI datasets are demonstrated, which motivate the proposed approach to compare distributions.
Test statistics are proposed for testing equality of two p-variate probability density functions. The statistics are based on the integrated square distance between two kernel-based density estimates and are two-sample versions of the statistic studied by Hall (1984, J. Multivariate Anal. 14 1-16). Particular emphasis is laid on the case where the two bandwidths are fixed and equal. Asymptotic distributional results and power calculations are supplemented by an empirical study based on univariate examples. Kernel density estimation is the most widely-used practical method for accurate nonparametric density estimation. However, long-standing worst-case theoretical results showing that its performance worsens exponentially with the dimension of the data have quashed its application to modern high-dimensional datasets for decades. In practice, it has been recognized that often such data have a much lower-dimensional intrinsic structure. We propose a small modification to kernel density estimation for estimating probability density functions on Riemannian submanifolds of Euclidean space. Using ideas from Riemannian geometry, we prove the consistency of this modified estimator and show that the convergence rate is determined by the intrinsic dimension of the submanifold. We conclude with empirical results demonstrating the behavior predicted by our theory.