Two-sample statistics based on anisotropic kernels

摘要

The paper introduces a new kernel-based Maximum Mean Discrepancy (MMD) statistic formeasuring 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 n data points and a set of n_R reference points, where n_R can be drastically smaller than n.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 ‖p?q‖√n→∞, and a finite-sample lower bound of the testing power is obtained. Applications to flow cytometry and diffusion MRI datasets are demonstrated, which motivate the proposed approach to compare distributions.