this talk 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 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 $\vert p-q \vert \sim$ o($n^{-1/2+\delta})$ for any $\delta>$ 0 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.