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 techniques and theory touch on spectral theory of laplacians and heat kernels, optimization, and linear algebra. applications to flow cytometry and diffusion mri datasets are demonstrated, which motivate the proposed approach to compare distributions.