Two-sample Statistics Based on Anisotropic Kernels
This work addresses the problem of comparing distributions in high-dimensional data for fields like biomedical imaging, though it appears incremental as it builds on existing kernel-based methods.
The paper introduces an anisotropic kernel-based Maximum Mean Discrepancy statistic for measuring distances between distributions using finite multivariate samples, proving test consistency and a finite-sample power lower bound, with applications in flow cytometry and diffusion MRI datasets.
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 $n$ 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\| \sqrt{n} \to \infty $, 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.