# Two-sample statistics based on anisotropic kernels

Published online

Journal Article

Abstract 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.

### Cited Authors

• Cheng, X; Cloninger, A; Coifman, RR

### Published Date

• December 10, 2019

• 32929389

• 32929389

• 2049-8772

• 2049-8764

### Digital Object Identifier (DOI)

• 10.1093/imaiai/iaz018

• en