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 @math data points and a set of @math reference points, where @math can be drastically smaller than @math . 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 @math for any @math , 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.
Pour des echantillons aleatoires independants: X 1 X 2 …X n1 distribue selon f(x) et Y 1 Y 2 …Y n2 distribue selon g(x) on presente un test qui possede les proprietes suivantes: la probabilite d'erreur de premiere espece ne depend pas de f; lorsque min (n 1 , n 2 )→∞, la statistique de test est asymptotiquement independante de la distribution sous Ho et la distribution limite est connue; le test est consistant contre des alternatives generales